Chapter 11 Financial Mathematics (Q & A)

Correct Option: (B) To compensate the lender for the risk and opportunity cost of lending money.

Explanation: Interest is charged primarily to compensate the lender for:

• The risk involved in lending (possibility of default), and
• The opportunity cost of forgoing other investment opportunities during the lending period.

Therefore, option (B) is the most appropriate choice.

Correct Option: (A) $\textsf{₹}\,10,000$

Explanation:

The principal amount is the original sum of money borrowed or invested before any interest is added.

Here, it is clearly stated that you borrow $\textsf{₹}\,10,000$, which directly refers to the principal amount.

Therefore, the correct answer is: Option (A).

Correct Option: (B) Higher returns for lenders/investors.

Explanation:

A higher interest rate implies that lenders or investors earn more from the money they lend or invest. It increases their income from interest, thus offering higher returns.

On the contrary:

• (A) is incorrect because a higher interest rate increases the cost for borrowers.
• (C) is incorrect because interest rate does not directly reduce lending risk.
• (D) is incorrect because the complexity of calculation is not related to the interest rate level.

Therefore, option (B) is correct.

Correct Option: (A) Both A and R are true and R is the correct explanation of A.

Explanation:

Assertion (A): Interest rates indeed influence borrowing and lending decisions. When rates change, they directly affect how much people borrow or save.

Reason (R): High interest rates make borrowing more expensive and saving more rewarding. Hence, individuals and businesses tend to borrow less and save more.

Since both statements are correct and Reason (R) clearly explains why Assertion (A) is true, option (A) is the correct choice.

Correct Option: (A) $\textsf{₹}\,50,000$

Explanation:

The principal amount is the original sum of money deposited or invested before any interest is added.

In this case, Ram deposits $\textsf{₹}\,50,000$, which is the amount on which interest will be calculated.

Therefore, the principal amount is $\textsf{₹}\,50,000$.

Correct Option: (B) year

Explanation:

The interest rate is generally expressed as a percentage per annum, i.e., per year. This is known as the annual interest rate.

Although interest can also be calculated monthly, quarterly, or for other time periods, the standard expression is per year unless otherwise specified.

Therefore, the correct answer is (B) year.

Correct Option: (A) principal

Explanation:

In simple interest, the interest is calculated only on the original sum of money borrowed or invested, which is called the principal.

Unlike compound interest, where interest is calculated on both the principal and the accumulated interest, simple interest does not consider interest on interest.

Hence, in simple interest, the interest is calculated only on the principal amount.

Correct Option: (B) $\textsf{₹}\,150$

Explanation:

We use the formula for simple interest:

Where,

$P = \textsf{₹}\,1000$, $R = 5\%$ per annum, $T = 3$ years

Substituting in equation (i):

$SI = \dfrac{1000 \times 5 \times 3}{100}$

$SI = \dfrac{15000}{100} = \textsf{₹}\,150$

Therefore, the simple interest earned is $\textsf{₹}\,150$.

Correct Option: (B) compound interest

Explanation:

In compound interest, the interest is calculated not just on the original principal, but also on the interest earned in the previous periods.

This means that each time period, the interest is added to the principal to form a new base amount on which the interest is calculated again.

Therefore, the interest is calculated on the principal amount plus the compound interest earned previously.

Correct Option: (C) $\textsf{₹}\,1102.50$

Given:

Principal (P) = $\textsf{₹}\,1000$

Rate (R) = 5% per annum

Time (T) = 2 years

Formula for compound amount:

Substituting the values:

$A = 1000 \left(1.05\right)^2$

$A = 1000 \times 1.1025$

$A = \textsf{₹}\,1102.50$

Therefore, the accumulated amount after 2 years is $\textsf{₹}\,1102.50$.

Correct Option: (D) $A = P + \dfrac{PRT}{100}$

Explanation:

In Simple Interest, the formula for interest is:

Where:

• P = Principal
• R = Rate of Interest per period
• T = Time (in same units as rate)

To find the accumulated amount (A) under simple interest, we add the interest to the principal:

Substituting equation (i) into equation (ii):

$A = P + \dfrac{PRT}{100}$

Hence, the correct formula for accumulated amount under simple interest is: $A = P + \dfrac{PRT}{100}$

Correct Option: (B) $A = P(1 + i)^n$

Explanation:

In Compound Interest, the interest is added to the principal after each compounding period, and future interest is calculated on the increased amount.

The standard formula to find the accumulated amount (A) is:

Where:

• P = Principal
• i = Interest rate per compounding period
• n = Total number of compounding periods

Hence, the correct formula for compound amount is: $A = P(1 + i)^n$

Correct Option: (A) Both A and R are true and R is the correct explanation of A.

Explanation:

Assertion (A): True. For a time period longer than one year, compound interest yields a higher accumulated amount than simple interest because it accumulates interest on the previously earned interest.

Reason (R): True. Compound interest is calculated on the principal and also on the accumulated interest, whereas simple interest is calculated only on the original principal amount.

Thus, the reason clearly explains why the assertion holds true.

Therefore, both Assertion and Reason are correct, and the Reason is the correct explanation of the Assertion.

Correct Option: (B) $\textsf{₹}\,25600$

Given:

Principal, $P = \textsf{₹}\,20,000$

Rate of Interest, $R = 7\%$ per annum

Time, $T = 4$ years

Using the Simple Interest formula:

Substituting the values:

$SI = \dfrac{560000}{100} = \textsf{₹}\,5600$

Total amount to be repaid:

Hence, the total amount to be repaid is: $\textsf{₹}\,25600$

Correct Option: (B) $\textsf{₹}\,26215$ (approx)

Given:

Principal, $P = \textsf{₹}\,20,000$

Rate of Interest, $R = 7\%$ per annum (compounded annually)

Time, $T = 4$ years

Using the Compound Interest formula:

Substituting the values:

Now calculating $(1.07)^4$ approximately:

$(1.07)^2 = 1.1449$, so

$(1.07)^4 = (1.1449)^2 \approx 1.311$

Therefore,

$A = 20000 \times 1.311 \approx \textsf{₹}\,26215$

Hence, the total amount to be repaid is approximately: $\textsf{₹}\,26215$

Correct Option: (C) accumulated

Explanation:

The simple interest rate and the compound interest rate are said to be equivalent when both produce the same accumulated amount (i.e., total amount including interest and principal) over the same time period.

This means, under both types of interest calculation, the final amount at the end of the time period remains equal, although the calculation methods differ.

Therefore, the correct word to complete the sentence is: accumulated.

Correct Option: (A) True

Explanation:

We are given the condition for equivalence of simple interest rate $r_s$ and compound interest rate $i$ over $t$ years:

This equation ensures that both types of interest yield the same amount over the same time period.

Therefore, if this equation holds true for all $t$, then the two rates are indeed equivalent for any time period.

Hence, the correct answer is: True.

Correct Option: (A) True

Explanation:

The effective rate of interest (also called effective annual rate or EAR) is the actual annual interest rate earned or paid on an investment or loan after taking into account the effect of compounding within the year.

It is especially relevant when interest is compounded more than once per year (e.g., quarterly or monthly), and it gives a more accurate picture than the nominal rate.

It is calculated using the formula:

Hence, the statement is true, and it applies to compound interest scenarios.

Correct Option: (B) 4%

Explanation:

We are given:

• Nominal annual interest rate = 8%
• Compounded semi-annually → 2 compounding periods per year

To find the interest rate per compounding period ($i$), we use the formula:

Therefore, the interest rate per compounding period is: 4%

Correct Option: (A) $EAR = (1 + j/m)^m - 1$

Explanation:

The Effective Annual Rate (EAR) is the actual annual interest rate taking compounding into account. When a nominal interest rate $j$ is compounded $m$ times per year, the formula for EAR is:

Here,

• $j$ is the nominal annual interest rate
• $m$ is the number of compounding periods per year

This formula correctly accounts for the effect of compounding multiple times a year.

Therefore, option (A) is the correct answer.

Correct Option: (C) 10.38%

Explanation:

We are given:

• Nominal annual interest rate ($j$) = 10% = 0.10
• Compounded quarterly → $m = 4$ times per year

To find the Effective Annual Rate (EAR), we use the formula:

Substituting the values:

Now calculate:

$= (1.025)^4 - 1 = 1.1038 - 1 = 0.1038$

$EAR = 0.1038 = 10.38\%$

Therefore, the effective annual rate is: 10.38%

Correct Option: (D) A is false but R is true.

Explanation:

Assertion (A): "The effective rate of interest is always equal to the nominal rate of interest."

This is false. The effective rate of interest is only equal to the nominal rate when interest is compounded annually. When compounding occurs more frequently (like quarterly or monthly), the effective rate becomes higher than the nominal rate.

Reason (R): "The effective rate considers the effect of compounding more frequently than annually, which increases the actual return."

This is true. More frequent compounding leads to a higher effective rate due to interest being calculated on accumulated amounts more often.

Therefore, the assertion is false, but the reason is true. Hence, the correct answer is:

Option (D).

Correct Option: (D) Both (B) and (C).

Explanation:

Bank A: Offers Simple Interest at 7% per annum, so effective interest rate for 1 year = 7%.

Bank B: Offers Compound Interest at 6.8% per annum compounded semi-annually.

To find the Effective Annual Rate (EAR), we use the formula:

Where:

• $r = 6.8\% = 0.068$
• $m = 2$ (since compounded semi-annually)

$= (1.034)^2 - 1 = 1.068156 - 1 = 0.068156 = 6.8156\%$

Conclusion: Since Bank A offers 7% and Bank B offers approximately 6.8156%, Bank A gives a better return over 1 year.

However, options (B) and (C) are both correct expressions of the EAR for Bank B. So the correct choice is:

Option (D).

Correct Option: (A) Bank A

Explanation:

From Question 23, we already calculated the effective annual rate (EAR) for Bank B:

Therefore, Bank B offers an effective rate of 6.8156%.

Bank A offers a simple interest rate of 7% per annum.

Comparison:

• Bank A: 7.0000%
• Bank B: 6.8156%

Since 7% > 6.8156%, Bank A offers a better interest rate over a 1-year period.

Hence, the correct answer is: Option (A).

Correct Option: (A) True

Explanation:

The Present Value (PV) is defined as the current worth of a future sum of money or stream of cash flows, given a specified rate of return or discount rate.

It reflects the idea that a certain amount of money today is worth more than the same amount in the future due to its potential earning capacity (time value of money).

The formula for Present Value is:

Where:

• $PV$ = Present Value
• $FV$ = Future Value
• $r$ = Rate of return (per period)
• $n$ = Number of periods

This concept is applicable to both simple and compound interest scenarios, but it is especially relevant in compound interest and discounted cash flow calculations.

Hence, the statement is: True.

Correct Option: (A) $\$10000 / (1 + 0.06)^3$

Explanation:

We are given:

• Future Value (FV) = $\textsf{₹}\,10,000$
• Time ($n$) = 3 years
• Annual Compound Interest Rate ($r$) = 6% = 0.06

We use the Present Value (PV) formula for compound interest:

Substituting the given values:

So, to get $\textsf{₹}\,10,000$ in 3 years at 6% compounded annually, you must invest:

$\dfrac{10000}{(1.06)^3} = \dfrac{10000}{1.191016} \approx \textsf{₹}\,8396.86$

Hence, the correct answer is: Option (A).

Correct Option: (A) True

Explanation:

Net Present Value (NPV) is a fundamental concept in financial decision-making. It helps determine the profitability of an investment or project.

Definition: NPV is the difference between the present value (PV) of cash inflows and the present value of cash outflows over a period of time.

Where:

• $R_t$ = Net cash inflow during the period $t$
• $r$ = Discount rate
• $t$ = Time period
• $C_0$ = Initial investment or cash outflow at time 0

This calculation accounts for the time value of money by discounting future cash flows to their present values and subtracting the initial outlay.

Hence, the statement is: True.

Correct Option: (B) The project is expected to generate returns greater than the required rate of return.

Explanation:

Net Present Value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows, discounted at the required rate of return.

If $NPV > 0$, it means:

• The project’s return is higher than the required rate of return.
• The investment is expected to generate surplus value over its cost.
• The project adds value to the firm and is considered financially viable.

Therefore, a positive NPV implies: The project is expected to generate returns greater than the required rate of return.

Correct Option: (A) True

Explanation:

Future Value (FV) refers to the value of a current asset or investment at a specified date in the future based on an assumed rate of growth or interest.

The formula for future value under compound interest is:

Where:

• $PV$ = Present Value (initial investment)
• $r$ = Interest rate per period
• $n$ = Number of periods

This concept applies to both simple and compound interest, though the formulas differ.

Thus, Future Value is indeed the value of an investment at a future date, making the statement True.

Correct Option: (B) $5000 \times (1 + 0.07)^5$

Explanation:

Since the interest is compounded annually, the formula for future value (FV) is:

Where:

• $P = 5000$ (Principal)
• $r = 0.07$ (Annual interest rate)
• $n = 5$ (Years)

Substituting the values in equation (i):

This gives the correct future value under compound interest.

Therefore, Option (B) is correct.

Correct Option: (A) Both A and R are true and R is the correct explanation of A.

Explanation:

Assertion (A): Present value (PV) and future value (FV) are indeed connected through both the interest rate and the time period. This is the core concept in the time value of money.

Reason (R): The formula for compound interest is:

Where:

• $\text{FV}$ = Future Value
• $\text{PV}$ = Present Value
• $i$ = Interest rate per period
• $n$ = Number of periods

Equation (i) clearly demonstrates how PV and FV are mathematically related through the interest rate and time period.

Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Correct Option: (B) $12000 / (1 + 0.05)^2$

Explanation:

To find the present value (PV) of a future cash inflow using compound interest, the formula is:

Where:

• $FV = 12000$ (Future Value)
• $r = 0.05$ (Annual rate of return)
• $n = 2$ years

Substituting in equation (i):

This will give the present value of the expected cash inflow discounted at 5% per annum over 2 years.

Therefore, Option (B) is correct.

Correct Option: (C) Present Value of Inflow - Initial Investment

Explanation:

Net Present Value (NPV) is a financial metric used to assess the profitability of an investment or project. The formula for NPV is:

This formula evaluates whether the project will generate more value than it costs:

• If NPV > 0 → The project is profitable.
• If NPV < 0 → The project results in a loss.
• If NPV = 0 → The project breaks even.

Therefore, the correct expression for calculating NPV is given in Option (C).

Correct Option: (A) True

Explanation:

An annuity is defined as a series of equal payments made at regular intervals over a specified period of time. These payments can be made monthly, quarterly, annually, etc., and may include interest and/or principal components depending on the context.

Examples of annuities include:

• Monthly pension payments
• Home mortgage payments
• Insurance premiums
• Fixed deposits with regular payouts

Therefore, the statement is True, making Option (A) the correct answer.

Correct Option: (C) end

Explanation:

In a regular annuity (also known as an ordinary annuity), the payments are made at the end of each time period. This is the most common form of annuity used in financial transactions such as loan repayments, mortgage EMIs, and investment payouts.

For example:

• If rent is paid at the end of every month, it's a regular annuity.
• If you invest in a bond that pays interest annually at the year-end, it’s also a regular annuity.

Hence, the correct answer is Option (C).

Correct Option: (C) $1000 \times \frac{(1.05)^3 - 1}{0.05}$

Explanation:

To find the future value (FV) of a regular annuity, we use the formula:

Where:

• $A = 1000$ (annual payment)
• $r = 0.05$ (annual interest rate)
• $n = 3$ (years)

Substituting the values:

This is exactly what is shown in Option (C), which is the correct choice.

Note: Other options are incorrect because:

• Option (A): Calculates compound value of a single payment.
• Option (B): Represents manually summing future values of each payment, but not in a formulaic or generalizable form.
• Option (D): Ignores interest and assumes simple total.

Conclusion: The correct formula for computing the future value of a regular annuity is given in Option (C).

Correct Option: (D) All of the above

Explanation:

A regular annuity is a series of equal payments made at regular intervals, typically at the end of each period. It has many practical applications in personal finance and business, such as:

• (A) Loan repayments (EMI calculations): When paying off a loan through EMIs, each payment is structured as part of a regular annuity.
• (B) Saving for a future goal: Making equal monthly or annual deposits into a savings plan for future use is a form of annuity.
• (C) Insurance premium payments: Regularly paying premiums for insurance coverage is also a typical annuity application.

Since all three options are valid applications of regular annuities,

Final Answer: Option (D)

Correct Option: (A) Both A and R are true and R is the correct explanation of A.

Explanation:

Assertion (A): The future value (FV) of an annuity is indeed calculated by summing up the future values of each individual payment made during the annuity period. Each payment grows over time as it earns interest.

Reason (R): This is also correct. Each payment earns interest from the moment it is made until the end of the annuity term, which results in different future values for each installment depending on how early the payment was made.

Thus, both statements are true, and the reason correctly explains the assertion.

Final Answer: Option (A)

Correct Option: (C) $\textsf{₹}\,500 \times (1.04)^2$

Explanation:

In a regular annuity, each deposit earns compound interest based on the number of years it remains in the account.

Since the first deposit is made at the end of year 1 and the total duration is 3 years, this amount will remain in the account for 2 more years (from the end of year 1 to the end of year 3).

Therefore, its future value at the end of 3 years is calculated as:

Hence, the value of the first deposit at the end of 3 years is Option (C).

Correct Option: (D) Both (A) and (C) are correct formulations.

Explanation:

The case involves a regular annuity where a person deposits $\textsf{₹}\,500$ at the end of each year for 3 years at an interest rate of 4% compounded annually.

Method 1: Term-wise accumulation

Each deposit earns compound interest for the number of years it remains in the account:

• The 1st deposit is invested for 2 years: $\textsf{₹}\,500 \times (1.04)^2$
• The 2nd deposit is invested for 1 year: $\textsf{₹}\,500 \times (1.04)$
• The 3rd deposit earns no interest: $\textsf{₹}\,500$

So, total future value is:

Method 2: Using Future Value formula of Ordinary Annuity

The formula is:

Where:
$P = \textsf{₹}\,500$, $r = 0.04$, $n = 3$

So,

Conclusion: Both expressions (i) and (iii) represent the future value of the annuity and are correct.

Correct Option: (A) True

Explanation:

Taxes are mandatory payments levied by the government on individuals and businesses. They are imposed on various economic activities such as:

• Income (e.g., income tax)
• Goods and services (e.g., GST, VAT)
• Property (e.g., property tax)

These taxes are essential to fund public goods and services such as education, healthcare, infrastructure, defense, etc. Hence, they are not voluntary and not limited to income alone.

Therefore, the correct answer is True.

Correct Option: (B) Goods and services only

Explanation:

Goods and Services Tax (GST) in India is a consumption-based indirect tax levied on the supply of goods and services. It is charged at every stage of the supply chain — from manufacturer to consumer — but is ultimately borne by the end consumer.

GST is not levied on income; income is taxed separately under the Income Tax Act. Also, GST applies to the supply of goods and services, not just their production.

Therefore, the correct answer is (B) Goods and services only.

Correct Option: (B) $\textsf{₹}\,1180$

Explanation:

Given:

• Price before GST = $\textsf{₹}\,1000$
• GST rate = 18%

Step 1: Calculate GST amount

GST = $18\%$ of $\textsf{₹}\,1000$ = $\frac{18}{100} \times 1000 = \textsf{₹}\,180$

Step 2: Add GST to the original price

Final Price = $\textsf{₹}\,1000 + \textsf{₹}\,180 = \textsf{₹}\,1180$

Therefore, the final price including GST is $\textsf{₹}\,1180$.

Correct Option: (A) True

Explanation:

Income Tax is a direct tax levied by the government on the income earned by individuals, Hindu Undivided Families (HUFs), firms, companies, and other entities.

It is assessed and paid annually based on the financial year's earnings and applicable tax slabs or rates defined under the Income Tax Act, 1961.

Therefore, the statement that "Income Tax is levied on the income of individuals or entities, usually on an annual basis" is true, and option (A) is the correct choice.

Correct Option: (A) Both A and R are true and R is the correct explanation of A.

Explanation:

Assertion (A) is true because taxes are a major source of revenue for the government, which helps fund various development and welfare programs.

Reason (R) is also true because the tax collected is used to provide essential public services such as building infrastructure, running healthcare systems, and ensuring access to education.

Since R directly explains why A is true, option (A) is the most appropriate.

Correct Option: (A) ₹12,500

Explanation:

The first ₹5,00,000 of taxable income is taxed as per the given slab rates:

• ₹0 to ₹2,50,000: 0% → ₹0 tax
• ₹2,50,001 to ₹5,00,000: 5% → 5% of ₹2,50,000 = ₹12,500

Total tax on the first ₹5,00,000 = ₹12,500

Therefore, the correct answer is option (A).

Correct Option: (D) Both (B) and (C) are correct

Explanation:

For a taxable income of ₹7,00,000, using the slab system:

• ₹0 – ₹2,50,000: 0% → ₹0
• ₹2,50,001 – ₹5,00,000: 5% → ₹12,500
• ₹5,00,001 – ₹7,00,000: 20% → 20% of ₹2,00,000 = ₹40,000

Total tax = ₹12,500 + ₹40,000 = ₹52,500

Option (B) expresses the breakdown and option (C) gives the final value. Hence, both (B) and (C) are correct.

Correct Option: (C) A monthly meter rent regardless of usage

Explanation:

Fixed charges are those that do not depend on how much of the service you use. A monthly meter rent is a typical example — it is charged irrespective of consumption.

• (A) and (D) are variable charges based on usage.
• (B) is also a variable component (tax) applied on consumption.
• (C) is the only fixed charge listed.

Therefore, the correct answer is option (C).

Correct Option: (A) True

Explanation:

A tariff rate refers to the price charged per unit of consumption for services such as electricity, water, gas, etc. It is usually expressed as:

• ₹ per kilowatt-hour (kWh) for electricity
• ₹ per kiloliter (KL) or liter for water

This rate can be fixed or variable (e.g., slab-wise), but the definition itself is correct.

Hence, the correct answer is option (A) True.

Correct Option: (A) True

Explanation:

A surcharge is an additional charge added to the regular bill amount. It is typically imposed due to:

• High consumption of resources
• Usage during peak demand periods
• Government-imposed levies or policy adjustments

It is meant to discourage excessive usage or to recover extra costs incurred by the provider.

Hence, the correct answer is option (A) True.

Correct Option: (C) $\textsf{₹}\,1200$

Given:

Consumption = 150 units

Tariff rate = $\textsf{₹}\,8$ per unit

To Find: Total energy charge (only consumption amount)

Solution:

Total energy charge = Units consumed $\times$ Rate per unit

Hence, the total energy charge is $\textsf{₹}\,1200$.

Correct Option: (D) Both (B) and (C) are correct

From Question 51:

• Total Energy Charge = ₹1200
• Fixed Charge = ₹100
• Surcharge = ₹120

Total Bill Amount (excluding taxes):

= ₹1200 (Energy) + ₹100 (Fixed) + ₹120 (Surcharge)

= ₹1420

Option (B) correctly includes all components of the bill.

Option (C) gives the same numerical calculation.

Hence, the correct answer is (D) Both (B) and (C) are correct.

Correct Option: (C) Compound Interest

Explanation:

• Fixed Charge – A standard charge included in many utility bills regardless of usage.
• Discount – Sometimes offered for early payment or special cases.
• Surcharge – An additional fee, often applied for late payment or peak usage.
• Compound Interest – This relates to finance and investment, not directly to billing or tariffs.

Therefore, Compound Interest is not typically a concept associated with bills and tariffs.

Correct Option: (A) True

Explanation:

A Service Charge is typically a fee added to a bill for services rendered. It may be:

• A fixed amount (e.g., ₹50 per service)
• Or calculated as a percentage of the bill (e.g., 10% of the total amount)

It is commonly applied in various industries such as hospitality, travel, and certain retail or personal services—not just in restaurants.

Hence, the statement is True.

Correct Option: (A) (i)-(d), (ii)-(a), (iii)-(b), (iv)-(c)

Matching the concepts:

• (i) Simple Interest – (d) Interest only on the principal amount.
• (ii) Compound Interest – (a) Interest on principal plus accumulated interest.
• (iii) Effective Rate – (b) Actual annual interest considering compounding.
• (iv) NPV (Net Present Value) – (c) Current value of future cash flows minus initial cost.

This matching accurately defines each financial term.

Correct Option: (B) 10%

Given: A sum doubles in 10 years at simple interest.

Let: Principal = $\textsf{₹}\,P$, Time = 10 years

Then Amount = $\textsf{₹}\,2P$

So, Interest = Amount − Principal = $\textsf{₹}\,2P - \textsf{₹}\,P = \textsf{₹}\,P$

Using Simple Interest formula: $$ I = \frac{P \times R \times T}{100} $$

Canceling $P$ from both sides (as $P ≠ 0$):

$1 = \frac{R \times 10}{100}$

Multiplying both sides by 100:

$100 = 10R$

Dividing both sides by 10:

$R = 10\%$

Conclusion: The annual rate of interest is 10%.

Correct Option: (A) Greater than the simple interest rate required to double the money in 10 years.

Explanation:

In simple interest (SI), to double the money in 10 years, the required rate is:

Let Principal = ₹P, then Interest = ₹P, Time = 10 years

Using SI formula:

\( I = \frac{P \times R \times T}{100} \Rightarrow P = \frac{P \times R \times 10}{100} \)

Solving: \( R = 10\% \)

In compound interest (CI), the formula is:

\( A = P(1 + r)^t \)

To double: \( 2P = P(1 + r)^{10} \Rightarrow (1 + r)^{10} = 2 \)

Taking 10th root: \( 1 + r = \sqrt[10]{2} \approx 1.0718 \Rightarrow r \approx 0.0718 \) or 7.18%

Conclusion: The compound interest rate required to double a sum in 10 years is approximately 7.18%, which is less than the 10% simple interest rate.

However, the question asks which rate is greater, and compares simple interest rate to compound interest rate. Since CI achieves doubling with a lower rate, it means:

The simple interest rate must be higher to achieve the same result.

Hence, the compound interest rate is less than the simple interest rate.

Correction: The correct option is: (B) Less than the simple interest rate required to double the money in 10 years.

Correct Option: (C) 12

Explanation:

The nominal rate is given as 6% compounded monthly.

This means that interest is calculated and added to the principal every month.

Since there are 12 months in a year, the number of compounding periods per year is:

12

Conclusion: There are 12 compounding periods in a year when the nominal rate is compounded monthly.

Correct Option: (C) Both (A) and (B)

Explanation:

The present value (PV) of a future amount is calculated using the formula:

\( PV = \frac{FV}{(1 + r)^n} \)

• \( FV \) = future value
• \( r \) = interest rate (discount rate)
• \( n \) = time period in years

From the formula, it's clear that:

• If the interest rate increases (\( r \) ↑), the denominator becomes larger, making PV smaller.
• If the time period increases (\( n \) ↑), the exponent increases, again making PV smaller.

So, both a higher interest rate and a longer time period will reduce the present value of a future amount.

Conclusion: The present value will be lower if both the interest rate is higher and the time period is longer.

Correct Option: (A) beginning

Explanation:

An annuity due is a type of annuity in which payments are made at the beginning of each period, rather than at the end.

This is different from an ordinary annuity, where payments are made at the end of each period.

Conclusion: In an annuity due, payments occur at the beginning of each period.

Correct Option: (C) Goods and Services Tax (GST)

Explanation:

The Goods and Services Tax (GST) is a value-added tax that is levied on the supply of goods and services at each stage of production and distribution.

• It ensures that tax is applied only on the value added at each stage.
• Businesses can claim input tax credit on the tax paid on purchases, effectively taxing only the value they add.

Other options:

• Income Tax – charged on income earned by individuals or entities.
• Wealth Tax – levied on the net wealth of individuals (now abolished in India).
• Property Tax – imposed on property ownership, usually by local governments.

Conclusion: GST is the tax levied on the value added at each stage of the supply chain.

Correct Option: (C) The amount of electrical energy used, typically measured in kWh.

Explanation:

In an electricity bill, "units consumed" refers to the total electrical energy used during the billing period. It is measured in kilowatt-hours (kWh), where:

1 unit = 1 kWh

• This value is calculated by taking the difference between the current and previous meter readings.
• The number of units consumed is then multiplied by the tariff rate to determine the energy charge.

Other options explained:

• (A) Total amount payable includes energy charge + fixed charges + taxes, not just units.
• (B) Fixed charge is a constant monthly charge, irrespective of usage.
• (D) Tariff rate is the cost per unit (kWh), not the unit itself.

Conclusion: "Units consumed" correctly refers to the electrical energy used, measured in kWh.

Correct Option: (C) Annuity Payment.

Explanation:

The given scenario involves repaying a loan through equal annual installments over a period of 3 years, with interest compounded annually at 8%. This is a classic case of an annuity payment.

• An annuity is a series of equal payments made at regular intervals over a specified period.
• Here, the loan is being repaid through 3 equal payments (installments) — this defines a loan amortization annuity.
• Each installment includes a portion of the principal and compound interest.

Other options:

• Simple Interest (A): Would apply if interest were not compounded and was calculated only on the principal.
• Compound Interest (B): While interest is compounded, the focus here is on structured payments, i.e., annuity.
• NPV (D): Net Present Value is used to evaluate investment profitability, not to calculate loan repayments.

Conclusion: The correct financial concept involved is Annuity Payment.

Correct Option: (D) Both (A) and (B).

Explanation:

The GST (Goods and Services Tax) amount on an item priced at ₹2500 with a GST rate of 12% is calculated as follows:

GST = ₹2500 × 12% = ₹2500 × 0.12 = ₹300

So:

• Option (A): ₹2500 × 0.12 is the correct formula.
• Option (B): ₹300 is the correct computed value.
• Option (C): ₹2500 + 300 gives the total price including GST (₹2800), but this is not the GST amount alone.

Conclusion: Both (A) and (B) correctly represent the GST amount — hence, the correct answer is (D).

Correct Option: (C) Effective Rate of Interest

Explanation:

• The Effective Rate of Interest (ERI) is used to compare interest rates that are compounded at different frequencies (e.g., annually, semi-annually, quarterly).
• It represents the actual interest earned or paid in a year, considering the effects of compounding.
• This allows fair comparison between financial products with varying compounding intervals.

Other Options:

• Simple Interest (A): Does not consider compounding at all.
• Compound Interest (B): Considers compounding but doesn't facilitate cross-comparison of rates with different frequencies.
• Nominal Rate of Interest (D): Is the stated rate, not adjusted for compounding.

Conclusion: Effective Rate of Interest is the correct concept used for comparing interest rates with different compounding frequencies.

Correct Option: (C) Net Present Value

Explanation:

• Net Present Value (NPV) is the sum of the present values of all future cash inflows and outflows associated with an investment, minus the initial investment cost.
• It is a key financial metric used in capital budgeting to evaluate the profitability of an investment or project.
• If NPV is positive, the investment is considered profitable; if negative, it is likely to result in a loss.

Other Options:

• Future Value (A): The value of an investment at a future date, not adjusted to today's value.
• Present Value (B): The value today of a single future cash flow, not the total net value of all cash flows.
• Accumulated Amount (D): Usually refers to the future sum of money including interest, not discounted back to present value.

Conclusion: The correct term is Net Present Value (NPV), which accurately describes the difference between the present value of future cash flows and the initial investment.

Correct Option: (A) True

Explanation:

• When the Net Present Value (NPV) of a project is zero, it means the present value of future cash inflows is exactly equal to the initial investment (cost).
• This implies that the project is expected to generate a return equal to the required rate of return (or discount rate).
• In such a case, the investor neither gains nor loses in terms of present value—it’s a break-even situation.

Why Other Options Are Incorrect:

• (B) False: Incorrect because a zero NPV does not imply a loss; it implies the investment meets the minimum acceptable return.
• (C) True, but it's still not a good project: Misleading — while the project may not generate excess returns, it’s not “bad” financially; it simply breaks even.
• (D) False, it means the project will lose money: Incorrect — a negative NPV indicates a loss, not zero.

Conclusion: A project with NPV = 0 meets the required return, making Option (A) the correct answer.

Correct Option: (C) Annuity

Explanation:

• A sinking fund is a fund established by making regular, periodic payments to accumulate a specified amount in the future.
• This concept is a classic example of an annuity, where equal payments are made at regular intervals and invested to earn interest (usually compounded).
• The goal is to accumulate a future sum, for example, to repay a debt, replace equipment, or fund future liabilities.

Why Other Options Are Incorrect:

• (A) Simple Interest: Does not account for reinvestment or compounding of payments.
• (B) Compound Interest: While compound interest is involved in the calculations, the key structure is that of an annuity.
• (D) Present Value: Sinking fund focuses on accumulating a future value, not discounting back to present value.

Conclusion: A sinking fund is an application of an annuity, making Option (C) the correct answer.

Correct Option: (D) All of the above

Explanation:

• A supply bill typically refers to any bill related to the provision of essential utility services.
• This includes:
  • Electricity bill – for electrical energy supply.
  • Water bill – for municipal or private water supply.
  • Piped gas bill – for domestic or industrial fuel gas supply.
• All of these are recurring bills generated for the consumption of basic utilities.

Conclusion: Since electricity, water, and piped gas are all considered supply services, Option (D) – "All of the above" – is correct.

Correct Option: (B) Present Value

Explanation:

• Present Value (PV) is the current worth of a future sum of money or stream of cash flows, given a specified rate of return (discount rate).
• It helps determine how much a future amount is worth in today's terms, considering the time value of money.

Why Other Options Are Incorrect:

• (A) Future Value: Calculates how much a present amount will grow to in the future, not its value today.
• (C) Accumulated Amount: Refers to the total future amount after compounding, not the value today.
• (D) Net Present Value (NPV): Involves evaluating an investment by subtracting the initial investment from the present value of future cash flows. It is more comprehensive than just calculating present value.

Conclusion: To find out what a future sum is worth today, you calculate its Present Value, making Option (B) the correct answer.

Correct Option: (B) ₹10

Explanation:

We are given:

• Principal (P) = ₹1000
• Rate (R) = 10% per annum
• Time (T) = 2 years

Simple Interest (SI):

SI = (P × R × T) / 100 = (1000 × 10 × 2) / 100 = ₹200

Compound Interest (CI):

CI = P × (1 + R/100)^T − P = 1000 × (1.10)² − 1000 = 1000 × 1.21 − 1000 = ₹210

Difference:

CI − SI = ₹210 − ₹200 = ₹10

Conclusion: The difference between compound and simple interest in this case is ₹10, so the correct answer is Option (B).

Correct Option: (B) principal repayment

Explanation:

• In an annuity-based loan repayment structure (like EMIs), each payment consists of two components:
• Interest: Charged on the remaining loan balance.
• Principal Repayment: The part that reduces the outstanding loan amount.
• As the loan progresses, the interest portion decreases, and the principal repayment portion increases.

Why Other Options Are Incorrect:

• (A) Future Value: Refers to the value of an investment at a future date, not related to the structure of loan repayments.
• (C) Surcharge: A tax or fee, not related to loan amortization.
• (D) Tax: Not a component of periodic loan repayments.

Conclusion: A part of each loan payment in an annuity goes towards principal repayment, making Option (B) the correct answer.

Correct Option: (B) The single lump sum amount today that is equivalent to the series of future payments.

Explanation:

• The present value of a regular annuity represents the value today (present value) of a series of equal payments made at regular intervals in the future, discounted using a specific interest rate.
• It answers the question: “What is the current worth of a set of future payments?”

Why Other Options Are Incorrect:

• (A): Refers to the future value of an annuity, not the present value.
• (C): Interest earned is not what the present value formula calculates directly.
• (D): The number of payments is a variable in the formula, not the result of it.

Conclusion: The formula for present value helps determine how much a series of future annuity payments is worth in today's terms, making Option (B) the correct answer.

Correct Option: (C) Greater than 12%

Explanation:

When interest is compounded more frequently than once a year, the Effective Annual Rate (EAR) becomes higher than the Nominal Annual Rate.

The formula to calculate the Effective Annual Rate (EAR) from the Nominal Annual Rate compounded monthly is:

Where:

• $r = 0.12$ (Nominal annual interest rate = 12%)
• $n = 12$ (compounded monthly, so 12 times a year)

Substituting the values into equation (i):

⇒ $\text{EAR} = (1 + 0.01)^{12} - 1$

⇒ $\text{EAR} = (1.01)^{12} - 1$

⇒ $\text{EAR} \approx 1.1268 - 1$

⇒ $\text{EAR} \approx 0.1268$ or 12.68%

Therefore, the annual effective rate is approximately 12.68%, which is greater than the nominal rate of 12%.

Correct Option: (A) True

Explanation:

Taxable income refers to the portion of an individual’s or entity’s income that is subject to taxation by the government.

This is not the gross income, but the income that remains after subtracting allowable deductions (like investments under section 80C, interest on home loans, etc.) and exemptions (such as House Rent Allowance, Leave Travel Allowance, etc.).

Hence, tax is calculated on the net taxable income rather than the gross total income.

Therefore, the given statement is true.

Correct Option: (D) All of the above

Explanation:

The total electricity bill is calculated by considering multiple factors, which include:

• Number of units consumed: The more electricity (in kWh or units) you use, the higher the bill will be.
• Applicable tariff rates: These rates vary by consumer category (residential, commercial, industrial) and slab. Higher consumption can move a user into a higher slab rate.
• Fixed charges and surcharges: Utilities also impose fixed monthly charges and other surcharges (e.g., fuel adjustment cost, electricity duty) regardless of consumption level.

All these components together determine the final amount payable.

Hence, option (D) is the correct answer.

Correct Option: (A) Positive

Explanation:

The Net Present Value (NPV) is calculated by discounting the future cash flows of a project at a specified required rate of return (also known as the discount rate).

The Internal Rate of Return (IRR) is the rate at which the NPV of all future cash flows becomes zero.

Now, if the IRR > required rate of return, it means that the project earns more than the minimum acceptable return. Hence, discounting the future cash flows at the required rate will result in a positive NPV.

Therefore, it implies:

Hence, option (A) is the correct answer.

Correct Option: (C) progressive

Explanation:

India follows a progressive tax system for income tax, where the tax rate increases with an increase in taxable income.

This means individuals with higher incomes pay taxes at higher rates, and those with lower incomes pay at lower rates, ensuring equity and fairness in the tax burden.

This can be contrasted with:

• Proportional tax (Option A): A flat rate of tax regardless of income.
• Regressive tax (Option B): Lower-income individuals pay a higher percentage of their income than higher-income individuals.
• Fixed tax (Option D): A constant tax amount regardless of income.

Hence, option (C) is the correct answer.

Correct Option: (B) $\textsf{₹}\,1416 / 1.18$

Explanation:

The final price of an item after GST is the sum of the original price and the GST amount.

If the GST rate is 18%, then the final price is 118% of the original price, or in other words, $1.18$ times the original price.

Let the price before GST be $x$. Then,

Dividing both sides by $1.18$:

So, the original price before GST is $\textsf{₹}\,\dfrac{1416}{1.18}$.

Hence, option (B) is the correct answer.

Correct Option: (C) Net Present Value (NPV)

Explanation:

Net Present Value (NPV) is a financial metric used to evaluate and compare different investment opportunities based on their expected future cash flows, discounted to the present value.

The formula for NPV is:

Key Features:

• Accounts for the time value of money.
• Useful for comparing projects with varying cash flow timings and amounts.
• A positive NPV indicates a profitable investment.

Other options are not suitable for this purpose:

• Simple Interest: Only considers interest on the principal, not useful for complex cash flows.
• Compound Interest: Deals with interest on accumulated interest, not suitable for evaluating investment profitability.
• Annuity: Refers to a fixed series of payments, not a tool for comparison of investment opportunities.

Therefore, the correct answer is (C) Net Present Value (NPV).

Correct Option: (B) Compound Interest (specifically, the future value formula)

Explanation:

The concept of present value is based on the time value of money, which is a core idea in compound interest.

The future value (FV) formula under compound interest is:

By rearranging this equation to solve for Present Value (PV), we get:

Where,

• $FV$ = Future Value
• $PV$ = Present Value
• $r$ = Rate of interest per period
• $n$ = Number of periods

This derivation shows that present value is directly obtained from the compound interest formula.

Hence, the correct answer is (B).

Correct Option: (A) $\textsf{₹}\,540 / 0.18$

Explanation:

We are given the GST amount and the GST rate. We need to find the original cost (i.e., the value before GST was added).

Let the original cost of the service be $x$.

GST is 18% of the original cost, i.e.,

We convert 18% to decimal:

Now solving for $x$:

Therefore, the original cost before GST was:

$\textsf{₹}\,540 \div 0.18 = \textsf{₹}\,3000$

Hence, the correct answer is (A).

Definition:

Simple Interest (SI) is the interest calculated on the principal amount for a specified period at a given rate of interest. It does not take into account any interest on previously earned interest.

Formula:

The formula to calculate Simple Interest is:

Where:

• $SI$ = Simple Interest
• $P$ = Principal amount
• $R$ = Rate of interest per annum
• $T$ = Time (in years)

Note: The units of $R$ and $T$ must be consistent, typically rate per annum and time in years.

Definition:

Compound Interest (CI) is the interest calculated on the initial principal as well as on the accumulated interest from previous periods. This means that interest is added to the principal at the end of each compounding period, and future interest is calculated on the new total.

Formula for Compound Interest (when compounded annually):

If the principal is $P$, the annual rate of interest is $R\%$, and the time is $T$ years, then:

Difference between SI and CI:

Given:

Principal, $P = \textsf{₹} 15,000$

Rate of interest, $R = 8\%$ per annum

Time, $T = 3$ years

To Find:

Simple Interest (SI)

Formula:

Substituting the values:

$SI = \dfrac{360000}{100}$

$SI = \textsf{₹} 3,600$

Final Answer:

The simple interest earned is $\textsf{₹} 3,600$.

Given:

Principal, $P = \textsf{₹} 10,000$

Rate of Interest, $R = 5\%$ per annum

Time, $T = 4$ years

To Find:

Accumulated Amount (A)

Step 1: Calculate Simple Interest (SI)

$SI = \dfrac{200000}{100} = \textsf{₹} 2,000$

Step 2: Calculate Accumulated Amount

$A = 10000 + 2000 = \textsf{₹} 12,000$

Final Answer:

The accumulated amount is $\textsf{₹} 12,000$.

Given:

Principal, $P = \textsf{₹} 20,000$

Rate of Interest, $R = 10\%$ per annum

Time, $T = 2$ years

To Find:

Compound Interest (CI)

Step 1: Calculate Amount (A)

$A = 20000 \left(1.1\right)^2$

$A = 20000 \times 1.21 = \textsf{₹} 24,200$

Step 2: Calculate Compound Interest (CI)

$CI = 24200 - 20000 = \textsf{₹} 4,200$

Final Answer:

The compound interest is $\textsf{₹} 4,200$.

Given:

Principal, $P = \textsf{₹} 5,000$

Rate of Interest, $R = 6\%$ per annum

Time, $T = 3$ years

To Find:

Accumulated Amount (A)

Step 1: Use the compound interest formula to find the amount (A)

$A = 5000 \left(1.06\right)^3$

$A = 5000 \times 1.191016 = \textsf{₹} 5,955.08$

Final Answer:

The accumulated amount is $\textsf{₹} 5,955.08$.

Definition:

Effective Rate of Interest (also called Effective Annual Rate) is the actual interest earned or paid on an investment or loan in one year, taking into account the effect of compounding over the year.

Formula:

Usefulness:

The effective rate is useful because:

• It helps in comparing different investment or loan options that have different compounding frequencies.
• It reflects the true cost of borrowing or the actual return on investment.
• It assists in making better financial decisions by providing a standardised interest rate.

Example:

If the nominal interest rate is $12\%$ compounded monthly, then:

$= \left(1 + 0.01\right)^{12} - 1 = (1.01)^{12} - 1$

$= 1.126825 - 1 = 0.126825 = 12.6825\%$

So, the effective rate is 12.68%, which is greater than the nominal rate.

Given:

Nominal Annual Interest Rate, $r = 8\% = 0.08$

Compounded Quarterly $\Rightarrow$ Number of Compounding Periods, $n = 4$

To Find:

Effective Annual Rate (EAR)

Step 1: Use the formula for Effective Rate

$= \left(1 + 0.02\right)^4 - 1$

$= (1.02)^4 - 1$

$= 1.08243216 - 1 = 0.08243216$

Final Answer:

The effective annual rate is 8.2432%

Given:

Effective Annual Rate (EAR) $= 10.25\% = 0.1025$

Compounded Half-Yearly $\Rightarrow n = 2$

To Find:

Nominal Annual Interest Rate $r$

Step 1: Use the formula for Effective Rate

Step 2: Substitute the values

$\Rightarrow \left(1 + \dfrac{r}{2} \right)^2 = 1.1025$

Take square root on both sides:

$1 + \dfrac{r}{2} = \sqrt{1.1025} = 1.05$

$\Rightarrow \dfrac{r}{2} = 1.05 - 1 = 0.05$

$\Rightarrow r = 2 \times 0.05 = 0.10 = 10\%$

Final Answer:

The nominal rate of interest is 10% per annum.

Definition:

Present Value (PV) is the current worth of a sum of money that is to be received in the future, discounted at a particular rate of compound interest.

It answers the question: “How much is a future amount worth today?”

Formula for Present Value (PV):

Where:

$PV$ = Present Value

$A$ = Future Amount

$r$ = Rate of compound interest per period

$n$ = Number of periods

Conclusion:

The higher the interest rate or the longer the time period, the lower the present value of a future amount.

Given:

Future Value ($A$) $= \textsf{₹} 12,100$

Rate of Compound Interest ($r$) $= 10\% = 0.10$

Time ($n$) $= 2$ years

To Find:

Present Value ($PV$)

Using the Present Value Formula:

Substitute the values:

$PV = \dfrac{12,100}{(1 + 0.10)^2}$

$PV = \dfrac{12,100}{(1.10)^2}$

$PV = \dfrac{12,100}{1.21}$

$PV = \textsf{₹} 10,000$

Final Answer:

The Present Value is $\textsf{₹} 10,000$.

Definition:

Future Value (FV) is the amount of money that an investment made today will grow to after earning interest for a specified number of periods at a given compound interest rate.

It answers the question: "What will be the value of today’s money in the future?"

Formula for Future Value (FV):

Where:

$FV$ = Future Value

$PV$ = Present Value

$r$ = Rate of compound interest per period

$n$ = Number of periods

Relation between Present Value and Future Value:

The Future Value is derived by compounding the Present Value over a number of time periods using the compound interest formula.

Alternatively, the Present Value is the discounted value of the Future Value:

Conclusion:

Future Value and Present Value are inversely related: as time or interest rate increases, the Present Value decreases for the same Future Value, and vice versa.

Given:

Present Value ($PV$) $= \textsf{₹} 8,000$

Rate of interest ($r$) $= 7\% = 0.07$

Time ($n$) $= 3$ years

To Find:

Future Value ($FV$)

Using the Formula:

Substituting the values:

$FV = 8,000 \times (1 + 0.07)^3$

$FV = 8,000 \times (1.07)^3$

$FV = 8,000 \times 1.225043$

$FV = \textsf{₹} 9,800.34$

Final Answer:

The Future Value is $\textsf{₹} 9,800.34$.

Definition of Annuity:

An annuity is a series of equal payments or receipts made at regular intervals over a specified period of time. These payments can be made weekly, monthly, quarterly, yearly, etc.

Definition of Regular Annuity:

A Regular Annuity (also known as an ordinary annuity) is a type of annuity in which the payments are made at the end of each period.

Examples include monthly deposits into a savings account, monthly rent payments, or insurance premiums.

Key Characteristics of a Regular Annuity:

• Equal payments
• Fixed time intervals
• Payments occur at the end of each period

Given:

Annual deposit = $\textsf{₹} 1,000$

Rate of interest ($r$) = $5\% = 0.05$

Number of years ($n$) = 3

To Find:

Accumulated amount at the end of 3 years

Using the formula for Future Value of an Ordinary Annuity:

Substituting the values:

$FV = 1,000 \times \frac{(1 + 0.05)^3 - 1}{0.05}$

$FV = 1,000 \times \frac{1.157625 - 1}{0.05}$

$FV = 1,000 \times \frac{0.157625}{0.05}$

$FV = 1,000 \times 3.1525$

$FV = \textsf{₹} 3,152.50$

Final Answer:

The accumulated amount at the end of 3 years is $\textsf{₹} 3,152.50$.

Definition of Tax:

Tax is a mandatory financial charge or levy imposed by a government on individuals or businesses to fund public expenditures and government activities. It is collected to support various services like education, healthcare, infrastructure, defense, etc.

The Two Main Types of Taxes:

1. Direct Tax:

A direct tax is a type of tax that is paid directly by an individual or organization to the government. It is imposed on income or wealth.

Examples: Income Tax, Wealth Tax, Corporate Tax.

2. Indirect Tax:

An indirect tax is a tax that is levied on goods and services rather than on income or profits. It is collected by an intermediary (like a retailer) from the person who bears the ultimate economic burden of the tax (the consumer).

Examples: Goods and Services Tax (GST), Customs Duty, Excise Duty.

Given:

Selling Price (excluding GST) = $\textsf{₹} 500$

GST rate = $18\%$

To Find:

GST amount

Solution:

GST is calculated as:

Substituting the values:

$\text{GST} = 500 \times \frac{18}{100}$

$\text{GST} = 500 \times 0.18$

$\text{GST} = \textsf{₹} 90$

Final Answer:

The GST amount is $\textsf{₹} 90$.

Given:

Price including GST = $\textsf{₹} 560$

GST Rate = $12\%$

To Find:

Original price before GST

Solution:

Let the original price before GST be $\textsf{₹} x$.

Then, price including GST = $x + 12\%$ of $x = x\left(1 + \frac{12}{100}\right) = x \times 1.12$

Dividing both sides by $1.12$:

$x = \frac{560}{1.12}$

$x = \textsf{₹} 500$

Final Answer:

The original price before GST is $\textsf{₹} 500$.

Meaning of Income Tax:

Income Tax is a direct tax imposed by the Government of India on the income earned by individuals, Hindu Undivided Families (HUFs), companies, firms, LLPs, and other entities during a financial year.

Key Features:

• It is governed by the Income Tax Act, 1961.
• The tax is levied on total income, which includes income from salaries, house property, business or profession, capital gains, and other sources.
• The tax rates vary based on the income slabs, residential status, and the age of the individual.
• There are provisions for deductions (e.g., under Section 80C, 80D, etc.) to reduce taxable income.
• Tax is collected by means such as Tax Deducted at Source (TDS), advance tax, and self-assessment tax.

Purpose:

The revenue collected through income tax is used by the government to fund infrastructure, education, health care, defense, and various social welfare schemes.

Final Note:

Every eligible taxpayer must file an Income Tax Return (ITR) annually to declare their income and taxes paid. Non-compliance can lead to penalties and legal consequences.

Given:

Taxable income = $\textsf{₹} 6,00,000$

Tax rate for income between $\textsf{₹} 5,00,001$ and $\textsf{₹} 10,00,000$ = $10\%$

To Find:

Tax liability for the portion of income falling between $\textsf{₹} 5,00,001$ and $\textsf{₹} 10,00,000$

Solution:

Income falling under $10\%$ slab = $\textsf{₹} 6,00,000 - \textsf{₹} 5,00,000 = \textsf{₹} 1,00,000$

Tax @ $10\%$ on $\textsf{₹} 1,00,000$ = $\frac{10}{100} \times \textsf{₹} 1,00,000 = \textsf{₹} 10,000$

Final Answer:

The tax liability is $\textsf{₹} 10,000$.

Definition:

A Tariff Rate refers to the rate or price charged by a utility service provider (such as electricity, water, or gas companies) for the consumption of services by a consumer.

Explanation:

Utility providers charge users based on how much of the service they consume. The tariff rate is generally expressed in terms of units:

• Electricity – per kilowatt-hour (kWh)
• Water – per kilolitre (kL) or litre
• Gas – per cubic meter (m³) or per unit

Types of Tariff Rates:

• Flat Rate Tariff: A single rate is charged for all units consumed.
• Slab Rate Tariff: Different rates are applied based on the number of units consumed (e.g., first 100 units at one rate, next 200 units at a higher rate).
• Time-of-Day Tariff: Higher rates during peak hours and lower during off-peak hours.

Importance:

Tariff rates help determine how much a consumer pays for utility services and encourage efficient usage by penalizing excessive consumption through higher rates in progressive slabs.

Fixed Charge:

This is a flat fee charged by the electricity provider regardless of the amount of electricity consumed. It is typically charged to cover costs related to meter maintenance, infrastructure, administrative expenses, and availability of power supply.

Key Features:

• Does not vary with consumption
• Charged even if no electricity is used
• Ensures minimum revenue for the utility provider

Usage Charge (also called Energy Charge):

This is the variable component of the electricity bill based on the actual amount of electricity consumed by the user. It is usually charged per kilowatt-hour (kWh).

Key Features:

• Varies with the number of units consumed
• Encourages efficient use of electricity
• Usually follows a slab or tier system (e.g., first 100 units at one rate, next 100 at a higher rate, etc.)

Conclusion:

The Fixed Charge is a constant fee for service availability and infrastructure, while the Usage Charge is based on actual consumption and encourages responsible usage of electricity.

Given:

Fixed charge = $\textsf{₹} 100$

Usage charge per unit = $\textsf{₹} 5$

Units consumed = 150

To Find:

Total bill amount (excluding taxes/surcharges)

Solution:

Step 1: Calculate usage charge.

Usage charge = 150 × $\textsf{₹} 5 = \textsf{₹} 750$

Step 2: Add fixed charge.

Total bill = Fixed charge + Usage charge

Final Answer:

Total bill amount = $\textsf{₹} 850$

Surcharge:

A surcharge is an additional charge added to the base amount of a bill. It is typically imposed by the government or the utility provider for various purposes such as fuel cost adjustment, late payment penalties, or funding infrastructure development.

Examples of Surcharges:

• Fuel Surcharge
• Late Payment Surcharge
• Electricity Duty or Tax

Service Charge:

A service charge is a fee levied to cover the cost of providing a particular service. It may be fixed or based on usage. This charge is usually included in bills like restaurant bills, electricity bills, internet bills, etc.

Purpose:

• Maintenance of infrastructure
• Operational and administrative costs
• To ensure quality of service

Conclusion:

Surcharge is an extra amount added due to specific reasons (like fuel cost), whereas Service Charge is a fee for the actual service provided. Both are mandatory components of a final bill amount.

Given:

Bill amount = $\textsf{₹} 800$

Service charge = $10\%$ of bill amount

To Find:

Total amount payable including service charge

Solution:

Step 1: Calculate $10\%$ of $\textsf{₹} 800$

Step 2: Add the service charge to the original bill amount

Final Answer:

Total amount payable = $\textsf{₹} 880$

Given:

Principal (P) = $\textsf{₹} 25,000$

Rate of interest (R) = $6\%$ per annum

Time period = From 15th March 2023 to 15th August 2023

To Find:

Simple Interest (SI)

Solution:

Step 1: Calculate the time in years

From 15th March to 15th August = 5 months = $\frac{5}{12}$ years

Step 2: Use the formula of Simple Interest:

Step 3: Substitute the values into the formula:

Simplifying:

$SI = \frac{25000 \times 6 \times 5}{100 \times 12} = \frac{750000}{1200} = \textsf{₹} 625$

Final Answer:

Simple Interest earned = $\textsf{₹} 625$

Given:

Time (T) = 10 years

The investment doubles, i.e., the amount becomes twice the principal.

To Find:

Rate of interest per annum (R)

Solution:

Let the Principal (P) = $\textsf{₹} 100$ (for ease of calculation)

Since it doubles, the Amount (A) = $\textsf{₹} 200$

Therefore, Simple Interest (SI) = A - P = $\textsf{₹} 200 - \textsf{₹} 100 = \textsf{₹} 100$

Using the formula for Simple Interest:

Substitute the known values:

Simplifying:

$100 = R \times 10$

⇒ $R = \frac{100}{10} = 10\%$

Final Answer:

Rate of interest per annum = $10\%$

Given:

Principal (P) = $\textsf{₹} 10,000$

Amount (A) = $\textsf{₹} 12,100$

Time (T) = 2 years

To Find:

Rate of interest per annum (R)

Solution:

Using the compound interest formula:

Substitute the known values:

Divide both sides by 10,000:

$\left(1 + \frac{R}{100}\right)^2 = \frac{12100}{10000} = 1.21$

Take square root on both sides:

$1 + \frac{R}{100} = \sqrt{1.21} = 1.1$

Now, subtract 1 from both sides:

$\frac{R}{100} = 0.1$

Multiply both sides by 100:

$R = 10\%$

Final Answer:

Rate of interest per annum = $10\%$

Given:

Nominal annual interest rate = $12\%$

Compounding frequency = Monthly = $12$ times a year

To Find:

Effective Annual Rate (EAR)

Solution:

We use the formula for Effective Annual Rate (EAR):

Where:

$r = 0.12$ (nominal annual rate in decimal)

$n = 12$ (number of compounding periods per year)

Substitute the values:

$EAR = \left(1 + \frac{0.12}{12}\right)^{12} - 1$

$EAR = \left(1 + 0.01\right)^{12} - 1$

$EAR = (1.01)^{12} - 1$

$EAR \approx 1.126825 - 1$

$EAR \approx 0.126825$

Convert to percentage:

$EAR \approx 12.6825\%$

Final Answer:

Effective Annual Rate = $12.68\%$ (approximately)

Given:

Principal (P) = $\textsf{₹} 5,000$

Rate of Interest (R) = $6\%$ per annum

Time (T) = 4 years

To Find:

Future Value (Amount)

Solution:

First, we calculate Simple Interest (SI) using the formula:

Substitute the values:

$SI = \frac{5000 \times 6 \times 4}{100}$

$SI = \frac{120000}{100} = \textsf{₹} 1,200$

Now, Future Value = Principal + Simple Interest

$A = \textsf{₹} 6,200$

Final Answer:

Future Value = $\textsf{₹} 6,200$

Given:

Principal (P) = $\textsf{₹} 10,000$

Rate of Interest (R) = $10\%$ per annum

Time = $1.5$ years

Compounded half-yearly $\Rightarrow$ $n = 2$ times per year

To Find:

Compound Interest (CI)

Solution:

Convert time in terms of number of half-years:

Rate per half-year = $\frac{10}{2} = 5\%$

Use Compound Interest formula for Amount (A):

Substitute the values:

$A = 10000 \left(1 + \frac{5}{100}\right)^3$

$A = 10000 \times (1.05)^3$

$A = 10000 \times 1.157625$

$A = \textsf{₹} 11,576.25$

Now, Compound Interest (CI) = Amount - Principal

$CI = \textsf{₹} 1,576.25$

Final Answer:

Compound Interest = $\textsf{₹} 1,576.25$

Given:

Future Value (A) = $\textsf{₹} 10,000$

Rate of Interest (R) = $8\%$ per annum

Time (T) = 3 years

To Find:

Present Value (P)

Solution:

We use the formula to find Present Value under simple interest:

Substitute the values:

$P = \frac{10000}{1 + \frac{8 \times 3}{100}} = \frac{10000}{1 + \frac{24}{100}}$

$P = \frac{10000}{1.24}$

$P = \textsf{₹} 8,064.52$

Final Answer:

Present Value = $\textsf{₹} 8,064.52$

Definition of Net Present Value (NPV):

The Net Present Value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows over a period of time.

It is a method used in capital budgeting to evaluate the profitability of an investment or project.

Formula:

Where:

• $C_t$ = Cash inflow at time $t$
• $C_0$ = Initial investment (cash outflow at time $t = 0$)
• $r$ = Discount rate
• $t$ = Time period
• $n$ = Number of periods

Interpretation of Positive NPV:

A positive NPV means that the present value of cash inflows is greater than the present value of cash outflows. This implies:

• The investment is expected to generate more value than its cost.
• The project or investment is profitable.
• It should be accepted as it adds value to the firm.

Conclusion:

NPV helps in making informed financial decisions. A positive NPV indicates a financially viable project.

Given:

• Initial Investment, $C_0 = \textsf{₹} 10,000$
• Cash inflow after one year, $C_1 = \textsf{₹} 12,000$
• Discount rate, $r = 10\% = 0.10$
• Time period, $t = 1$ year

To Find:

Net Present Value (NPV)

Formula:

Substitute the values:

Conclusion:

Since NPV is positive ($\textsf{₹} 909.09$), the project is financially viable and should be accepted.

Given:

• Annual deposit = $\textsf{₹} 500$
• Rate of interest, $r = 6\% = 0.06$ per annum
• Number of periods, $n = 2$ years
• Type of annuity: Annuity Due (payment at the beginning of each period)

To Find:

The accumulated amount at the end of 2 years.

Formula for Annuity Due (Future Value):

Substitute the values:

Conclusion:

The accumulated amount at the end of 2 years is $\textsf{₹} 1,091.80$.

Definition:

Input Tax Credit (ITC) under GST refers to the credit a registered taxpayer can claim for the tax paid on purchases (inputs) which are used in the course of business for making taxable supplies.

Explanation:

When a business purchases goods or services, it pays GST on those purchases. If these goods or services are used to make further taxable supplies, then the business can reduce its tax liability by claiming the credit of the tax paid on inputs. This mechanism ensures that the tax is collected only on the value addition at each stage of the supply chain.

Example:

Suppose a manufacturer pays $\textsf{₹} 10,000$ as GST on raw materials (input) and collects $\textsf{₹} 15,000$ as GST on the finished product (output). The manufacturer can claim ITC of $\textsf{₹} 10,000$ and needs to pay only $\textsf{₹} 5,000$ to the government.

Conditions for claiming ITC:

• The buyer must possess a valid tax invoice.
• The goods/services must have been received.
• Supplier must have paid the tax to the government.
• The buyer must have filed the GST returns.

Conclusion:

ITC is a fundamental feature of the GST regime, promoting transparency and removing the cascading effect of taxes.

Given:

• Price of the product (excluding GST) = $\textsf{₹} 800$
• GST rate = $28\%$

To Find:

The final price of the product including GST.

Formula:

Calculate GST:

Final Calculation:

Conclusion:

The final price of the product including $28\%$ GST is $\textsf{₹} 1,024$.

Definition:

Tax slabs in income tax refer to the categorization of income ranges with different tax rates applied to each range. This system ensures a progressive tax structure, where individuals with higher income pay a higher percentage of tax.

Explanation:

The Income Tax Department of India has defined specific slabs for different categories of taxpayers such as individuals (below 60 years), senior citizens (60–80 years), and super senior citizens (above 80 years). Each slab specifies the income range and the corresponding tax rate.

Example (Old Regime – for individuals below 60 years as per previous structure):

Note: The New Tax Regime introduced optional slabs with lower rates and no exemptions/deductions.

Conclusion:

Tax slabs are a mechanism to promote equity in taxation by taxing different income levels at proportionate rates. Taxpayers must choose the regime and compute taxes accordingly.

Given:

• Water consumption = 250 units
• Tariff rate = $\textsf{₹} 15$ per unit

To Find:

Total water consumption charge

Formula:

Calculation:

Conclusion:

The total water consumption charge is $\textsf{₹} 3,750$.

Given:

• Invoice Amount = $\textsf{₹} 1,500$
• Service Charge = $5\%$
• Surcharge = $2\%$

To Find:

Total amount including service charge and surcharge

Step 1: Calculate Service Charge

Step 2: Calculate Surcharge

Step 3: Calculate Total Amount

Conclusion:

The total amount including service charge and surcharge is $\textsf{₹} 1,605$.

Given:

• Compound Interest Rate = $10\%$ per annum
• Time = 3 years

To Find:

Equivalent Simple Interest Rate for 3 years

Assume:

Let the Principal be $\textsf{₹} 100$ (for ease of calculation)

Step 1: Calculate Compound Amount after 3 years

Using the formula:

Substituting the values:

Step 2: Calculate Compound Interest

Step 3: Find Equivalent Simple Interest Rate

Use the formula for Simple Interest:

Substitute known values:

Simplifying:

Conclusion:

The equivalent simple interest rate for $10\%$ compound interest over 3 years is approximately $11.03\%$ per annum.

Given:

• Time = 8 years
• Amount becomes triple of the principal

To Find:

Rate of compound interest (approximate)

Assume:

Let the Principal $P = \textsf{₹} 100$ (for ease of calculation)

Then, Amount $A = 3P = \textsf{₹} 300$

Step 1: Use Compound Interest Formula

Substitute values:

Divide both sides by 100:

Step 2: Take 8th root on both sides

To isolate $r$, take the 8th root of 3:

Now calculate $3^{1/8}$:

$3^{1/8} \approx 1.147$

Step 3: Solve for r

Multiply both sides by 100:

Conclusion:

The approximate compound interest rate is $14.7\%$ per annum.

Given:

• Annuity Payment (R) = $\textsf{₹} 2,000$ per year
• Time period (n) = 3 years
• Discount Rate (r) = $8\%$ per annum
• Payments are made at the end of each year

To Find:

Present Value (PV) of the annuity

Formula:

Present Value of Ordinary Annuity (payments at end of each period):

Step 1: Substitute the values:

$R = 2,000$, $r = 0.08$, $n = 3$

Simplify inside the brackets:

$PV = 2000 \times \frac{1 - (1.08)^{-3}}{0.08}$

Now calculate $(1.08)^{-3} \approx 0.7938$

Step 2: Calculate the fraction

$\frac{1 - 0.7938}{0.08} = \frac{0.2062}{0.08} \approx 2.5775$

Step 3: Multiply with annuity payment

Conclusion:

The Present Value of the annuity is approximately $\textsf{₹} 5,155$.

Given:

• Selling Price including GST = $\textsf{₹} 1,000$
• GST Rate = $18\%$

To Find:

The amount of GST included in the selling price

Step 1: Let the price before GST be $\textsf{₹} x$

Then the final price after adding $18\%$ GST becomes:

That is, $x(1 + \frac{18}{100}) = 1000$

$x \times 1.18 = 1000$

Step 2: Solve for $x$

$x = \frac{1000}{1.18} \approx 847.46$

Step 3: Calculate GST

GST = Selling Price - Price before GST

Conclusion:

The GST amount charged is approximately $\textsf{₹} 152.54$.

Given:

• Electricity meter reading in June = 1250 units
• Electricity meter reading in July = 1420 units

To Find:

The number of units consumed in July

Solution:

Units consumed = July reading - June reading

$= 170$ units

Conclusion:

The number of units consumed in July is 170 units.

Explanation:

The concepts of nominal interest rate and effective interest rate are crucial in finance as they help in understanding the true cost of borrowing or the true return on an investment, especially when interest is compounded more frequently than once a year.

1. Nominal Interest Rate:

The nominal interest rate (also known as the stated rate or annual percentage rate, APR) is the advertised or quoted interest rate on a loan or investment for a given period, typically an annual period, without taking into account the effect of compounding.

It represents the simple interest rate over the period, even if interest is calculated and added to the principal more than once within that period. For example, if a loan has a nominal interest rate of 12% per annum, and interest is compounded monthly, it means the annual rate is 12%, but 1% ($12\% / 12$ months) is applied each month.

2. Effective Interest Rate:

The effective interest rate (also known as the annual effective rate or annual equivalent rate, AER) is the true annual rate of interest paid on a loan or earned on an investment, considering the effect of compounding over the given period. It takes into account how frequently the interest is compounded within a year.

When interest is compounded more frequently (e.g., semi-annually, quarterly, monthly, or daily), the actual interest earned or paid will be higher than the nominal rate because the interest itself starts earning interest. The effective rate reflects this impact of compounding.

The formula to calculate the effective interest rate is:

Where:

• $i$ = nominal annual interest rate (as a decimal)
• $n$ = number of compounding periods per year

3. Key Differences:

4. Example:

Suppose you have two investment options for $\textsf{₹} 10,000$:

Option A: Nominal interest rate of $5\%$ per annum, compounded annually.

Option B: Nominal interest rate of $5\%$ per annum, compounded quarterly.

Let's calculate the effective interest rate for each option:

For Option A:

$i = 0.05$ (5%)

$n = 1$ (compounded annually)

Using the formula:

$\text{Effective Rate}_A = \left(1 + \frac{0.05}{1}\right)^1 - 1$

$\text{Effective Rate}_A = (1 + 0.05) - 1$

$\text{Effective Rate}_A = 1.05 - 1$

$\text{Effective Rate}_A = 0.05 = 5\%$

In this case, the nominal rate and effective rate are the same because the interest is compounded only once a year.

For Option B:

$i = 0.05$ (5%)

$n = 4$ (compounded quarterly, as there are 4 quarters in a year)

Using the formula:

$\text{Effective Rate}_B = \left(1 + \frac{0.05}{4}\right)^4 - 1$

$\text{Effective Rate}_B = (1 + 0.0125)^4 - 1$

$\text{Effective Rate}_B = (1.0125)^4 - 1$

$\text{Effective Rate}_B \approx 1.050945 - 1$

$\text{Effective Rate}_B \approx 0.050945 = 5.0945\%$

Conclusion:

Even though both options have the same nominal interest rate of $5\%$, the effective interest rate for Option B ($5.0945\%$) is higher than that for Option A ($5\%$). This is because the interest in Option B is compounded more frequently (quarterly), allowing the interest to earn interest sooner.

Therefore, to truly compare the profitability of these two investments, one should look at the effective interest rate, which clearly shows that Option B would yield a slightly higher return over a year despite having the same nominal rate.

Solution:

Given:

• Principal amount (P) = $\textsf{₹} 50,000$
• Time (T) = 9 months
• Annual Interest Rate (R) = $12\%$ per annum
• Compounding Frequency = Quarterly

To Find:

Compound Interest (CI)

Since the interest is compounded quarterly, we need to adjust the annual rate and the time period to match the compounding frequency.

1. Calculate the interest rate per quarter ($i$):

Annual Interest Rate = $12\%$

Number of quarters in a year = 4

$i = \frac{12\%}{4} = 3\%$ per quarter

In decimal form, $i = \frac{3}{100} = 0.03$

2. Calculate the number of compounding periods ($n$):

Total time = 9 months

Since there are 3 months in a quarter, the number of compounding periods is:

$n = \frac{9 \text{ months}}{3 \text{ months/quarter}} = 3$ quarters

3. Calculate the Amount (A) using the compound interest formula:

Substituting the values:

$A = 50000(1 + 0.03)^3$

$A = 50000(1.03)^3$

First, calculate $(1.03)^3$:

$(1.03)^2 = 1.03 \times 1.03 = 1.0609$

$(1.03)^3 = 1.0609 \times 1.03$

$\begin{array}{cc}& & 1 & . & 0 & 6 & 0 & 9 \\ \times & & 1 & . & 0 & 3 \\ \hline && 0 & 3 & 1 & 8 & 2 & 7 \\ 0 & 0 & 0 & 0 & 0 & 0 & \times \\ 1 & 0 & 6 & 0 & 9 & \times & \times \\ \hline 1 & . & 0 & 9 & 2 & 7 & 2 & 7 \\ \hline \end{array}$

So, $(1.03)^3 = 1.092727$

Now, calculate $A$:

$A = 50000 \times 1.092727$

$A = 54636.35$

4. Calculate the Compound Interest (CI):

$CI = \textsf{₹} 54636.35 - \textsf{₹} 50000$

$CI = \textsf{₹} 4636.35$

The compound interest on $\textsf{₹} 50,000$ for 9 months at $12\%$ per annum compounded quarterly is $\textsf{₹} 4636.35$.

Solution:

Given:

• Effective Annual Rate = $5.0625\%$
• Compounding Frequency = Annually

To Find:

Nominal Interest Rate

The formula for the effective interest rate is given by:

Where:

• $i$ = nominal annual interest rate (as a decimal)
• $n$ = number of compounding periods per year

In this problem:

• Effective Rate = $5.0625\% = 0.050625$ (as a decimal)
• Since the interest is compounded annually, the number of compounding periods per year ($n$) is 1.

Substitute these values into the formula (i):

$0.050625 = \left(1 + \frac{i}{1}\right)^1 - 1$

$0.050625 = (1 + i) - 1$

$0.050625 = 1 + i - 1$

$0.050625 = i$

To express this as a percentage, multiply by 100:

$i = 0.050625 \times 100\% = 5.0625\%$

Conclusion:

When interest is compounded annually, the nominal interest rate is equal to the effective annual rate.

Therefore, the nominal rate compounded annually is $5.0625\%$.

Solution:

Given:

• Future Value (FV) = $\textsf{₹} 5,000$
• Time (T) = 1 year
• Annual Interest Rate (R) = $5\%$ per annum
• Compounding Frequency = Half-yearly

To Find:

Present Value (PV)

The formula for the Present Value when compounded is derived from the compound interest formula:

Rearranging to find PV:

Where:

• FV = Future Value
• PV = Present Value
• $i$ = interest rate per compounding period
• $n$ = total number of compounding periods

1. Calculate the interest rate per compounding period ($i$):

The annual interest rate is $5\%$. Since it is compounded half-yearly, there are 2 compounding periods in a year.

$i = \frac{\text{Annual Rate}}{\text{Number of compounding periods per year}} = \frac{5\%}{2} = 2.5\%$ per half-year

As a decimal, $i = 0.025$

2. Calculate the total number of compounding periods ($n$):

The time period is 1 year. Since it's compounded half-yearly, there are 2 periods per year.

$n = \text{Time in years} \times \text{Number of compounding periods per year} = 1 \text{ year} \times 2 = 2$ periods

3. Substitute the values into the Present Value formula (ii):

$PV = \frac{5000}{(1 + 0.025)^2}$

$PV = \frac{5000}{(1.025)^2}$

Calculate $(1.025)^2$:

$\begin{array}{cc}& & 1 & . & 0 & 2 & 5 \\ \times & & 1 & . & 0 & 2 & 5 \\ \hline && & & & 5 & 1 & 2 & 5 \\ && & 2 & 0 & 5 & 0 & \times \\ && 0 & 0 & 0 & 0 & \times & \times \\ & 1 & 0 & 2 & 5 & \times & \times & \times \\ \hline 1 & . & 0 & 5 & 0 & 6 & 2 & 5 \\ \hline \end{array}$

So, $(1.025)^2 = 1.050625$

Now, calculate PV:

$PV = \frac{5000}{1.050625}$

$PV \approx 4758.8095238$

Rounding to two decimal places (for currency):

$PV \approx \textsf{₹} 4758.81$

The Present Value of $\textsf{₹} 5,000$ due in 1 year at $5\%$ per annum compounded half-yearly is approximately $\textsf{₹} 4758.81$.

Problem Statement:

Calculate the value of an investment after 2 years at $7\%$ p.a. simple interest, given an initial investment of $\textsf{₹} 20,000$.

Given:

Principal (P) $= \textsf{₹} 20,000$

Rate (R) $= 7\%$ p.a.

Time (T) $= 2$ years

To Find:

The Value of investment after 2 years, which is the Amount (A).

Formulae Used:

Simple Interest (S.I.) is given by the formula:

Amount (A) is the sum of Principal and Simple Interest:

Solution:

First, calculate the Simple Interest (S.I.) using equation (1):

$\text{S.I.} = \frac{20000 \times 7 \times 2}{100}$

$\text{S.I.} = 200 \times 7 \times 2$

$\text{S.I.} = 1400 \times 2$

Now, calculate the Amount (A) using equation (2) and the calculated Simple Interest:

$\text{A} = P + \text{S.I.}$

$\text{A} = \textsf{₹} 20000 + \textsf{₹} 2800$

Therefore, the value of the investment after 2 years is $\textsf{₹} 22,800$.

Problem Statement:

Define an annuity due and explain how its future value differs from a regular annuity.

To Define and Explain:

Definition of Annuity Due.

Comparison of its Future Value with a Regular Annuity.

Definition:

An annuity is a series of equal payments or receipts made at regular intervals. Annuities are generally classified based on the timing of these payments.

An Annuity Due is a type of annuity where payments are made at the beginning of each period. Examples include rent payments (paid at the start of the month) or insurance premiums. Because payments are made at the beginning of the period, each payment has an additional period to earn interest compared to a regular annuity.

Difference in Future Value from a Regular Annuity:

To understand the difference, let's first briefly define a regular annuity:

A Regular Annuity (or Ordinary Annuity) is a series of equal payments made at the end of each period. Most loan payments, like car loans or mortgages, are examples of regular annuities, where payments are made after a period of using the asset.

The key difference in their future values stems directly from the timing of the payments:

1. Timing of Payments:

* In an Annuity Due, payments occur at the start of each period.

* In a Regular Annuity, payments occur at the end of each period.

2. Interest Earned:

* Since payments in an annuity due are made at the beginning of the period, each payment has one extra period to earn interest compared to a corresponding payment in a regular annuity.

* For example, if the first payment of a regular annuity is made at the end of year 1, the first payment of an annuity due is made at the beginning of year 1 (which is effectively today). This "today's" payment earns interest for the entire first period.

3. Future Value Comparison:

* Consequently, the future value of an annuity due will always be greater than the future value of a regular annuity, assuming the same payment amount, interest rate, and number of periods.

* This is because every payment in an annuity due earns interest for an additional period. The future value (FV) of an annuity due can be calculated by multiplying the future value of a regular annuity by $ (1 + r) $, where $r$ is the interest rate per period.

Let's denote:

$Pmt$ = Payment amount per period

$r$ = Interest rate per period

$n$ = Number of periods

The formula for the Future Value of a Regular Annuity ($FV_{Reg}$) is:

The formula for the Future Value of an Annuity Due ($FV_{Due}$) is:

From the formulas, it is clear that:

This confirms that an annuity due has a higher future value because its payments earn interest for one more period.

Problem Statement:

Given a tax amount of $\textsf{₹} 50,000$ for an income slab taxed at $20\%$, find the income within that specific slab.

Given:

Tax Rate for the slab $= 20\%$

Tax Amount for this slab $= \textsf{₹} 50,000$

To Find:

The Income within this slab.

Formulae Used:

The relationship between Tax Amount, Tax Rate, and Income is given by:

Rearranging the formula to find Income:

Or equivalently:

Solution:

Let the income within this slab be $I$.

Using the formula from equation (2) or (3):

$I = \frac{\textsf{₹} 50,000}{20/100}$

$I = \textsf{₹} 50,000 \times \frac{100}{20}$

Simplify the fraction $\frac{100}{20}$:

$\frac{\cancel{100}^{5}}{\cancel{20}_{1}} = 5$

Now, substitute this value back into the equation for $I$:

$I = \textsf{₹} 50,000 \times 5$

$I = \textsf{₹} 250,000$

Thus, the income within this slab is $\textsf{₹} 250,000$.

Problem Statement:

Calculate the total water bill given a fixed charge, a per kilolitre consumption charge, and the total consumption in kilolitres.

Given:

Fixed Charge $= \textsf{₹} 50$

Consumption Charge per kilolitre $= \textsf{₹} 10$

Total Consumption $= 10$ kilolitres

To Find:

The Total Bill (excluding other charges).

Solution:

The total water bill consists of two components: the fixed charge and the consumption charge.

Step 1: Calculate the consumption charge.

The consumption charge is calculated by multiplying the consumption charge per kilolitre by the total kilolitres consumed.

$\text{Consumption Charge} = 10 \text{ kilolitres} \times \textsf{₹} 10/\text{kilolitre}$

Step 2: Calculate the total bill.

The total bill is the sum of the fixed charge and the consumption charge.

$\text{Total Bill} = \textsf{₹} 50 + \textsf{₹} 100$

$\text{Total Bill} = \textsf{₹} 150$

Therefore, the total water bill (excluding other charges) is $\textsf{₹} 150$.

Problem Statement:

Define the concept of 'Present Value of an Annuity' and explain why it is calculated.

To Define and Explain:

Definition of Present Value of an Annuity.

Reasons for calculating the Present Value of an Annuity.

Concept of 'Present Value of an Annuity':

An annuity is a series of equal payments or receipts made at fixed intervals over a specified period. Examples include regular pension payments, loan repayments (like mortgages or car loans), or insurance policy payouts.

The Present Value of an Annuity (PVA) is the current worth of a series of future payments or receipts, discounted back to the present time at a specific interest (or discount) rate. In simpler terms, it answers the question: "How much money would I need to invest today, at a given interest rate, to be able to receive a series of equal payments in the future?"

This concept is based on the fundamental principle of the time value of money, which states that a rupee today is worth more than a rupee in the future due to its potential earning capacity (i.e., interest). Therefore, future cash flows must be discounted to their equivalent value today.

The formula for the Present Value of a Regular Annuity ($PVA$) is generally given by:

Where:

• $Pmt$ = Payment amount per period
• $r$ = Interest rate (or discount rate) per period
• $n$ = Total number of periods

Why is it Calculated?

The Present Value of an Annuity is calculated for various financial and investment decisions. Its primary purpose is to allow for a fair comparison of future cash flows in today's terms. Here are some key reasons why it is calculated:

1. Loan Amortization and Valuation:

* It is used to determine the principal amount of a loan (like a mortgage or car loan) given fixed periodic payments, the interest rate, and the loan term. The loan amount is essentially the present value of all future loan payments.

* It helps in calculating how much a borrower can afford to borrow based on their ability to make regular payments.

2. Investment and Retirement Planning:

* Individuals or institutions use it to figure out how much money they need to save or invest today to provide a stream of regular income in retirement (e.g., pension plans) or for future expenses.

* It helps in evaluating investment opportunities that promise fixed periodic returns.

3. Valuation of Bonds and Other Financial Instruments:

* The price of a bond is often calculated as the present value of its future interest payments (coupon payments, which form an annuity) plus the present value of its face value (a single lump sum) at maturity.

* Similar valuation can be applied to other financial instruments that provide periodic cash flows.

4. Legal Settlements and Insurance Payouts:

* In legal cases, a lump-sum settlement might be determined by calculating the present value of a stream of future payments that would otherwise be made to the plaintiff (e.g., for lost income or medical expenses).

* Insurance companies use it to determine the lump-sum equivalent of future periodic annuity payments from life insurance or structured settlements.

5. Capital Budgeting Decisions:

* Businesses use present value concepts to evaluate potential projects or investments. When a project is expected to generate a series of constant cash inflows, the present value of these inflows helps determine the project's profitability and feasibility.

In essence, calculating the Present Value of an Annuity enables financial professionals, businesses, and individuals to make informed decisions by bringing all future cash flows back to a common point in time (the present) for accurate comparison and valuation.

Problem Statement:

Calculate the total bill including a government surcharge, given usage charge, fixed charge, and the surcharge percentage on their total.

Given:

Usage Charge $= \textsf{₹} 300$

Fixed Charge $= \textsf{₹} 120$

Government Surcharge $= 5\%$ on the total of usage and fixed charges

To Find:

The Total Amount with the surcharge.

Solution:

Step 1: Calculate the total of usage and fixed charges.

Let $Base \text{ Charge}$ be the sum of usage and fixed charges.

$\text{Base Charge} = \textsf{₹} 300 + \textsf{₹} 120$

Step 2: Calculate the government surcharge amount.

The surcharge is $5\%$ of the Base Charge.

$\text{Surcharge Amount} = \textsf{₹} 420 \times \frac{5}{100}$

$\text{Surcharge Amount} = \textsf{₹} 420 \times 0.05$

Step 3: Calculate the total amount with the surcharge.

The total amount is the sum of the Base Charge and the Surcharge Amount.

$\text{Total Amount} = \textsf{₹} 420 + \textsf{₹} 21$

$\text{Total Amount} = \textsf{₹} 441$

Therefore, the total amount with the surcharge is $\textsf{₹} 441$.

Problem Statement:

Calculate the compound interest on $\textsf{₹} 50,000$ for 3 years at $10\%$ per annum. Compare this with the simple interest for the same period and rate, and determine the difference in interest earned.

Given:

Principal (P) $= \textsf{₹} 50,000$

Time (T or n) $= 3$ years

Rate (R or i) $= 10\%$ per annum

To Find:

1. Compound Interest (C.I.)

2. Simple Interest (S.I.)

3. Difference in Interest Earned ($C.I. - S.I.$)

Formulae Used:

1. Simple Interest (S.I.):

2. Amount (A) with Compound Interest:

3. Compound Interest (C.I.):

Solution:

Part 1: Calculate Simple Interest (S.I.)

Using formula (1):

$\text{S.I.} = \frac{50000 \times 10 \times 3}{100}$

$\text{S.I.} = 500 \times 10 \times 3$

$\text{S.I.} = 5000 \times 3$

Part 2: Calculate Compound Interest (C.I.)

First, calculate the Amount (A) using formula (2):

$\text{A} = 50000 \left(1 + \frac{10}{100}\right)^3$

$\text{A} = 50000 \left(1 + 0.1\right)^3$

$\text{A} = 50000 (1.1)^3$

We need to calculate $(1.1)^3$:

$1.1 \times 1.1 = 1.21$

$1.21 \times 1.1 = 1.331$

So, $(1.1)^3 = 1.331$

Now, substitute this value back into the amount calculation:

$\text{A} = 50000 \times 1.331$

$\text{A} = \textsf{₹} 66,550$

Next, calculate the Compound Interest (C.I.) using formula (3):

$\text{C.I.} = \text{A} - P$

$\text{C.I.} = \textsf{₹} 66,550 - \textsf{₹} 50,000$

Part 3: Calculate the Difference in Interest Earned

Difference $= \text{C.I.} - \text{S.I.}$

Difference $= \textsf{₹} 16,550 - \textsf{₹} 15,000$

Conclusion:

The Simple Interest earned is $\textsf{₹} 15,000$.

The Compound Interest earned is $\textsf{₹} 16,550$.

The difference in interest earned (Compound Interest minus Simple Interest) is $\textsf{₹} 1,550$.

Problem Statement:

Calculate the accumulated amount of $\textsf{₹} 1,00,000$ invested for 5 years at $8\%$ per annum compounded quarterly.

Given:

Principal (P) $= \textsf{₹} 1,00,000$

Time (t) $= 5$ years

Annual Rate (R) $= 8\%$ per annum $= 0.08$

Compounding Frequency (m) $= \text{quarterly} = 4$ times a year

To Find:

The Accumulated Amount (A).

Formula Used:

The formula for the accumulated amount when interest is compounded $m$ times a year is:

Where:

• $A$ = Accumulated Amount / Future Value
• $P$ = Principal Amount
• $R$ = Annual Interest Rate (as a decimal)
• $m$ = Number of times interest is compounded per year
• $t$ = Time in years

Solution:

First, determine the interest rate per compounding period ($\frac{R}{m}$) and the total number of compounding periods ($mt$).

Rate per period ($\frac{R}{m}$):

$\frac{R}{m} = \frac{0.08}{4} = 0.02$

Total number of periods ($mt$):

$mt = 4 \times 5 = 20$ periods

Now, substitute these values into formula (1):

$\text{A} = 1,00,000 \left(1 + 0.02\right)^{20}$

$\text{A} = 1,00,000 (1.02)^{20}$

To calculate $(1.02)^{20}$, we typically use a financial calculator or a logarithm table. For this calculation:

$(1.02)^{20} \approx 1.485947396$

Now, multiply this by the principal:

$\text{A} = 1,00,000 \times 1.485947396$

$\text{A} = \textsf{₹} 1,48,594.7396$

Rounding to two decimal places for currency:

$\text{A} \approx \textsf{₹} 1,48,594.74$

Therefore, the accumulated amount after 5 years, compounded quarterly, is $\textsf{₹} 1,48,594.74$.

Problem Statement:

Calculate the present value of $\textsf{₹} 1,50,000$ to be received in 5 years, given a required rate of return of $12\%$ per annum compounded annually.

Given:

Future Value (FV) $= \textsf{₹} 1,50,000$

Number of years (n) $= 5$ years

Annual Rate of Return (r) $= 12\%$ per annum $= 0.12$

Compounding Frequency: Annually

To Find:

The Present Value (PV) of the amount.

Formula Used:

The formula for the Present Value (PV) of a single future amount, compounded annually, is:

Where:

• $PV$ = Present Value
• $FV$ = Future Value
• $r$ = Annual Interest Rate (as a decimal)
• $n$ = Number of compounding periods (years in this case)

Solution:

Substitute the given values into formula (1):

$\text{PV} = \frac{1,50,000}{(1 + 0.12)^5}$

$\text{PV} = \frac{1,50,000}{(1.12)^5}$

First, calculate $(1.12)^5$:

$(1.12)^5 \approx 1.7623416832$

Now, perform the division:

$\text{PV} = \frac{1,50,000}{1.7623416832}$

$\text{PV} \approx \textsf{₹} 85,119.576$

Rounding to two decimal places for currency:

$\text{PV} \approx \textsf{₹} 85,119.58$

Therefore, the present value of $\textsf{₹} 1,50,000$ received in 5 years, at a $12\%$ annual compound rate, is approximately $\textsf{₹} 85,119.58$.

Problem Statement:

Calculate the accumulated value of an annuity where $\textsf{₹} 5,000$ is invested at the end of each year for 3 years, with an interest rate of $7\%$ per annum compounded annually.

Given:

Periodic Payment (Pmt) $= \textsf{₹} 5,000$ (invested at the end of each year, indicating a Regular Annuity)

Number of Periods (n) $= 3$ years

Annual Interest Rate (r) $= 7\%$ per annum $= 0.07$

Compounding Frequency: Annually

To Find:

The Accumulated Value (Future Value) of this annuity at the end of the 3rd year.

Formula Used:

The formula for the Future Value of a Regular Annuity ($FV_{Reg}$) is:

Where:

• $FV_{Reg}$ = Future Value of the Regular Annuity
• $Pmt$ = Payment amount per period
• $r$ = Interest rate per period (as a decimal)
• $n$ = Total number of periods

Solution:

Substitute the given values into formula (1):

$\text{FV}_{Reg} = 5000 \times \left[ \frac{(1+0.07)^3 - 1}{0.07} \right]$

$\text{FV}_{Reg} = 5000 \times \left[ \frac{(1.07)^3 - 1}{0.07} \right]$

First, calculate $(1.07)^3$:

$1.07 \times 1.07 = 1.1449$

$1.1449 \times 1.07 = 1.225043$

So, $(1.07)^3 = 1.225043$

Now, substitute this value back into the formula:

$\text{FV}_{Reg} = 5000 \times \left[ \frac{1.225043 - 1}{0.07} \right]$

$\text{FV}_{Reg} = 5000 \times \left[ \frac{0.225043}{0.07} \right]$

$\text{FV}_{Reg} = 5000 \times 3.2149$ (approximately)

$\text{FV}_{Reg} = \textsf{₹} 16074.5$ (approximately)

Let's use a more precise value for the fraction:

$\frac{0.225043}{0.07} \approx 3.2149$

$\text{FV}_{Reg} = 5000 \times 3.2149$

$\text{FV}_{Reg} = \textsf{₹} 16074.50$

Therefore, the accumulated value of this annuity at the end of the 3rd year is approximately $\textsf{₹} 16,074.50$.

Problem Statement:

Calculate the Net Present Value (NPV) of a project requiring an initial investment of $\textsf{₹} 50,000$ and generating specified cash flows over 3 years, with a discount rate of $10\%$ per annum. Based on the NPV, determine whether the company should accept or reject the project.

Given:

Initial Investment ($CF_0$) $= \textsf{₹} 50,000$ (This is an outflow, so it will be treated as negative)

Cash Flow at end of Year 1 ($CF_1$) $= \textsf{₹} 20,000$

Cash Flow at end of Year 2 ($CF_2$) $= \textsf{₹} 25,000$

Cash Flow at end of Year 3 ($CF_3$) $= \textsf{₹} 30,000$

Discount Rate (r) $= 10\%$ per annum $= 0.10$

To Find:

1. The Net Present Value (NPV) of the project.

2. Whether to Accept or Reject the project based on NPV.

Formula Used:

The Net Present Value (NPV) is calculated as the sum of the present values of all future cash flows minus the initial investment. The formula for NPV is:

Where:

• $CF_0$ = Initial Investment (at time 0, usually negative)
• $CF_t$ = Cash flow at the end of year $t$
• $r$ = Discount rate
• $n$ = Number of years

Solution:

Step 1: Calculate the Present Value (PV) of each cash inflow.

PV of Cash Flow at Year 1 ($CF_1$):

$PV_1 = \frac{\textsf{₹} 20,000}{(1+0.10)^1}$

$PV_1 = \frac{\textsf{₹} 20,000}{1.10}$

PV of Cash Flow at Year 2 ($CF_2$):

$PV_2 = \frac{\textsf{₹} 25,000}{(1+0.10)^2}$

$PV_2 = \frac{\textsf{₹} 25,000}{(1.10)^2}$

$PV_2 = \frac{\textsf{₹} 25,000}{1.21}$

PV of Cash Flow at Year 3 ($CF_3$):

$PV_3 = \frac{\textsf{₹} 30,000}{(1+0.10)^3}$

$PV_3 = \frac{\textsf{₹} 30,000}{(1.10)^3}$

$PV_3 = \frac{\textsf{₹} 30,000}{1.331}$

Step 2: Calculate the Net Present Value (NPV).

$\text{NPV} = \text{Initial Investment} + \text{PV of CF}_1 + \text{PV of CF}_2 + \text{PV of CF}_3$

$\text{NPV} = -\textsf{₹} 50,000 + \textsf{₹} 18,181.82 + \textsf{₹} 20,661.16 + \textsf{₹} 22,539.44$

$\text{NPV} = -\textsf{₹} 50,000 + \textsf{₹} (18,181.82 + 20,661.16 + 22,539.44)$

$\text{NPV} = -\textsf{₹} 50,000 + \textsf{₹} 61,382.42$

$\text{NPV} = \textsf{₹} 11,382.42$

Decision Rule for NPV:

• If NPV > 0: Accept the project. The project is expected to add value to the firm.
• If NPV < 0: Reject the project. The project is expected to diminish firm value.
• If NPV = 0: Be indifferent. The project is expected to just cover the cost of capital.

Conclusion:

The calculated Net Present Value (NPV) of the project is $\textsf{₹} 11,382.42$.

Since the NPV is positive ($\textsf{₹} 11,382.42 > 0$), the project is expected to generate a return higher than the required rate of $10\%$. Therefore, the company should accept the project.

Problem Statement:

Calculate the effective annual rate of interest for two given nominal rates with different compounding frequencies and determine which rate is more beneficial for an investor.

Given:

i) Nominal Rate ($R_1$) $= 6\%$ p.a. compounded monthly

ii) Nominal Rate ($R_2$) $= 7\%$ p.a. compounded half-yearly

To Find:

1. The Effective Annual Rate (EAR) for both nominal rates.

2. Which rate is more beneficial for an investor.

Formula Used:

The formula for the Effective Annual Rate (EAR) is:

Where:

• $R$ = Nominal annual interest rate (as a decimal)
• $m$ = Number of times interest is compounded per year

Solution:

Part i) Calculate EAR for $6\%$ p.a. compounded monthly

Here, $R = 0.06$ and $m = 12$ (monthly compounding).

Using formula (1):

$\text{EAR}_1 = \left(1 + \frac{0.06}{12}\right)^{12} - 1$

$\text{EAR}_1 = \left(1 + 0.005\right)^{12} - 1$

$\text{EAR}_1 = (1.005)^{12} - 1$

Calculating $(1.005)^{12} \approx 1.06167781186$

$\text{EAR}_1 = 1.06167781186 - 1$

$\text{EAR}_1 = 0.06167781186$

Converting to percentage:

Part ii) Calculate EAR for $7\%$ p.a. compounded half-yearly

Here, $R = 0.07$ and $m = 2$ (half-yearly compounding).

Using formula (1):

$\text{EAR}_2 = \left(1 + \frac{0.07}{2}\right)^2 - 1$

$\text{EAR}_2 = \left(1 + 0.035\right)^2 - 1$

$\text{EAR}_2 = (1.035)^2 - 1$

Calculating $(1.035)^2 = 1.035 \times 1.035 = 1.071225$

$\text{EAR}_2 = 1.071225 - 1$

$\text{EAR}_2 = 0.071225$

Converting to percentage:

Comparison and Conclusion:

Comparing the two effective annual rates:

$\text{EAR}_1 \approx 6.1678\%$

$\text{EAR}_2 = 7.1225\%$

Since $7.1225\% > 6.1678\%$, the effective annual rate of $7\%$ p.a. compounded half-yearly is higher than the effective annual rate of $6\%$ p.a. compounded monthly.

For an investor, a higher effective annual rate means that their investment will grow more over a year. Therefore, the rate of $7\%$ p.a. compounded half-yearly is more beneficial for an investor.

Problem Statement:

Calculate the Present Value of an annuity of $\textsf{₹} 3,000$ per year for 3 years, with payments made at the end of each year, if the interest rate is $9\%$ per annum compounded annually.

Given:

Periodic Payment (Pmt) $= \textsf{₹} 3,000$ (payments made at the end of each year, indicating a Regular Annuity)

Number of Periods (n) $= 3$ years

Annual Interest Rate (r) $= 9\%$ per annum $= 0.09$

Compounding Frequency: Annually

To Find:

The Present Value (PV) of this annuity.

Formula Used:

The formula for the Present Value of a Regular Annuity ($PVA_{Reg}$) is:

Where:

• $PVA_{Reg}$ = Present Value of the Regular Annuity
• $Pmt$ = Payment amount per period
• $r$ = Interest rate per period (as a decimal)
• $n$ = Total number of periods

Solution:

Substitute the given values into formula (1):

$\text{PVA}_{Reg} = 3000 \times \left[ \frac{1 - (1+0.09)^{-3}}{0.09} \right]$

$\text{PVA}_{Reg} = 3000 \times \left[ \frac{1 - (1.09)^{-3}}{0.09} \right]$

First, calculate $(1.09)^{-3}$:

$(1.09)^{-3} = \frac{1}{(1.09)^3}$

$(1.09)^3 = 1.09 \times 1.09 \times 1.09 = 1.1881 \times 1.09 = 1.295029$

So, $(1.09)^{-3} = \frac{1}{1.295029} \approx 0.772183498$

Now, substitute this value back into the formula:

$\text{PVA}_{Reg} = 3000 \times \left[ \frac{1 - 0.772183498}{0.09} \right]$

$\text{PVA}_{Reg} = 3000 \times \left[ \frac{0.227816502}{0.09} \right]$

$\text{PVA}_{Reg} = 3000 \times 2.531294466$

$\text{PVA}_{Reg} = \textsf{₹} 7593.883398$

Rounding to two decimal places for currency:

$\text{PVA}_{Reg} \approx \textsf{₹} 7,593.88$

Therefore, the present value of this annuity is approximately $\textsf{₹} 7,593.88$.

Problem Statement:

Calculate the total income tax payable by an individual with a taxable income of $\textsf{₹} 8,50,000$ based on the provided progressive tax slab rates (ignoring cess and surcharge).

Given:

Taxable Income $= \textsf{₹} 8,50,000$

Income Tax Rates:

To Find:

The Total Income Tax Payable.

Solution:

We need to calculate the tax for each slab that the individual's income falls into. The taxable income is $\textsf{₹} 8,50,000$.

Step 1: Tax on the first slab (Up to $\textsf{₹} 2,50,000$)

Taxable amount in this slab $= \textsf{₹} 2,50,000$

$\text{Tax}_1 = \textsf{₹} 0$

Step 2: Tax on the second slab ($\textsf{₹} 2,50,001$ to $\textsf{₹} 5,00,000$)

This slab covers income from $\textsf{₹} 2,50,001$ up to $\textsf{₹} 5,00,000$.

Amount taxable in this slab $= \textsf{₹} 5,00,000 - \textsf{₹} 2,50,000 = \textsf{₹} 2,50,000$

$\text{Tax}_2 = \textsf{₹} 2,50,000 \times \frac{5}{100}$

$\text{Tax}_2 = \textsf{₹} 12,500$

Step 3: Tax on the third slab ($\textsf{₹} 5,00,001$ to $\textsf{₹} 10,00,000$)

The individual's total income is $\textsf{₹} 8,50,000$. The income falling into this slab is the portion above $\textsf{₹} 5,00,000$ up to $\textsf{₹} 8,50,000$.

Amount taxable in this slab $= \textsf{₹} 8,50,000 - \textsf{₹} 5,00,000 = \textsf{₹} 3,50,000$

$\text{Tax}_3 = \textsf{₹} 3,50,000 \times \frac{20}{100}$

$\text{Tax}_3 = \textsf{₹} 70,000$

Step 4: Tax on the fourth slab (Above $\textsf{₹} 10,00,000$)

The individual's income of $\textsf{₹} 8,50,000$ does not exceed $\textsf{₹} 10,00,000$.

Taxable amount in this slab $= \textsf{₹} 0$

$\text{Tax}_4 = \textsf{₹} 0$

Step 5: Calculate the Total Income Tax Payable

Total Tax Payable = $\text{Tax}_1 + \text{Tax}_2 + \text{Tax}_3 + \text{Tax}_4$

Total Tax Payable = $\textsf{₹} 0 + \textsf{₹} 12,500 + \textsf{₹} 70,000 + \textsf{₹} 0$

Total Tax Payable = $\textsf{₹} 82,500$

Conclusion:

The total income tax payable by the individual is $\textsf{₹} 82,500$.

Problem Statement:

Calculate the net GST payable by a shopkeeper who buys goods for $\textsf{₹} 1,00,000$ and sells them for $\textsf{₹} 1,50,000$, with an $18\%$ GST rate applicable on both transactions.

Given:

Purchase Price of Goods $= \textsf{₹} 1,00,000$

Selling Price of Goods $= \textsf{₹} 1,50,000$

GST Rate $= 18\%$

To Find:

The Net GST Payable by the shopkeeper to the government.

Concept Used:

Input Tax Credit (ITC): This is the credit a taxpayer receives for the GST paid on purchases. When a business pays GST on inputs (goods or services bought), they can use this credit to reduce the GST they have to pay on outputs (goods or services sold) to the government.

Net GST Payable = Output GST - Input GST Credit

Solution:

Step 1: Calculate the Input GST (GST paid on purchases).

Input GST = Purchase Price $\times$ GST Rate

Input GST = $\textsf{₹} 1,00,000 \times 18\%$

Input GST = $\textsf{₹} 1,00,000 \times \frac{18}{100}$

This $\textsf{₹} 18,000$ is the Input Tax Credit available to the shopkeeper.

Step 2: Calculate the Output GST (GST collected on sales).

Output GST = Selling Price $\times$ GST Rate

Output GST = $\textsf{₹} 1,50,000 \times 18\%$

Output GST = $\textsf{₹} 1,50,000 \times \frac{18}{100}$

Step 3: Calculate the Net GST Payable to the government.

Net GST Payable = Output GST - Input GST (Input Tax Credit)

Net GST Payable = $\textsf{₹} 27,000 - \textsf{₹} 18,000$

Net GST Payable = $\textsf{₹} 9,000$

Conclusion:

The net GST payable by the shopkeeper to the government after claiming Input Tax Credit is $\textsf{₹} 9,000$.

Problem Statement:

Calculate the total electricity bill amount based on tiered energy charges, fixed charge, a surcharge on energy charge, and other charges, for a total consumption of 350 units.

Given:

• Fixed Charge: $\textsf{₹} 120$
• Energy Charge Slabs:
  • First 100 units at $\textsf{₹} 4$ per unit
  • Next 200 units at $\textsf{₹} 6$ per unit
  • Above 300 units at $\textsf{₹} 8$ per unit
• Surcharge: $10\%$ on total energy charge
• Other Charges: $\textsf{₹} 50$ (Service Charge, etc.)
• Total Units Consumed: 350 units

To Find:

The Total Electricity Bill Amount.

Solution:

Step 1: Calculate the Energy Charge based on units consumed in each slab.

Total units consumed are 350 units.

a) First 100 units:

Charge for first 100 units = $100 \text{ units} \times \textsf{₹} 4/\text{unit}$

Units remaining to be charged = $350 - 100 = 250$ units.

b) Next 200 units (from the remaining 250 units):

Charge for next 200 units = $200 \text{ units} \times \textsf{₹} 6/\text{unit}$

Units remaining to be charged = $250 - 200 = 50$ units.

c) Units above 300 (the remaining 50 units):

These 50 units fall into the "Above 300 units" slab (since $100 + 200 = 300$ units have already been accounted for).

Charge for remaining 50 units = $50 \text{ units} \times \textsf{₹} 8/\text{unit}$

d) Total Energy Charge:

Total Energy Charge = $\text{Charge}_1 + \text{Charge}_2 + \text{Charge}_3$

Total Energy Charge = $\textsf{₹} 400 + \textsf{₹} 1,200 + \textsf{₹} 400$

Step 2: Calculate the Surcharge.

Surcharge is $10\%$ on the total energy charge.

Surcharge Amount = $10\% \text{ of } \textsf{₹} 2,000$

Surcharge Amount = $\frac{10}{100} \times \textsf{₹} 2,000$

Surcharge Amount = $0.10 \times \textsf{₹} 2,000$

Step 3: Calculate the Total Electricity Bill.

Total Electricity Bill = Fixed Charge + Total Energy Charge + Surcharge Amount + Other Charges

Total Electricity Bill = $\textsf{₹} 120 + \textsf{₹} 2,000 + \textsf{₹} 200 + \textsf{₹} 50$

Total Electricity Bill = $\textsf{₹} 2,370$

Conclusion:

The total electricity bill amount is $\textsf{₹} 2,370$.

Problem Statement:

Compare two investment options: Option A with simple interest and Option B with compound interest, for an initial investment of $\textsf{₹} 40,000$ over 5 years. Determine which option yields a higher return and by what amount.

Given:

Initial Investment (Principal, P) $= \textsf{₹} 40,000$

Time (T or n) $= 5$ years

Option A: Rate (R) $= 9\%$ p.a. Simple Interest

Option B: Rate (r) $= 8\%$ p.a. Compounded Annually

To Find:

1. Accumulated Amount for Option A ($A_A$).

2. Accumulated Amount for Option B ($A_B$).

3. Which option yields a higher return.

4. The difference in returns.

Formulae Used:

1. Simple Interest (S.I.):

2. Amount (A) with Simple Interest:

3. Amount (A) with Compound Interest:

Solution:

Calculation for Option A (Simple Interest):

Using formula (1) to find Simple Interest:

$\text{S.I.} = \frac{40000 \times 9 \times 5}{100}$

$\text{S.I.} = 400 \times 9 \times 5$

$\text{S.I.} = 400 \times 45$

$\text{S.I.} = \textsf{₹} 18,000$

Using formula (2) to find the Accumulated Amount ($A_A$):

$A_A = P + \text{S.I.}$

$A_A = \textsf{₹} 40,000 + \textsf{₹} 18,000$

Calculation for Option B (Compound Interest):

Using formula (3) to find the Accumulated Amount ($A_B$):

$A_B = 40000 \left(1 + \frac{8}{100}\right)^5$

$A_B = 40000 (1 + 0.08)^5$

$A_B = 40000 (1.08)^5$

Calculate $(1.08)^5$:

$(1.08)^5 \approx 1.4693280768$

Now, substitute this value back into the amount calculation:

$A_B = 40000 \times 1.4693280768$

$A_B = \textsf{₹} 58,773.123072$

Rounding to two decimal places:

Comparison of Returns:

Amount from Option A ($A_A$) = $\textsf{₹} 58,000$

Amount from Option B ($A_B$) = $\textsf{₹} 58,773.12$

Since $\textsf{₹} 58,773.12 > \textsf{₹} 58,000$, Option B yields a higher return.

Difference in Returns:

Difference = $A_B - A_A$

Difference = $\textsf{₹} 58,773.12 - \textsf{₹} 58,000$

Difference = $\textsf{₹} 773.12$

Conclusion:

Option B (8% p.a. compounded annually) yields a higher return at the end of 5 years. The difference in interest earned, favoring Option B, is $\textsf{₹} 773.12$.

Problem Statement:

Calculate the Present Value of an investment that promises three future cash flows: $\textsf{₹} 10,000$ at the end of year 1, $\textsf{₹} 15,000$ at the end of year 2, and $\textsf{₹} 20,000$ at the end of year 3. The applicable discount rate is $11\%$ per annum compounded annually.

Given:

• Cash Flow at end of Year 1 ($CF_1$) $= \textsf{₹} 10,000$
• Cash Flow at end of Year 2 ($CF_2$) $= \textsf{₹} 15,000$
• Cash Flow at end of Year 3 ($CF_3$) $= \textsf{₹} 20,000$
• Discount Rate (r) $= 11\%$ per annum $= 0.11$
• Compounding Frequency: Annually

To Find:

The Total Present Value (PV) of the investment.

Formula Used:

The formula for the Present Value (PV) of a single future cash flow is:

Where:

• $PV$ = Present Value
• $FV$ = Future Value (the cash flow at time $n$)
• $r$ = Discount rate per period (as a decimal)
• $n$ = Number of periods (years in this case)

To find the total present value of multiple uneven cash flows, we sum the present values of each individual cash flow.

Solution:

Step 1: Calculate the Present Value (PV) of each individual cash flow.

a) PV of Cash Flow at Year 1 ($CF_1 = \textsf{₹} 10,000$):

$PV_1 = \frac{\textsf{₹} 10,000}{(1+0.11)^1}$

$PV_1 = \frac{\textsf{₹} 10,000}{1.11}$

b) PV of Cash Flow at Year 2 ($CF_2 = \textsf{₹} 15,000$):

$PV_2 = \frac{\textsf{₹} 15,000}{(1+0.11)^2}$

$PV_2 = \frac{\textsf{₹} 15,000}{(1.11)^2}$

$PV_2 = \frac{\textsf{₹} 15,000}{1.2321}$

c) PV of Cash Flow at Year 3 ($CF_3 = \textsf{₹} 20,000$):

$PV_3 = \frac{\textsf{₹} 20,000}{(1+0.11)^3}$

$PV_3 = \frac{\textsf{₹} 20,000}{(1.11)^3}$

$PV_3 = \frac{\textsf{₹} 20,000}{1.367631}$

Step 2: Sum the Present Values to get the Total Present Value.

Total PV = $PV_1 + PV_2 + PV_3$

Total PV = $\textsf{₹} 9,009.01 + \textsf{₹} 12,174.34 + \textsf{₹} 14,624.50$

Total PV = $\textsf{₹} 35,807.85$

Conclusion:

The Present Value of the investment is approximately $\textsf{₹} 35,807.85$.

Problem Statement:

Calculate the total amount accumulated at the end of 3 years in a recurring deposit scheme where $\textsf{₹} 2,000$ is deposited at the beginning of each year, and the interest rate is $6\%$ per annum compounded annually.

Given:

Periodic Payment (Pmt) $= \textsf{₹} 2,000$ (deposited at the beginning of each year, indicating an Annuity Due)

Number of Periods (n) $= 3$ years

Annual Interest Rate (r) $= 6\%$ per annum $= 0.06$

Compounding Frequency: Annually

To Find:

The Total Amount Accumulated (Future Value) of this annuity at the end of 3 years.

Formula Used:

The formula for the Future Value of an Annuity Due ($FV_{Due}$) is:

Where:

• $FV_{Due}$ = Future Value of the Annuity Due
• $Pmt$ = Payment amount per period
• $r$ = Interest rate per period (as a decimal)
• $n$ = Total number of periods

Solution:

Substitute the given values into formula (1):

$\text{FV}_{Due} = 2000 \times \left[ \frac{(1+0.06)^3 - 1}{0.06} \right] \times (1+0.06)$

$\text{FV}_{Due} = 2000 \times \left[ \frac{(1.06)^3 - 1}{0.06} \right] \times (1.06)$

First, calculate $(1.06)^3$:

$(1.06)^2 = 1.06 \times 1.06 = 1.1236$

$(1.06)^3 = 1.1236 \times 1.06 = 1.191016$

Now, substitute this value back into the formula:

$\text{FV}_{Due} = 2000 \times \left[ \frac{1.191016 - 1}{0.06} \right] \times 1.06$

$\text{FV}_{Due} = 2000 \times \left[ \frac{0.191016}{0.06} \right] \times 1.06$

$\text{FV}_{Due} = 2000 \times 3.1836 \times 1.06$

$\text{FV}_{Due} = 2000 \times 3.374616$

$\text{FV}_{Due} = \textsf{₹} 6,749.232$

Rounding to two decimal places for currency:

$\text{FV}_{Due} \approx \textsf{₹} 6,749.23$

Conclusion:

The total amount accumulated at the end of 3 years is approximately $\textsf{₹} 6,749.23$.

Problem Statement:

Compare the effective annual rates of interest for two nominal rates: $10\%$ per annum compounded semi-annually and $9.8\%$ per annum compounded monthly. Determine which option is better for borrowing money.

Given:

Option 1: Nominal Rate ($R_1$) $= 10\%$ p.a. $= 0.10$, Compounded Semi-annually ($m_1 = 2$)

Option 2: Nominal Rate ($R_2$) $= 9.8\%$ p.a. $= 0.098$, Compounded Monthly ($m_2 = 12$)

To Find:

1. The Effective Annual Rate (EAR) for both options.

2. Which option is better for borrowing money.

Formula Used:

The formula for the Effective Annual Rate (EAR) is:

Where:

• $R$ = Nominal annual interest rate (as a decimal)
• $m$ = Number of times interest is compounded per year

Solution:

Part 1: Calculate EAR for Option 1 ($10\%$ p.a. compounded semi-annually)

Here, $R_1 = 0.10$ and $m_1 = 2$.

Using formula (1):

$\text{EAR}_1 = \left(1 + \frac{0.10}{2}\right)^2 - 1$

$\text{EAR}_1 = \left(1 + 0.05\right)^2 - 1$

$\text{EAR}_1 = (1.05)^2 - 1$

$\text{EAR}_1 = 1.1025 - 1$

$\text{EAR}_1 = 0.1025$

Converting to percentage:

Part 2: Calculate EAR for Option 2 ($9.8\%$ p.a. compounded monthly)

Here, $R_2 = 0.098$ and $m_2 = 12$.

Using formula (1):

$\text{EAR}_2 = \left(1 + \frac{0.098}{12}\right)^{12} - 1$

$\text{EAR}_2 = \left(1 + 0.0081666667\right)^{12} - 1$

$\text{EAR}_2 = (1.0081666667)^{12} - 1$

Calculating $(1.0081666667)^{12} \approx 1.102568$

$\text{EAR}_2 = 1.102568 - 1$

$\text{EAR}_2 = 0.102568$

Converting to percentage:

Comparison and Conclusion:

Comparing the two effective annual rates:

$\text{EAR}_1 = 10.25\%$

$\text{EAR}_2 \approx 10.2568\%$

For borrowing money, an individual wants to pay the least amount of interest. This means the option with the lower effective annual rate is more beneficial for a borrower.

In this case, $10.25\% < 10.2568\%$.

Therefore, Option 1 ($10\%$ p.a. compounded semi-annually) is better for borrowing money as it results in a slightly lower effective annual interest rate.

Problem Statement:

Calculate the Present Value of cash flows for Project B using a $8\%$ per annum discount rate. Then, compare its Net Present Value (NPV) with Project A (from Question 5, which had an NPV of $\textsf{₹} 11,382.42$ with a $10\%$ discount rate) and recommend which project should be chosen.

Given:

For Project B (using cash flows from Question 5):

• Initial Investment ($CF_0$) $= \textsf{₹} 50,000$ (outflow)
• Cash Flow at end of Year 1 ($CF_1$) $= \textsf{₹} 20,000$
• Cash Flow at end of Year 2 ($CF_2$) $= \textsf{₹} 25,000$
• Cash Flow at end of Year 3 ($CF_3$) $= \textsf{₹} 30,000$
• Discount Rate (r) $= 8\%$ per annum $= 0.08$

From Question 5:

• NPV of Project A $= \textsf{₹} 11,382.42$ (calculated with a $10\%$ discount rate for the same cash flows)

To Find:

1. The Net Present Value (NPV) of Project B.

2. Which project (Project A or Project B) should be chosen.

Formula Used:

The Net Present Value (NPV) is calculated as the sum of the present values of all future cash flows minus the initial investment. The formula for NPV is:

For this specific project:

Where:

• $CF_0$ = Initial Investment (negative as it's an outflow)
• $CF_t$ = Cash flow at the end of year $t$
• $r$ = Discount rate

Solution:

Step 1: Calculate the Present Value (PV) of each cash inflow for Project B.

a) PV of Cash Flow at Year 1 ($CF_1 = \textsf{₹} 20,000$):

$PV_1 = \frac{\textsf{₹} 20,000}{(1+0.08)^1}$

$PV_1 = \frac{\textsf{₹} 20,000}{1.08}$

b) PV of Cash Flow at Year 2 ($CF_2 = \textsf{₹} 25,000$):

$PV_2 = \frac{\textsf{₹} 25,000}{(1+0.08)^2}$

$PV_2 = \frac{\textsf{₹} 25,000}{(1.08)^2}$

$PV_2 = \frac{\textsf{₹} 25,000}{1.1664}$

c) PV of Cash Flow at Year 3 ($CF_3 = \textsf{₹} 30,000$):

$PV_3 = \frac{\textsf{₹} 30,000}{(1+0.08)^3}$

$PV_3 = \frac{\textsf{₹} 30,000}{(1.08)^3}$

$PV_3 = \frac{\textsf{₹} 30,000}{1.259712}$

Step 2: Calculate the Net Present Value (NPV) of Project B.

$\text{NPV}_B = - \textsf{₹} 50,000 + \textsf{₹} 18,518.52 + \textsf{₹} 21,433.47 + \textsf{₹} 23,814.73$

$\text{NPV}_B = - \textsf{₹} 50,000 + \textsf{₹} 63,766.72$

Comparison and Recommendation:

NPV of Project A (from Question 5) $= \textsf{₹} 11,382.42$

NPV of Project B $= \textsf{₹} 13,766.72$

Both projects have a positive NPV, which generally means both are acceptable. However, when choosing between mutually exclusive projects (assuming a company can only undertake one), the project with the higher positive NPV is preferred as it is expected to add more value to the firm.

Comparing the two NPVs:

$\textsf{₹} 13,766.72$ (Project B) $> \textsf{₹} 11,382.42$ (Project A)

Therefore, Project B yields a higher Net Present Value.

Conclusion:

The Net Present Value (NPV) of Project B is $\textsf{₹} 13,766.72$. Since Project B has a higher positive NPV compared to Project A ($\textsf{₹} 11,382.42$), the company should choose Project B.

Problem Statement:

Calculate the amount of each monthly installment and the total amount paid by a customer for a television set, given its price, GST rate, a down payment, and the number of interest-free installments.

Given:

Television Set Price (before GST) $= \textsf{₹} 30,000$

GST Rate $= 18\%$

Down Payment $= \textsf{₹} 10,000$

Number of Installments $= 3$ equal monthly installments

Interest on Installments $= 0\%$

To Find:

1. The Amount of each Installment.

2. The Total Amount Paid by the customer for the television.

Solution:

Step 1: Calculate the Total Price of the Television including GST.

First, calculate the GST amount:

$\text{GST Amount} = \text{Television Set Price} \times \text{GST Rate}$

$\text{GST Amount} = \textsf{₹} 30,000 \times 18\%$

$\text{GST Amount} = \textsf{₹} 30,000 \times \frac{18}{100}$

$\text{GST Amount} = \textsf{₹} 5,400$

Now, calculate the Total Price:

$\text{Total Price} = \text{Television Set Price} + \text{GST Amount}$

$\text{Total Price} = \textsf{₹} 30,000 + \textsf{₹} 5,400$

Step 2: Calculate the Remaining Amount to be Paid after the Down Payment.

$\text{Remaining Amount} = \text{Total Price} - \text{Down Payment}$

$\text{Remaining Amount} = \textsf{₹} 35,400 - \textsf{₹} 10,000$

Step 3: Calculate the Amount of each Installment.

Since the installments are interest-free and equal, the remaining amount is simply divided by the number of installments.

$\text{Installment Amount} = \frac{\text{Remaining Amount}}{\text{Number of Installments}}$

$\text{Installment Amount} = \frac{\textsf{₹} 25,400}{3}$

$\text{Installment Amount} \approx \textsf{₹} 8,466.666...$

Step 4: Calculate the Total Amount Paid by the Customer.

The total amount paid is the sum of the down payment and the total of all installments.

$\text{Total Amount Paid} = \text{Down Payment} + (\text{Installment Amount} \times \text{Number of Installments})$

$\text{Total Amount Paid} = \textsf{₹} 10,000 + (\textsf{₹} 8,466.67 \times 3)$

$\text{Total Amount Paid} = \textsf{₹} 10,000 + \textsf{₹} 25,400.01$ (Due to rounding in installment calculation, this is slightly off)

Alternatively, since there's $0\%$ interest, the total amount paid should simply be the Total Price calculated in Step 1.

$\text{Total Amount Paid} = \text{Total Price (including GST)}$

$\text{Total Amount Paid} = \textsf{₹} 35,400$

Conclusion:

The amount of each monthly installment is approximately $\textsf{₹} 8,466.67$.

The total amount paid by the customer for the television is $\textsf{₹} 35,400$.

Problem Statement:

Calculate the total income tax payable by an individual with a total income of $\textsf{₹} 12,00,000$, using the provided tax slab rates and including a $4\%$ health and education cess on the calculated tax amount.

Given:

Total Income $= \textsf{₹} 12,00,000$

Income Tax Rates:

Health and Education Cess $= 4\%$ on the tax amount.

To Find:

The Total Income Tax Payable by the individual (including cess).

Solution:

We will calculate the tax liability for each applicable slab based on the individual's total income of $\textsf{₹} 12,00,000$.

Step 1: Calculate Tax for the First Slab (Nil Rate).

Income up to $\textsf{₹} 2,50,000$ is exempt from tax.

Taxable amount in this slab $= \textsf{₹} 2,50,000$

Step 2: Calculate Tax for the Second Slab ($5\%$).

This slab covers income from $\textsf{₹} 2,50,001$ to $\textsf{₹} 5,00,000$.

Amount taxable in this slab $= \textsf{₹} 5,00,000 - \textsf{₹} 2,50,000 = \textsf{₹} 2,50,000$

$\text{Tax}_2 = \textsf{₹} 2,50,000 \times \frac{5}{100}$

Step 3: Calculate Tax for the Third Slab ($20\%$).

This slab covers income from $\textsf{₹} 5,00,001$ to $\textsf{₹} 10,00,000$.

Amount taxable in this slab $= \textsf{₹} 10,00,000 - \textsf{₹} 5,00,000 = \textsf{₹} 5,00,000$

$\text{Tax}_3 = \textsf{₹} 5,00,000 \times \frac{20}{100}$

Step 4: Calculate Tax for the Fourth Slab ($30\%$).

This slab covers income above $\textsf{₹} 10,00,000$. The individual's total income is $\textsf{₹} 12,00,000$.

Amount taxable in this slab $= \textsf{₹} 12,00,000 - \textsf{₹} 10,00,000 = \textsf{₹} 2,00,000$

$\text{Tax}_4 = \textsf{₹} 2,00,000 \times \frac{30}{100}$

Step 5: Calculate the Total Income Tax (before Cess).

Total Income Tax = $\text{Tax}_1 + \text{Tax}_2 + \text{Tax}_3 + \text{Tax}_4$

Total Income Tax = $\textsf{₹} 0 + \textsf{₹} 12,500 + \textsf{₹} 1,00,000 + \textsf{₹} 60,000$

Step 6: Calculate Health and Education Cess.

Cess is $4\%$ on the Total Income Tax calculated in Step 5.

$\text{Cess Amount} = \textsf{₹} 1,72,500 \times 4\%$

$\text{Cess Amount} = \textsf{₹} 1,72,500 \times \frac{4}{100}$

Step 7: Calculate the Total Income Tax Payable (including Cess).

Total Tax Payable = Total Income Tax + Cess Amount

Total Tax Payable = $\textsf{₹} 1,72,500 + \textsf{₹} 6,900$

Total Tax Payable = $\textsf{₹} 1,79,400$

Conclusion:

The total income tax payable by the individual, including health and education cess, is $\textsf{₹} 1,79,400$.

Problem Statement:

Calculate the total water bill for a month, considering fixed charges, tiered consumption charges, a sanitation surcharge on consumption, and meter rent, for a total consumption of 22 kilolitres.

Given:

• Fixed Charge: $\textsf{₹} 80$ per month
• Consumption Charge Slabs:
  • First 5 kilolitres (kL) at $\textsf{₹} 10$ per kL
  • Next 10 kL at $\textsf{₹} 18$ per kL
  • Above 15 kL at $\textsf{₹} 25$ per kL
• Sanitation Surcharge: $15\%$ of the consumption charge
• Meter Rent: $\textsf{₹} 30$ per month
• Total Consumption in a month: 22 kilolitres

To Find:

The Total Water Bill for the month.

Solution:

Step 1: Calculate the Consumption Charge based on units consumed in each slab.

Total units consumed are 22 kilolitres.

a) First 5 kilolitres:

Charge for first 5 kL = $5 \text{ kL} \times \textsf{₹} 10/\text{kL}$

Remaining units to be charged = $22 - 5 = 17$ kL.

b) Next 10 kilolitres (from the remaining 17 kL):

Charge for next 10 kL = $10 \text{ kL} \times \textsf{₹} 18/\text{kL}$

Units remaining to be charged = $17 - 10 = 7$ kL.

c) Units above 15 kL (the remaining 7 kL):

These 7 kL fall into the "Above 15 kL" slab (since $5 + 10 = 15$ kL have already been accounted for).

Charge for remaining 7 kL = $7 \text{ kL} \times \textsf{₹} 25/\text{kL}$

d) Total Consumption Charge:

Total Consumption Charge = $\text{Charge}_1 + \text{Charge}_2 + \text{Charge}_3$

Total Consumption Charge = $\textsf{₹} 50 + \textsf{₹} 180 + \textsf{₹} 175$

Step 2: Calculate the Sanitation Surcharge.

Surcharge is $15\%$ of the total consumption charge.

Surcharge Amount = $15\% \text{ of } \textsf{₹} 405$

Surcharge Amount = $\frac{15}{100} \times \textsf{₹} 405$

Surcharge Amount = $0.15 \times \textsf{₹} 405$

Step 3: Calculate the Total Water Bill.

Total Water Bill = Fixed Charge + Total Consumption Charge + Sanitation Surcharge + Meter Rent

Total Water Bill = $\textsf{₹} 80 + \textsf{₹} 405 + \textsf{₹} 60.75 + \textsf{₹} 30$

Total Water Bill = $\textsf{₹} 575.75$

Conclusion:

The total water bill for the month is $\textsf{₹} 575.75$.

Problem Statement:

Calculate the Present Value of an investment that pays $\textsf{₹} 5,000$ at the beginning of each year for 3 years, if the discount rate is $7\%$ per annum compounded annually.

Given:

Periodic Payment (Pmt) $= \textsf{₹} 5,000$ (payments made at the beginning of each year, indicating an Annuity Due)

Number of Periods (n) $= 3$ years

Annual Discount Rate (r) $= 7\%$ per annum $= 0.07$

Compounding Frequency: Annually

To Find:

The Present Value (PV) of this Annuity Due.

Formula Used:

The formula for the Present Value of an Annuity Due ($PVA_{Due}$) is:

Alternatively, since an annuity due's first payment occurs immediately (at time 0), its PV is the sum of the initial payment and the PV of a regular annuity for $(n-1)$ periods:

Where:

• $PVA_{Due}$ = Present Value of the Annuity Due
• $Pmt$ = Payment amount per period
• $r$ = Discount rate per period (as a decimal)
• $n$ = Total number of periods

Solution:

We will use formula (1) for this calculation:

$\text{PVA}_{Due} = 5000 \times \left[ \frac{1 - (1+0.07)^{-3}}{0.07} \right] \times (1+0.07)$

$\text{PVA}_{Due} = 5000 \times \left[ \frac{1 - (1.07)^{-3}}{0.07} \right] \times (1.07)$

First, calculate $(1.07)^{-3}$:

$(1.07)^{-3} = \frac{1}{(1.07)^3}$

$(1.07)^3 = 1.07 \times 1.07 \times 1.07 = 1.1449 \times 1.07 = 1.225043$

So, $(1.07)^{-3} = \frac{1}{1.225043} \approx 0.81629787$

Now, substitute this value back into the formula:

$\text{PVA}_{Due} = 5000 \times \left[ \frac{1 - 0.81629787}{0.07} \right] \times 1.07$

$\text{PVA}_{Due} = 5000 \times \left[ \frac{0.18370213}{0.07} \right] \times 1.07$

$\text{PVA}_{Due} = 5000 \times 2.62431614 \times 1.07$

$\text{PVA}_{Due} = 5000 \times 2.80801827$

$\text{PVA}_{Due} = \textsf{₹} 14040.09135$

Rounding to two decimal places for currency:

$\text{PVA}_{Due} \approx \textsf{₹} 14,040.09$

Conclusion:

The Present Value of this annuity due is approximately $\textsf{₹} 14,040.09$.

Problem Statement:

Derive the formula for the accumulated amount after $n$ years for an investment of $\textsf{₹} P$ at $r\%$ per annum compounded annually, which is $A = P(1 + r)^n$. Explain each term in the formula.

Derivation of the Formula:

Let:

• P = Principal amount (initial investment)
• r = Annual interest rate (as a decimal, e.g., for $10\%$, $r=0.10$)
• n = Number of years (compounding periods)
• A = Accumulated amount (future value) after $n$ years

When interest is compounded annually, it means that the interest earned in each year is added to the principal, and then the interest for the next year is calculated on this new, larger principal (principal + accumulated interest).

Year 1:

Interest earned in Year 1 ($I_1$) = $P \times r$

Amount at the end of Year 1 ($A_1$) = Principal + Interest in Year 1

$A_1 = P + P \times r$

Year 2:

The principal for Year 2 is the amount accumulated at the end of Year 1, i.e., $A_1 = P(1+r)$.

Interest earned in Year 2 ($I_2$) = $A_1 \times r = P(1+r) \times r$

Amount at the end of Year 2 ($A_2$) = Amount at end of Year 1 + Interest in Year 2

$A_2 = A_1 + A_1 \times r$

$A_2 = A_1 (1 + r)$

Substitute $A_1 = P(1+r)$ into the equation:

$A_2 = P(1 + r) (1 + r)$

Year 3:

The principal for Year 3 is the amount accumulated at the end of Year 2, i.e., $A_2 = P(1+r)^2$.

Interest earned in Year 3 ($I_3$) = $A_2 \times r = P(1+r)^2 \times r$

Amount at the end of Year 3 ($A_3$) = Amount at end of Year 2 + Interest in Year 3

$A_3 = A_2 + A_2 \times r$

$A_3 = A_2 (1 + r)$

Substitute $A_2 = P(1+r)^2$ into the equation:

$A_3 = P(1 + r)^2 (1 + r)$

Generalization for n Years:

Observing the pattern:

• After 1 year, the amount is $P(1+r)^1$
• After 2 years, the amount is $P(1+r)^2$
• After 3 years, the amount is $P(1+r)^3$

We can generalize this pattern. For $n$ years, the accumulated amount (A) will be:

This formula applies when the interest is compounded annually. If compounded $m$ times a year, the rate becomes $r/m$ and the periods become $n \times m$, leading to $A = P(1 + r/m)^{nm}$.

Explanation of Each Term in the Formula $A = P(1 + r)^n$:

1. A (Accumulated Amount / Future Value):

• This is the final value of the investment at the end of the specified period ($n$ years).
• It represents the initial principal plus all the compound interest earned over the investment term. This is what the investor will receive back or what the loan will amount to.

2. P (Principal / Initial Investment):

• This is the original sum of money invested or borrowed.
• It is the starting amount upon which interest is calculated.

3. r (Annual Interest Rate):

• This is the nominal annual rate of interest.
• It must be expressed as a decimal in the formula (e.g., if the rate is $10\%$, then $r = 0.10$).
• This rate is applied to the principal (and accumulated interest) each year.

4. n (Number of Years):

• This represents the total number of compounding periods. Since the interest is compounded annually in this derivation, $n$ directly corresponds to the number of years the money is invested or borrowed.
• The exponent $n$ signifies that the factor $(1+r)$ is multiplied by itself $n$ times, reflecting the compounding effect of interest over the years.

5. (1 + r):

• This term is the growth factor per period.
• It signifies that for every $\textsf{₹} 1$ invested, it grows to $\textsf{₹} (1+r)$ by the end of one year (or one compounding period). For example, if $r=0.10$, $\textsf{₹} 1$ becomes $\textsf{₹} 1.10$.

Problem Statement:

Calculate the total bill amount for a customer who buys 5 units of Item A (costing $\textsf{₹} 1,000$ each with $12\%$ GST) and 3 units of Item B (costing $\textsf{₹} 1,500$ each with $18\%$ GST).

Given:

• Item A:
  • Unit Cost (exclusive of tax) $= \textsf{₹} 1,000$
  • GST Rate $= 12\%$
  • Quantity Purchased $= 5$ units
• Item B:
  • Unit Cost (exclusive of tax) $= \textsf{₹} 1,500$
  • GST Rate $= 18\%$
  • Quantity Purchased $= 3$ units

To Find:

The Total Bill Amount including GST.

Solution:

Part 1: Calculate the Total Cost for Item A (including GST).

a) Cost of 5 units of Item A (exclusive of tax):

Cost of Item A (base) $= \text{Unit Cost} \times \text{Quantity}$

Cost of Item A (base) $= \textsf{₹} 1,000 \times 5 = \textsf{₹} 5,000$

b) GST on 5 units of Item A:

GST_A $= \text{Cost of Item A (base)} \times \text{GST Rate}_A$

GST_A $= \textsf{₹} 5,000 \times 12\%$

GST_A $= \textsf{₹} 5,000 \times \frac{12}{100} = \textsf{₹} 600$

c) Total Cost of 5 units of Item A (including GST):

Total Cost_A $= \text{Cost of Item A (base)} + \text{GST}_A$

Total Cost_A $= \textsf{₹} 5,000 + \textsf{₹} 600$

Part 2: Calculate the Total Cost for Item B (including GST).

a) Cost of 3 units of Item B (exclusive of tax):

Cost of Item B (base) $= \text{Unit Cost} \times \text{Quantity}$

Cost of Item B (base) $= \textsf{₹} 1,500 \times 3 = \textsf{₹} 4,500$

b) GST on 3 units of Item B:

GST_B $= \text{Cost of Item B (base)} \times \text{GST Rate}_B$

GST_B $= \textsf{₹} 4,500 \times 18\%$

GST_B $= \textsf{₹} 4,500 \times \frac{18}{100} = \textsf{₹} 810$

c) Total Cost of 3 units of Item B (including GST):

Total Cost_B $= \text{Cost of Item B (base)} + \text{GST}_B$

Total Cost_B $= \textsf{₹} 4,500 + \textsf{₹} 810$

Part 3: Calculate the Total Bill Amount.

Total Bill Amount = Total Cost of Item A + Total Cost of Item B

Total Bill Amount = $\textsf{₹} 5,600 + \textsf{₹} 5,310$

Total Bill Amount = $\textsf{₹} 10,910$

Conclusion:

The total bill amount including GST is $\textsf{₹} 10,910$.

Problem Statement:

Calculate the amount paid at the end of the third year and the total interest paid for a loan of $\textsf{₹} 30,000$ at $10\%$ per annum simple interest, with partial repayments of $\textsf{₹} 10,000$ at the end of the first and second years.

Given:

Initial Loan Amount (Principal, P) $= \textsf{₹} 30,000$

Simple Interest Rate (R) $= 10\%$ per annum

Repayment at End of Year 1 $= \textsf{₹} 10,000$

Repayment at End of Year 2 $= \textsf{₹} 10,000$

Repayment at End of Year 3 $= \text{Remaining amount}$

To Find:

1. The Amount Paid at the End of the Third Year.

2. The Total Interest Paid.

Concept Used:

For simple interest with partial payments, the interest for each period is typically calculated on the outstanding principal at the beginning of that period. Payments reduce the principal, thereby reducing the base for future interest calculations.

Solution:

We will track the outstanding principal and interest year by year.

Year 1:

• Principal at beginning of Year 1 ($P_0$) = $\textsf{₹} 30,000$
• Interest for Year 1 ($I_1$) = $P_0 \times R \times 1 \text{ year}$

$\text{I}_1 = \textsf{₹} 30,000 \times 0.10 = \textsf{₹} 3,000$

... (i)

• Amount Due at end of Year 1 = $P_0 + I_1 = \textsf{₹} 30,000 + \textsf{₹} 3,000 = \textsf{₹} 33,000$
• Payment Made at end of Year 1 = $\textsf{₹} 10,000$
• Outstanding Principal at end of Year 1 ($P_1$) = $\textsf{₹} 33,000 - \textsf{₹} 10,000 = \textsf{₹} 23,000$

Year 2:

• Principal at beginning of Year 2 ($P_1$) = $\textsf{₹} 23,000$
• Interest for Year 2 ($I_2$) = $P_1 \times R \times 1 \text{ year}$

$\text{I}_2 = \textsf{₹} 23,000 \times 0.10 = \textsf{₹} 2,300$

... (ii)

• Amount Due at end of Year 2 = $P_1 + I_2 = \textsf{₹} 23,000 + \textsf{₹} 2,300 = \textsf{₹} 25,300$
• Payment Made at end of Year 2 = $\textsf{₹} 10,000$
• Outstanding Principal at end of Year 2 ($P_2$) = $\textsf{₹} 25,300 - \textsf{₹} 10,000 = \textsf{₹} 15,300$

Year 3:

• Principal at beginning of Year 3 ($P_2$) = $\textsf{₹} 15,300$
• Interest for Year 3 ($I_3$) = $P_2 \times R \times 1 \text{ year}$

$\text{I}_3 = \textsf{₹} 15,300 \times 0.10 = \textsf{₹} 1,530$

... (iii)

• Amount Paid at end of Year 3 (Remaining Amount) = $P_2 + I_3$
• Amount Paid at end of Year 3 = $\textsf{₹} 15,300 + \textsf{₹} 1,530 = \textsf{₹} 16,830$

Total Interest Paid:

Total Interest Paid = Interest for Year 1 + Interest for Year 2 + Interest for Year 3

Total Interest Paid = $\textsf{₹} 3,000 + \textsf{₹} 2,300 + \textsf{₹} 1,530$

Total Interest Paid = $\textsf{₹} 6,830$

Conclusion:

The amount paid at the end of the third year is $\textsf{₹} 16,830$.

The total interest paid is $\textsf{₹} 6,830$.

Problem Statement:

Calculate the compound interest on $\textsf{₹} 60,000$ for 2 years at $9\%$ per annum, compounded half-yearly.

Given:

Principal (P) $= \textsf{₹} 60,000$

Time (t) $= 2$ years

Annual Rate (R) $= 9\%$ per annum $= 0.09$

Compounding Frequency (m) $= \text{half-yearly} = 2$ times a year

To Find:

The Compound Interest (C.I.).

Formulae Used:

1. Amount (A) with Compound Interest:

2. Compound Interest (C.I.):

Where:

• $P$ = Principal Amount
• $R$ = Annual Interest Rate (as a decimal)
• $m$ = Number of times interest is compounded per year
• $t$ = Time in years

Solution:

Step 1: Determine the rate per period and total number of periods.

Rate per compounding period ($\frac{R}{m}$) = $\frac{0.09}{2} = 0.045$

Total number of compounding periods ($mt$) = $2 \times 2 = 4$ periods

Step 2: Calculate the Accumulated Amount (A) using formula (1).

$\text{A} = 60000 \left(1 + 0.045\right)^{4}$

$\text{A} = 60000 (1.045)^4$

Calculate $(1.045)^4$:

$(1.045)^2 = 1.045 \times 1.045 = 1.092025$

$(1.045)^4 = (1.092025)^2 = 1.092025 \times 1.092025 \approx 1.1925186$

Substitute this value back into the amount calculation:

$\text{A} = 60000 \times 1.1925186$

$\text{A} = \textsf{₹} 71,551.116$

Rounding to two decimal places for currency:

Step 3: Calculate the Compound Interest (C.I.) using formula (2).

$\text{C.I.} = \text{A} - P$

$\text{C.I.} = \textsf{₹} 71,551.12 - \textsf{₹} 60,000$

$\text{C.I.} = \textsf{₹} 11,551.12$

Conclusion:

The compound interest on $\textsf{₹} 60,000$ for 2 years at $9\%$ per annum, compounded half-yearly, is $\textsf{₹} 11,551.12$.

Problem Statement:

Calculate the accumulated value of an annuity of $\textsf{₹} 1,500$ paid at the end of each quarter for 1 year, if the interest rate is $8\%$ per annum compounded quarterly.

Given:

Periodic Payment (Pmt) $= \textsf{₹} 1,500$ (paid at the end of each quarter, indicating a Regular Annuity)

Time (t) $= 1$ year

Annual Interest Rate (R) $= 8\%$ per annum $= 0.08$

Compounding Frequency (m) $= \text{quarterly} = 4$ times a year

To Find:

The Accumulated Value (Future Value) of this annuity.

Formula Used:

The formula for the Future Value of a Regular Annuity ($FV_{Reg}$) is:

Where:

• $Pmt$ = Payment amount per period
• $r'$ = Interest rate per period (calculated as $R/m$)
• $n$ = Total number of periods (calculated as $t \times m$)

Solution:

Step 1: Determine the interest rate per period and total number of periods.

Interest rate per period ($r'$):

$r' = \frac{R}{m} = \frac{0.08}{4} = 0.02$

Total number of periods ($n$):

$n = t \times m = 1 \text{ year} \times 4 \text{ quarters/year} = 4 \text{ periods}$

Step 2: Calculate the Future Value of the Regular Annuity using formula (1).

$\text{FV}_{Reg} = 1500 \times \left[ \frac{(1+0.02)^4 - 1}{0.02} \right]$

$\text{FV}_{Reg} = 1500 \times \left[ \frac{(1.02)^4 - 1}{0.02} \right]$

First, calculate $(1.02)^4$:

$(1.02)^2 = 1.02 \times 1.02 = 1.0404$

$(1.02)^4 = (1.0404)^2 = 1.0404 \times 1.0404 = 1.08243216$

Now, substitute this value back into the formula:

$\text{FV}_{Reg} = 1500 \times \left[ \frac{1.08243216 - 1}{0.02} \right]$

$\text{FV}_{Reg} = 1500 \times \left[ \frac{0.08243216}{0.02} \right]$

$\text{FV}_{Reg} = 1500 \times 4.121608$

$\text{FV}_{Reg} = \textsf{₹} 6182.412$

Rounding to two decimal places for currency:

$\text{FV}_{Reg} \approx \textsf{₹} 6,182.41$

Conclusion:

The accumulated value of the annuity at the end of 1 year is approximately $\textsf{₹} 6,182.41$.

Problem Statement:

Calculate the total electricity bill amount based on a fixed charge, tiered energy charges, a regulatory surcharge on energy charges, for a total consumption of 280 units.

Given:

• Fixed Charge: $\textsf{₹} 150$
• Energy Charge Slabs:
  • First 200 units at $\textsf{₹} 5$ per unit
  • Above 200 units at $\textsf{₹} 7$ per unit
• Total Units Consumed: 280 units
• Regulatory Charge: $5\%$ on total energy charge

To Find:

The Total Electricity Bill Amount.

Solution:

Step 1: Calculate the Energy Charge based on units consumed in each slab.

Total units consumed are 280 units.

a) First 200 units:

Charge for first 200 units = $200 \text{ units} \times \textsf{₹} 5/\text{unit}$

Units remaining to be charged = $280 - 200 = 80$ units.

b) Remaining 80 units (above 200 units):

Charge for remaining 80 units = $80 \text{ units} \times \textsf{₹} 7/\text{unit}$

c) Total Energy Charge:

Total Energy Charge = $\text{Charge}_1 + \text{Charge}_2$

Total Energy Charge = $\textsf{₹} 1,000 + \textsf{₹} 560$

Step 2: Calculate the Regulatory Charge.

Regulatory Charge is $5\%$ on the total energy charge.

Regulatory Charge Amount = $5\% \text{ of } \textsf{₹} 1,560$

Regulatory Charge Amount = $\frac{5}{100} \times \textsf{₹} 1,560$

Regulatory Charge Amount = $0.05 \times \textsf{₹} 1,560$

Step 3: Calculate the Total Electricity Bill.

Total Electricity Bill = Fixed Charge + Total Energy Charge + Regulatory Charge Amount

Total Electricity Bill = $\textsf{₹} 150 + \textsf{₹} 1,560 + \textsf{₹} 78$

Total Electricity Bill = $\textsf{₹} 1,788$

Conclusion:

The total electricity bill amount is $\textsf{₹} 1,788$.

Problem Statement:

Explain the concept of Net Present Value (NPV) as a criterion for investment decisions. Discuss its advantages and disadvantages compared to other methods.

Concept of Net Present Value (NPV):

The Net Present Value (NPV) is a capital budgeting technique used to determine the profitability of a project or investment. It measures the difference between the present value of future cash inflows and the present value of the initial investment (cash outflows) over a specified period.

The core idea behind NPV is the time value of money, which states that a rupee received today is worth more than a rupee received in the future due to its potential earning capacity. Therefore, all future cash flows (both inflows and outflows) associated with a project are discounted back to their present value using a specified discount rate (often the cost of capital or required rate of return).

The general formula for NPV is:

Where:

• $CF_t$ = Net cash flow at time $t$. ($CF_0$ is usually the initial investment, which is a cash outflow and thus a negative value).
• $r$ = Discount rate (cost of capital or required rate of return).
• $t$ = Time period (ranging from 0 to $n$).
• $n$ = Total number of periods.

If the initial investment is $I_0$ (at time $t=0$) and subsequent cash inflows are $CF_1, CF_2, ..., CF_n$:

NPV Decision Rule:

• If NPV > 0: Accept the project. The project is expected to generate a return greater than the cost of capital, thereby adding value to the firm and increasing shareholder wealth.
• If NPV < 0: Reject the project. The project is expected to generate a return less than the cost of capital, leading to a decrease in firm value.
• If NPV = 0: Indifferent. The project is expected to generate a return exactly equal to the cost of capital, neither adding nor detracting from firm value. In practice, such projects are often rejected due to the inherent risks and uncertainties not captured in the calculation.

For mutually exclusive projects (where only one can be chosen), the project with the highest positive NPV is typically selected.

Advantages of NPV:

1. Considers Time Value of Money: This is its fundamental strength. NPV explicitly discounts future cash flows back to their present value, making it a theoretically sound method as it accounts for the opportunity cost of capital.

2. Considers All Cash Flows: Unlike some other methods (e.g., payback period), NPV takes into account all cash flows generated by the project over its entire life, providing a comprehensive view of profitability.

3. Provides a Direct Measure of Value Added: A positive NPV indicates the actual increase in the wealth of the shareholders (or the value added to the firm) by undertaking the project. It directly answers how much value a project will create.

4. Consistent with Shareholder Wealth Maximization: The primary objective of a firm is to maximize shareholder wealth. By accepting projects with positive NPVs, the firm aligns its investment decisions directly with this objective.

5. Absolute Measure: NPV provides an absolute monetary value of a project's profitability, which can be easily understood and compared (for mutually exclusive projects).

6. Reinvestment Assumption: NPV implicitly assumes that intermediate cash flows are reinvested at the discount rate (cost of capital), which is generally considered a more realistic assumption than assuming reinvestment at the project's internal rate of return (IRR).

Disadvantages of NPV:

1. Requires Estimation of Discount Rate: Determining the appropriate discount rate (cost of capital) can be challenging and subjective. An incorrect discount rate can lead to flawed NPV results and incorrect decisions.

2. Requires Accurate Cash Flow Forecasts: NPV heavily relies on accurate forecasts of future cash flows, which are often difficult to predict with certainty, especially for long-term projects. Errors in forecasting can significantly impact the NPV.

3. Does Not Indicate Relative Profitability/Efficiency: NPV is an absolute measure (in currency units), not a relative one. It might not be suitable for comparing projects of different sizes or initial investments. A small project with a small positive NPV might be more capital-efficient than a large project with a large positive NPV. (This limitation is sometimes addressed by using the Profitability Index, which is a ratio).

4. Complexity: For non-finance professionals, the concept of discounting and present value might be less intuitive than simpler methods like payback period or accounting rate of return.

5. Difficulty with Mutually Exclusive Projects (of different lives): While useful for mutually exclusive projects, comparing projects with significantly different lifespans using NPV can sometimes be misleading, although methods like Equivalent Annual Annuity (EAA) can help address this.

Problem Statement:

Calculate the value of a car at the end of 3 years and the total depreciation over these 3 years, given its initial cost of $\textsf{₹} 10,00,000$ and a depreciation rate of $15\%$ per annum on a reducing balance basis.

Given:

Initial Cost (P) $= \textsf{₹} 10,00,000$

Depreciation Rate (r) $= 15\%$ per annum $= 0.15$

Time (n) $= 3$ years

Depreciation Method: Reducing Balance Basis (Compound Depreciation)

To Find:

1. The Value of the Car at the end of 3 years (A).

2. The Total Depreciation over these 3 years.

Formulae Used:

1. Value after $n$ years (A) with Reducing Balance Depreciation:

2. Total Depreciation:

Where:

• $P$ = Initial Cost
• $r$ = Depreciation Rate (as a decimal)
• $n$ = Number of years
• $A$ = Value after $n$ years

Solution:

Step 1: Calculate the Value of the Car at the end of 3 years (A).

Using formula (1):

$\text{A} = 10,00,000 (1 - 0.15)^3$

$\text{A} = 10,00,000 (0.85)^3$

Calculate $(0.85)^3$:

$(0.85)^2 = 0.85 \times 0.85 = 0.7225$

$(0.85)^3 = 0.7225 \times 0.85 = 0.614125$

Now, substitute this value back into the amount calculation:

$\text{A} = 10,00,000 \times 0.614125$

The value of the car at the end of 3 years is $\textsf{₹} 6,14,125$.

Step 2: Calculate the Total Depreciation over these 3 years.

Using formula (2):

$\text{Total Depreciation} = \text{Initial Cost} - \text{Value after 3 years}$

$\text{Total Depreciation} = \textsf{₹} 10,00,000 - \textsf{₹} 6,14,125$

$\text{Total Depreciation} = \textsf{₹} 3,85,875$

Conclusion:

The value of the car at the end of 3 years is $\textsf{₹} 6,14,125$.

The total depreciation over these 3 years is $\textsf{₹} 3,85,875$.