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| Simple Interest: Calculation Formula and Application | Accumulation with Simple Interest (Calculating Amount) | Problems based on Simple Interest |
Simple Interest
Simple Interest: Calculation Formula and Application
Definition
Simple Interest (SI) is the most basic method of calculating interest. In this method, the interest earned or paid is calculated strictly on the original principal amount. This means that the interest earned in any given period is not added to the principal for the purpose of calculating interest in the next period. The principal remains constant throughout the entire term of the investment or loan.
Consequently, the amount of simple interest earned or paid remains the same for each unit of time (e.g., each year, each month), assuming the interest rate and the principal do not change. Simple interest is thus a linear function of time.
Calculation Formula
The simple interest amount is directly proportional to the principal amount, the interest rate, and the time period. This direct relationship leads to a straightforward formula for calculating simple interest.
The formula for Simple Interest (SI) is:
$\mathbf{SI = P \times r \times t}$
Where:
- SI = The Simple Interest amount earned or paid, in monetary terms.
- P = The Principal amount, which is the initial sum of money borrowed, lent, or invested.
- r = The annual rate of interest, expressed as a decimal. If the rate is given as a percentage (say, R%), you must convert it to a decimal by dividing by 100: $r = \frac{R}{100}$.
- t = The time period for which the money is borrowed or invested, expressed in years.
Critical Requirement: Consistency of Units
It is paramount that the time period ($t$) and the interest rate ($r$) are in consistent units. The formula $SI = Prt$ assumes $r$ is an annual rate and $t$ is the number of years.
- If the rate is given annually (per annum or p.a.), but the time is given in months or days, you must convert the time into years.
- If time is in months, divide by 12: $t\$ (in years) = $\frac{Number\$ of\$ Months}{12}$. Example: 9 months = $\frac{9}{12} = 0.75$ years.
- If time is in days, divide by the number of days in a year (usually 365 for "exact simple interest" or sometimes 360 for "ordinary simple interest" in specific financial contexts). For standard calculations, use 365. Example: 73 days = $\frac{73}{365} = \frac{1}{5} = 0.2$ years.
- Conversely, if the rate is given for a period other than a year (e.g., per month), and the time is in years, you would need to convert the time into months, or convert the monthly rate into an annual rate. Stick to annual rate and years unless specified otherwise.
Application
While compound interest is more prevalent in modern finance, simple interest is still applied in certain situations and is foundational for understanding interest concepts.
- Short-Term Lending: Simple interest is sometimes used for very short-term loans or borrowings, such as certain payday loans (though often with extremely high rates effectively compounding) or calculations on bills of exchange for periods less than a year.
- Specific Bonds: Some bonds might pay simple interest annually on the face value.
- Basic Calculations: It is often used in educational settings to introduce the concept of interest before moving to the complexity of compounding.
- Partial Period Interest: In some cases, when an investment is withdrawn before completing a full interest period (especially for fixed deposits), simple interest might be calculated for the incomplete period based on the contracted annual rate.
It is important to note that for investments or loans extending over multiple periods (especially years), simple interest results in a significantly lower return for the lender/investor and a lower cost for the borrower compared to compound interest, because it ignores the power of "interest on interest".
Example 1. Calculate the simple interest earned on a principal of $\textsf{₹}\$ 8,000$ for 4 years at an annual interest rate of 6%.
Answer:
Given:
- Principal (P) = $\textsf{₹}\$ 8,000$
- Annual Rate (R) = 6%. Convert to decimal: $r = \frac{6}{100} = 0.06$.
- Time (t) = 4 years.
To Find:
- Simple Interest (SI).
Formula:
$SI = P \times r \times t$
Solution:
Substitute the given values into the formula:
$SI = 8000 \times 0.06 \times 4$
$SI = 8000 \times (6/100) \times 4$
$SI = (\cancel{8000}^{80} \times 6 / \cancel{100}) \times 4$
$SI = 80 \times 6 \times 4$
$SI = 480 \times 4$
$SI = 1920$
The simple interest earned is $\textsf{₹}\$ 1,920$.
Example 2. Find the simple interest on $\textsf{₹}\$ 25,000$ for 18 months at an annual rate of 10%.
Answer:
Given:
- Principal (P) = $\textsf{₹}\$ 25,000$
- Annual Rate (R) = 10%. Convert to decimal: $r = \frac{10}{100} = 0.10$.
- Time = 18 months. Convert to years: $t = \frac{18}{12} = \frac{3}{2} = 1.5$ years.
To Find:
- Simple Interest (SI).
Formula:
$SI = P \times r \times t$
Solution:
Substitute the given values into the formula:
$SI = 25000 \times 0.10 \times 1.5$
$SI = 25000 \times \frac{10}{100} \times \frac{15}{10}$
$SI = 25000 \times \frac{1}{10} \times \frac{15}{10}$
$SI = \cancel{25000}^{250} \times \frac{1}{\cancel{10}} \times \frac{15}{\cancel{10}}$
$SI = 250 \times 15$
Let's perform the multiplication:
$\begin{array}{cc}& & 2 & 5 & 0 \\ \times & & & 1 & 5 \\ \hline & 1 & 2 & 5 & 0 \\ 2 & 5 & 0 & \times \\ \hline 3 & 7 & 5 & 0 \\ \hline \end{array}$$SI = 3750$
The simple interest is $\textsf{₹}\$ 3,750$.
Example 3. A sum of $\textsf{₹}\$ 12,000$ was lent for 146 days at 5% per annum simple interest. Find the interest earned.
Answer:
Given:
- Principal (P) = $\textsf{₹}\$ 12,000$
- Annual Rate (R) = 5%. Convert to decimal: $r = \frac{5}{100} = 0.05$.
- Time = 146 days. Convert to years (using 365 days in a year): $t = \frac{146}{365}$ years.
To Find:
- Simple Interest (SI).
Formula:
$SI = P \times r \times t$
Solution:
Substitute the given values into the formula:
$SI = 12000 \times 0.05 \times \frac{146}{365}$
$SI = 12000 \times \frac{5}{100} \times \frac{146}{365}$
Notice that 146 and 365 are both divisible by 73:
$146 = 2 \times 73$
$365 = 5 \times 73$
So, the fraction $\frac{146}{365}$ simplifies to $\frac{2}{5}$.
$SI = 12000 \times \frac{5}{100} \times \frac{2}{5}$
$SI = \cancel{12000}^{120} \times \frac{\cancel{5}}{ \cancel{100}} \times \frac{2}{\cancel{5}}$
$SI = 120 \times \frac{1}{10} \times 2$
$SI = \cancel{120}^{12} \times \frac{1}{\cancel{10}} \times 2$
$SI = 12 \times 1 \times 2 = 24$
The simple interest earned is $\textsf{₹}\$ 240$.
Summary for Competitive Exams
Simple Interest (SI): Calculated ONLY on the original Principal. Interest amount is constant per period.
Formula: $\mathbf{SI = P \times r \times t}$
- P: Principal
- r: Annual interest rate (as a decimal, $\frac{R\%}{100}$)
- t: Time in years. (Convert months by $\div 12$, days by $\div 365$).
Key Point: Simple interest does not account for compounding ("interest on interest"). Primarily used for short durations or as a basic concept.
Accumulation with Simple Interest (Calculating Amount)
Concept
When dealing with interest, the total value of an investment or loan at the end of the term is often required. This total value includes the initial principal amount plus the total interest earned or charged over the period. This final sum is known as the Amount (A), or sometimes the Accumulated Value or Future Value (FV) in the context of simple interest.
Under simple interest, calculating the accumulated amount is straightforward because the interest calculation method is simple and linear.
Formula Derivation
By definition, the final Amount is the sum of the initial Principal and the total Simple Interest accrued:
$Amount = Principal + Simple\$ Interest$
Using the symbols we defined earlier:
$A = P + SI$
We know the formula for calculating Simple Interest is $SI = P \times r \times t$.
Substitute this expression for $SI$ into the Amount formula:
$A = P + (P \times r \times t)$
Now, we can see that the Principal ($P$) is a common factor in both terms on the right side of the equation ($P$ and $Prt$). We can factor out $P$:
$A = P(1 \times P^{-1} \times P + r \times t \times P \times P^{-1})$
$A = P(1 + r t)$
Thus, the formula for the Accumulated Amount (A) under simple interest is:
$\mathbf{A = P(1 + rt)}$
Where:
- A = The final Amount or Accumulated Value.
- P = The original Principal amount.
- r = The annual interest rate (as a decimal).
- t = The time period in years.
This formula allows you to directly calculate the total value at the end of the term without first calculating the simple interest separately, although both methods yield the same result ($A = P + SI$).
Remember, just like with the SI formula, it is crucial that the units of the rate ($r$) and time ($t$) are consistent (typically annual rate and time in years).
Example 1. Find the amount accumulated if $\textsf{₹}\$ 15,000$ is invested for 3 years at a simple interest rate of 9% per annum.
Answer:
Given:
- Principal (P) = $\textsf{₹}\$ 15,000$
- Annual Rate (R) = 9%. Convert to decimal: $r = \frac{9}{100} = 0.09$.
- Time (t) = 3 years.
To Find:
- Amount (A) accumulated.
Formula:
$A = P(1 + rt)$
Solution:
Substitute the given values into the formula:
$A = 15000 (1 + (0.09 \times 3))$
$A = 15000 (1 + 0.27)$
$A = 15000 (1.27)$
Let's perform the multiplication:
$\begin{array}{cc}& & 1 & 5 & 0 & 0 & 0 \\ \times & & & & 1 & . & 2 & 7 \\ \hline & & 1 & 0 & 5 & 0 & 0 & 0 \\ & 3 & 0 & 0 & 0 & 0 & \times \\ 1 & 5 & 0 & 0 & 0 & \times & \times \\ \hline 1 & 9 & 0 & 5 & 0 & . & 0 & 0 \\ \hline \end{array}$$A = 19050$
The accumulated amount after 3 years is $\textsf{₹}\$ 19,050$.
Alternate Solution:
First calculate Simple Interest (SI):
$SI = Prt = 15000 \times 0.09 \times 3 = 15000 \times 0.27 = \textsf{₹}\$ 4050$
Then calculate Amount (A):
$A = P + SI = 15000 + 4050 = \textsf{₹}\$ 19050$.
Both methods give the same result.
Example 2. What amount will be required to repay a loan of $\textsf{₹}\$ 50,000$ taken for 9 months at a simple interest rate of 12% per annum?
Answer:
Given:
- Principal (P) = $\textsf{₹}\$ 50,000$
- Annual Rate (R) = 12%. Convert to decimal: $r = \frac{12}{100} = 0.12$.
- Time = 9 months. Convert to years: $t = \frac{9}{12} = \frac{3}{4} = 0.75$ years.
To Find:
- Amount (A) to be repaid.
Formula:
$A = P(1 + rt)$
Solution:
Substitute the given values into the formula:
$A = 50000 (1 + (0.12 \times 0.75))$
$A = 50000 (1 + (0.12 \times \frac{3}{4}))$
$0.12 \times \frac{3}{4} = \frac{0.12 \times 3}{4} = \frac{0.36}{4} = 0.09$
$A = 50000 (1 + 0.09)$
$A = 50000 (1.09)$
$A = 50000 \times 1.09$
Let's perform the multiplication:
$\begin{array}{cc}& & 5 & 0 & 0 & 0 & 0 \\ \times & & & & 1 & . & 0 & 9 \\ \hline & & 4 & 5 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \times \\ 5 & 0 & 0 & 0 & 0 & \times & \times \\ \hline 5 & 4 & 5 & 0 & 0 & . & 0 & 0 \\ \hline \end{array}$$A = 54500$
The amount required to repay the loan is $\textsf{₹}\$ 54,500$.
Summary for Competitive Exams
Amount (A): Total value at the end of the term = Principal + Simple Interest.
Formula Derivation: $A = P + SI$, and $SI = Prt$. Substituting SI gives $A = P + Prt$. Factoring out P gives $\mathbf{A = P(1 + rt)}$.
Formula: $\mathbf{A = P(1 + rt)}$
- A: Final Amount / Accumulated Value
- P: Principal
- r: Annual interest rate (decimal)
- t: Time in years
This formula calculates the total value directly under simple interest conditions.
Problems based on Simple Interest
In the previous sections, we learned the fundamental formulas for calculating Simple Interest (SI) and the total Accumulated Amount (A) when the Principal (P), Rate (r), and Time (t) are known.
The two primary formulas are:
$\mathbf{SI = P \times r \times t}$
$\mathbf{A = P(1 + rt)}$
Often, in practical problems, we might be given the Simple Interest (SI) or the Amount (A) and asked to find one of the other variables (Principal, Rate, or Time). Since these are algebraic equations, we can rearrange them to solve for any unknown variable, provided the other relevant variables are known.
Rearranging Formulas to Find Unknowns
Finding Principal (P)
If you know the Simple Interest (SI), Rate (r), and Time (t), you can find the Principal using the formula $SI = Prt$.
Divide both sides by $(rt)$:
$P = \frac{SI}{rt}$
Alternatively, if you know the Amount (A), Rate (r), and Time (t), you can use the formula $A = P(1 + rt)$.
Divide both sides by $(1 + rt)$:
$P = \frac{A}{1 + rt}$
Finding Rate (r)
If you know the Simple Interest (SI), Principal (P), and Time (t), you can find the Rate (in decimal form) using the formula $SI = Prt$.
Divide both sides by $(Pt)$:
$r = \frac{SI}{Pt}$
Remember that this calculation gives the rate as a decimal (e.g., 0.05). To convert it to a percentage (R%), multiply the decimal by 100 ($R\% = r \times 100$).
If you know the Amount (A), Principal (P), and Time (t), you can also find the rate. Start with $A = P(1 + rt)$:
Divide by P: $\frac{A}{P} = 1 + rt$
Subtract 1: $\frac{A}{P} - 1 = rt$
Combine terms on the left: $\frac{A - P}{P} = rt$
Divide by t: $r = \frac{A - P}{Pt}$. Note that $A - P$ is the Simple Interest (SI), so this formula is identical to $r = \frac{SI}{Pt}$.
Finding Time (t)
If you know the Simple Interest (SI), Principal (P), and Rate (r), you can find the Time (in the unit corresponding to the rate, usually years) using the formula $SI = Prt$.
Divide both sides by $(Pr)$:
$t = \frac{SI}{Pr}$
This formula gives the time in years (assuming r is the annual rate). If the question requires time in months or days, you would convert the resulting years accordingly (e.g., $Time\$ in\$ Months = t \times 12$).
If you know the Amount (A), Principal (P), and Rate (r), you can find the time using the derived formula from the Amount calculation:
$t = \frac{A - P}{Pr}$. Again, $A-P = SI$, so this is the same formula.
Ensure that the units of the rate ($r$) and the calculated time ($t$) are consistent. If $r$ is an annual rate, $t$ will be in years.
Worked Examples
Example 1. At what annual rate of simple interest will $\textsf{₹}\$ 900$ amount to $\textsf{₹}\$ 1080$ in 4 years?
Answer:
Given:
- Principal (P) = $\textsf{₹}\$ 900$
- Amount (A) = $\textsf{₹}\$ 1080$
- Time (t) = 4 years
To Find:
- Annual Rate of Simple Interest (R%).
Solution:
First, we need to find the Simple Interest (SI) earned over the 4 years. The SI is the difference between the final Amount and the initial Principal.
$SI = A - P$
$SI = \textsf{₹}\$ 1080 - \textsf{₹}\$ 900$
Let's perform the subtraction:
$\begin{array}{cc} & 1 & 0 & 8 & 0 \\ - & & 9 & 0 & 0 \\ \hline & & 1 & 8 & 0 \\ \hline \end{array}$$SI = \textsf{₹}\$ 180$
Now we have the Simple Interest (SI), Principal (P), and Time (t). We can use the formula $SI = Prt$ to find the decimal rate $r$.
$180 = 900 \times r \times 4$
Combine the terms with $r$:
$180 = (900 \times 4) \times r$
$180 = 3600 \times r$
Now, solve for $r$ by dividing both sides by 3600:
$r = \frac{180}{3600}$
Simplify the fraction:
$r = \frac{\cancel{180}^{1}}{\cancel{3600}^{20}}$
$r = \frac{1}{20}$
This is the annual rate in decimal form. To convert it to a percentage (R%), multiply by 100:
$R\% = r \times 100\% = \frac{1}{20} \times 100\%$
$R\% = \frac{100}{20}\%$
$R\% = 5\%$
The annual rate of simple interest is 5%.
Alternate Solution (Using Amount Formula):
We can directly use the amount formula $A = P(1+rt)$ to find $r$.
Substitute the given values:
$1080 = 900(1 + r \times 4)$
$1080 = 900(1 + 4r)$
Divide both sides by 900:
$\frac{1080}{900} = 1 + 4r$
Simplify the fraction on the left:
$\frac{\cancel{1080}^{12}}{\cancel{900}^{10}} = \frac{12}{10} = \frac{6}{5} = 1.2$
So, $1.2 = 1 + 4r$
Subtract 1 from both sides:
$1.2 - 1 = 4r$
$0.2 = 4r$
Divide by 4:
$r = \frac{0.2}{4} = 0.05$
Convert decimal to percentage: $R\% = 0.05 \times 100\% = 5\%$.
Both methods yield the same result.
Example 2. In how many years will a sum of $\textsf{₹}\$ 4,500$ yield a simple interest of $\textsf{₹}\$ 1,350$ at 6% per annum?
Answer:
Given:
- Principal (P) = $\textsf{₹}\$ 4,500$
- Simple Interest (SI) = $\textsf{₹}\$ 1,350$
- Annual Rate (R) = 6%. Convert to decimal: $r = \frac{6}{100} = 0.06$.
To Find:
- Time (t) in years.
Formula:
$SI = P \times r \times t$
Solution:
Substitute the given values into the formula:
$1350 = 4500 \times 0.06 \times t$
Combine the known numerical terms on the right side:
$1350 = (4500 \times 0.06) \times t$
$4500 \times 0.06 = 4500 \times \frac{6}{100} = 45 \times 6 = 270$
So,
$1350 = 270 \times t$
Now, solve for $t$ by dividing both sides by 270:
$t = \frac{1350}{270}$
Simplify the fraction:
$t = \frac{135}{27}$
We can divide 135 by 27. $27 \times 5 = (30-3) \times 5 = 150 - 15 = 135$.
$t = 5$
The time required is 5 years.
Example 3. A sum of money doubles itself in 8 years at simple interest. Find the rate of interest.
Answer:
Given:
- Let the Principal be $P$.
- The sum doubles itself, meaning the Amount (A) at the end of the term is twice the Principal: $A = 2P$.
- Time (t) = 8 years.
To Find:
- Annual Rate of Simple Interest (R%).
Solution:
First, calculate the Simple Interest (SI) earned. The SI is the difference between the Amount and the Principal.
$SI = A - P$
$SI = 2P - P$
$SI = P$
So, when a sum doubles at simple interest, the total simple interest earned is equal to the original principal amount.
Now, use the formula $SI = Prt$:
$P = P \times r \times 8$
$P = 8P \times r$
Since the Principal ($P$) is the initial sum of money, it is assumed to be a non-zero value. Therefore, we can divide both sides of the equation by $P$:
$\frac{P}{P} = \frac{8Pr}{P}$
(Dividing both sides by P)
$1 = 8 \times r$
Now, solve for $r$ by dividing both sides by 8:
$r = \frac{1}{8}$
Convert the decimal rate $r$ to a percentage (R%):
$R\% = r \times 100\% = \frac{1}{8} \times 100\%$
$R\% = \frac{100}{8}\% = \frac{50}{4}\% = \frac{25}{2}\% = 12.5\%$
The annual rate of simple interest is 12.5%.
Example 4. What principal will amount to $\textsf{₹}\$ 6,500$ in 3 years at 7.5% per annum simple interest?
Answer:
Given:
- Amount (A) = $\textsf{₹}\$ 6,500$
- Time (t) = 3 years
- Annual Rate (R) = 7.5%. Convert to decimal: $r = \frac{7.5}{100} = 0.075$.
To Find:
- Principal (P).
Formula:
$A = P(1 + rt)$
Solution:
Substitute the given values into the formula:
$6500 = P(1 + (0.075 \times 3))$
Calculate the term inside the parenthesis:
$0.075 \times 3 = 0.225$
So,
$6500 = P(1 + 0.225)$
$6500 = P(1.225)$
Now, solve for $P$ by dividing both sides by 1.225:
$P = \frac{6500}{1.225}$
To perform the division, we can remove the decimal by multiplying the numerator and denominator by 1000:
$P = \frac{6500 \times 1000}{1.225 \times 1000} = \frac{6500000}{1225}$
We can simplify the fraction by dividing the numerator and denominator by common factors. Both end in 0 or 5, so they are divisible by 5.
$1225 \div 5 = 245$
$6500000 \div 5 = 1300000$
$P = \frac{1300000}{245}$
Again, both are divisible by 5.
$245 \div 5 = 49$
$1300000 \div 5 = 260000$
$P = \frac{260000}{49}$
Now, divide 260000 by 49. Since 49 is $7^2$, we can check for divisibility by 7, but it's unlikely to be exact for typical problems unless P is a round number. Let's check the original calculation again: $0.075 \times 3 = 0.225$. $1 + 0.225 = 1.225$. $\frac{6500}{1.225}$.
Let's use fraction form for the rate: $7.5\% = \frac{7.5}{100} = \frac{15/2}{100} = \frac{15}{200} = \frac{3}{40}$. So $r = 3/40$.
$6500 = P(1 + \frac{3}{40} \times 3)$
$6500 = P(1 + \frac{9}{40})$
$6500 = P(\frac{40}{40} + \frac{9}{40})$
$6500 = P(\frac{49}{40})$
Solve for P:
$P = 6500 \times \frac{40}{49}$
$P = \frac{260000}{49}$
It seems the principal might not be a whole number in this case. Let's re-read the question to ensure no misinterpretation. "What principal will amount to $\textsf{₹}\$ 6,500$ in 3 years at 7.5% per annum simple interest?". The setup seems correct. The calculation $\frac{260000}{49}$ is approximately 5306.12. In some contexts, problems might intentionally have non-integer results, or there might be a slight typo in the numbers. Assuming the numbers are correct as given:
The principal amount is $\textsf{₹}\$ \frac{260000}{49} \approx \textsf{₹}\$ 5306.12$ (rounded to two decimal places).
Calculation check:
$P = \frac{6500}{1.225}$
$\begin{array}{r} 5306.122\dots\phantom{)} \\ 1225{\overline{\smash{\big)}\,6500000.000\dots\phantom{)}}} \\ \underline{-~\phantom{(}(6125)\phantom{000.000)}} \\ 37500\phantom{0.000\dots)} \\ \underline{-~\phantom{()}(36750)\phantom{0.000)}} \\ 7500\phantom{.000\dots)} \\ \underline{-~\phantom{()}(7350)\phantom{.000)}} \\ 1500\phantom{0\dots)} \\ \underline{-~\phantom{()}(1225)\phantom{0\dots)}} \\ 2750\phantom{\dots)} \\ \underline{-~\phantom{()}(2450)\dots)} \\ 300\dots \end{array}$So, the calculated value $\textsf{₹}\$ \frac{260000}{49}$ or approximately $\textsf{₹}\$ 5306.12$ is correct based on the given numbers.
The principal that will amount to $\textsf{₹}\$ 6,500$ in 3 years at 7.5% p.a. simple interest is $\textsf{₹}\$ \frac{260000}{49}$.
Summary for Competitive Exams
The basic simple interest formulas are $\mathbf{SI = Prt}$ and $\mathbf{A = P(1+rt)}$. These can be rearranged to find any missing variable.
Finding P: $P = \frac{SI}{rt}$ or $P = \frac{A}{1+rt}$
Finding r: $r = \frac{SI}{Pt}$ (gives decimal rate; multiply by 100 for %)
Finding t: $t = \frac{SI}{Pr}$ (gives time in years if r is annual)
Important: Always ensure rate ($r$) is in decimal form and time ($t$) is in years when using these formulas. If given A and asked for SI, calculate $SI = A-P$. If given SI and asked for A, calculate $A = P+SI$.
Common Problem Types: Calculating SI or A (direct application), Finding Rate, Finding Time, Finding Principal, Problems involving "sum doubles/triples" (where $SI=P$ or $SI=2P$).