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Chapter 8 Comparing Quantities (Concepts)
Welcome back to the essential topic of Comparing Quantities. This chapter significantly builds upon the foundational methods learned in Class 7, extending our toolkit to handle more complex real-world applications, particularly those involving financial calculations over time. While we will reinforce our understanding of basic comparison tools like ratios, proportions, and percentages, the major focus will shift towards understanding how quantities grow or decrease exponentially, introducing the crucial concept of Compound Interest. Mastering these advanced techniques is vital for developing strong financial literacy and making informed decisions in everyday economic contexts.
We begin by revisiting and strengthening our core skills. This includes refining our ability to calculate percentage increase or decrease ($\text{Percentage Change} = \frac{\text{Amount of Change}}{\text{Original Amount}} \times 100\%$), applying these concepts to scenarios like population shifts or value fluctuations. We will also tackle more intricate problems involving discounts (calculating sale prices after a percentage reduction from the marked price), profit and loss calculations (reinforcing that Profit % or Loss % is always calculated on the Cost Price (CP): $Profit \% = \frac{SP - CP}{CP} \times 100\%$, $Loss \% = \frac{CP - SP}{CP} \times 100\%$, where SP is Selling Price), and understanding how taxes such as Sales Tax or Value Added Tax (VAT) are calculated as a percentage of the selling price and added to the final bill amount (often involving $\textsf{₹}$).
The principal new concept introduced in this chapter, representing a significant leap in understanding financial growth, is Compound Interest (CI). This stands in contrast to the Simple Interest (SI) studied earlier. Recall that under SI, the interest is calculated only on the original principal amount ($P$) throughout the loan or investment period ($SI = \frac{P \times R \times T}{100}$). Compound Interest, however, operates differently and more realistically reflects how many investments and loans function. In CI, the interest earned during each compounding period (e.g., a year) is added back to the principal. Consequently, the interest for the next period is calculated on this new, larger principal (original principal plus accumulated interest). This phenomenon is often described as earning "interest on interest", leading to significantly faster growth of money compared to simple interest over the same period and rate.
To calculate the final amount accumulated under compound interest, we learn a powerful formula. When the interest is compounded annually (once per year), the total Amount ($A$) after $n$ years is given by: $$ A = P \left(1 + \frac{R}{100}\right)^n $$ Where:
- $P$ = Principal (the initial sum of money)
- $R$ = Annual Rate of Interest (as a percentage)
- $n$ = Number of years
The chapter further explores scenarios where the compounding doesn't just happen annually. Interest can be compounded more frequently, such as half-yearly (twice a year) or quarterly (four times a year). In these cases, we must adjust the rate and the number of periods in the formula accordingly:
- Compounded Half-yearly: The annual rate $R$ is halved ($\frac{R}{2}$), and the number of compounding periods becomes twice the number of years ($2n$). The formula becomes $A = P \left(1 + \frac{R/2}{100}\right)^{2n}$.
- Compounded Quarterly: The annual rate $R$ is divided by four ($\frac{R}{4}$), and the number of periods becomes four times the number of years ($4n$). The formula becomes $A = P \left(1 + \frac{R/4}{100}\right)^{4n}$.
Finally, the principles underlying compound interest are extended to other real-world applications involving exponential growth or decay. Problems related to population growth often follow a similar formula, where the population increases by a certain percentage each year. Conversely, the concept of depreciation, where the value of an asset (like a car or machinery) decreases over time by a fixed percentage, can also be modeled using a formula analogous to compound interest, but involving subtraction or a negative rate within the bracket: $V_{final} = V_{initial} \left(1 - \frac{R}{100}\right)^n$. Mastering these concepts provides indispensable tools for navigating personal finance, understanding loans, investments, economic growth, inflation, and depreciation.
Percentage and its Conversion
In mathematics, comparing quantities is a frequent task. Sometimes quantities are easy to compare directly, but often we need a standard way to compare parts to a whole, or to compare different quantities relative to their original sizes. Percentage provides such a standard method, based on the concept of 'per hundred'.
Understanding Percentage
The term Percentage comes from the Latin phrase 'per centum', which means 'per hundred' or 'out of a hundred'. A percentage is a way to express a part of a whole as a number out of 100. The symbol used to represent percentage is '%'.
For example, if you score 80 marks out of 100 in a test, it means you scored 80 percent (80%). This can be written as $\frac{80}{100}$. If you score 15 marks out of 20, to express this as a percentage, you find the equivalent fraction out of 100: $\frac{15}{20} = \frac{15 \times 5}{20 \times 5} = \frac{75}{100}$, which is 75%.
Percentage is fundamentally a fraction where the denominator is 100. For any percentage value $p\%$, it represents the fraction $\frac{p}{100}$.
Percentages are used in various contexts, such as:
- Calculating interest rates on loans or savings.
- Expressing discounts on prices.
- Reporting results of surveys or elections.
- Comparing performance or proportions.
Converting Fractions, Decimals, and Ratios to Percentage
To compare quantities using percentages, we often need to convert other forms of numbers (like fractions, decimals, or ratios) into percentages.
1. Converting a Fraction to a Percentage:
Since a percentage is a fraction out of 100, to convert any fraction $\frac{a}{b}$ into a percentage, we need to find an equivalent fraction with 100 as the denominator. This is achieved by multiplying the fraction by 100 and then adding the percentage symbol $\%$.
Percentage $= (\text{Given Fraction}) \times 100\%$
... (i)
Example 1. Convert $\frac{3}{4}$ into a percentage.
Answer:
Given fraction: $\frac{3}{4}$.
Using the conversion formula:
Percentage $= \frac{3}{4} \times 100\%$
Calculate the value by multiplying the fraction by 100:
$= \frac{3}{\cancel{4}_{\normalsize 1}} \times \cancel{100}^{\normalsize 25}\%$
$= (3 \times 25)\% = 75\%$
So, the fraction $\frac{3}{4}$ is equivalent to 75%.
2. Converting a Decimal to a Percentage:
A decimal is another way of representing a fraction with a denominator as a power of 10. To convert a decimal into a percentage, multiply the decimal number by 100 and add the percentage symbol $\%$. Multiplying by 100 is equivalent to moving the decimal point two places to the right.
Percentage $= (\text{Given Decimal}) \times 100\%$
... (ii)
Example 2. Convert 0.25 into a percentage.
Answer:
Given decimal: 0.25.
Using the conversion rule:
Percentage $= (0.25 \times 100)\% = 25\%$
So, the decimal 0.25 is equivalent to 25%.
Example 3. Convert 1.5 into a percentage.
Answer:
Given decimal: 1.5.
Using the conversion rule:
Percentage $= (1.5 \times 100)\% = 150\%$
So, the decimal 1.5 is equivalent to 150%. This indicates a quantity that is more than the original whole.
3. Converting a Ratio to a Percentage:
A ratio $a:b$ is equivalent to the fraction $\frac{a}{b}$. Therefore, to convert a ratio to a percentage, first write the ratio as a fraction and then convert the fraction to a percentage by multiplying by 100%.
Ratio $a:b = \frac{a}{b}$
Percentage $= (\frac{a}{b} \times 100)\%$
... (iii)
Example 4. Convert the ratio 2:5 into a percentage.
Answer:
Given ratio: 2:5.
Write the ratio as a fraction: $\frac{2}{5}$.
Convert the fraction to a percentage:
Percentage $= (\frac{2}{5} \times 100)\%$
$= (\frac{2}{\cancel{5}_{\normalsize 1}} \times \cancel{100}^{\normalsize 20})\%$
$= (2 \times 20)\% = 40\%$
So, the ratio 2:5 is equivalent to 40%.
Converting Percentage to Fraction, Decimal, or Ratio
We can also convert a percentage back into a fraction, decimal, or ratio.
1. Converting Percentage to Fraction:
To convert a percentage $p\%$ into a fraction, remove the $\%$ sign and write the number as the numerator of a fraction with 100 as the denominator. Then, simplify the fraction to its lowest terms if required.
Percentage $p\% = \frac{p}{100}$
... (iv)
Example 5. Convert 60% into a fraction.
Answer:
Given percentage: 60%.
Remove the $\%$ sign and write as a fraction with denominator 100:
$60\% = \frac{60}{100}$
Simplify the fraction:
$\frac{60}{100} = \frac{\cancel{60}^{\normalsize 3}}{\cancel{100}_{\normalsize 5}} = \frac{3}{5}$
So, 60% is equal to the fraction $\frac{3}{5}$.
2. Converting Percentage to Decimal:
To convert a percentage $p\%$ to a decimal, remove the $\%$ sign and divide the number by 100 (which is equivalent to moving the decimal point two places to the left).
Percentage $p\% = \frac{p}{100} = p \div 100$
... (v)
Example 6. Convert 45% into a decimal.
Answer:
Given percentage: 45%.
Remove the $\%$ sign and divide by 100:
$45\% = \frac{45}{100} = 0.45$
So, 45% is equal to the decimal 0.45.
Example 7. Convert 8% into a decimal.
Answer:
Given percentage: 8%.
Remove the $\%$ sign and divide by 100:
$8\% = \frac{8}{100} = 0.08$
So, 8% is equal to the decimal 0.08.
3. Converting Percentage to Ratio:
To convert a percentage to a ratio, first convert the percentage to a fraction, and then express the simplified fraction as a ratio.
Percentage $p\% = \frac{p}{100} = \text{Fraction}$
Fraction (simplified) = Ratio $a:b$
Example 8. Convert 75% into a ratio.
Answer:
Given percentage: 75%.
Convert the percentage to a fraction:
Fraction $= \frac{75}{100}$
Simplify the fraction:
$= \frac{\cancel{75}^{\normalsize 3}}{\cancel{100}_{\normalsize 4}} = \frac{3}{4}$
Express the simplified fraction as a ratio:
The ratio is 3:4
So, 75% is equivalent to the ratio 3:4.
Finding a Percentage of Given Quantity
Percentage is often used to represent a part of a whole. For example, if a shirt has 20% discount, it means 20% of the original price is subtracted from the price. To find the actual amount of the discount, we need to calculate the percentage of the original price.
Calculating a Percentage of a Quantity
To find a percentage of a given quantity, we convert the percentage into its fractional or decimal form and then multiply it by the given quantity. Remember that $p\%$ is equivalent to the fraction $\frac{p}{100}$.
Percentage of Quantity $= (\text{Given Percentage}) \times (\text{Given Quantity})$
Using the fractional form of percentage:
Percentage of Quantity $= \frac{\text{Percentage Value}}{100} \times \text{Quantity}$
... (i)
Using the decimal form of percentage:
Percentage of Quantity $= (\text{Percentage as a Decimal}) \times \text{Quantity}$
... (ii)
Both methods give the same result. Choose the method that seems easier based on the numbers involved.
Example 1. Find 20% of $\textsf{₹}500$.
Answer:
Given percentage = 20%, Given quantity = $\textsf{₹}500$.
We need to find 20% of $\textsf{₹}500$.
Method 1: Using the fractional form.
$20\% = \frac{20}{100}$
$20\%$ of $\textsf{₹}500 = \frac{20}{100} \times 500$
[Using Formula (i)]
Simplify the expression:
$= \frac{\cancel{20}^{\normalsize 1}}{\cancel{100}_{\normalsize 5}} \times 500 = \frac{1}{5} \times 500$
$= \frac{500}{5} = 100$
Method 2: Using the decimal form.
$20\% = 0.20$
$20\%$ of $\textsf{₹}500 = 0.20 \times 500$
[Using Formula (ii)]
$= 100$
Both methods give the same result. So, 20% of $\textsf{₹}500$ is $\textsf{₹}100$.
Example 2. What is 15% of 250 kg?
Answer:
Given percentage = 15%, Given quantity = 250 kg.
We need to find 15% of 250 kg.
Using the fractional form:
$15\%$ of 250 kg $= \frac{15}{100} \times 250$
Simplify the expression:
$= \frac{15}{\cancel{100}_{\normalsize 10}} \times \cancel{250}^{\normalsize 25}$
$= \frac{15 \times 25}{10}$
Further simplification by cancelling 5 from 15 and 10, or 25 and 10:
$= \frac{3 \times \cancel{5}}{\cancel{10}_{\normalsize 2}} \times 25 = \frac{3 \times 25}{2}$
$= \frac{75}{2} = 37.5$
So, 15% of 250 kg is 37.5 kg.
Finding the Whole Quantity When a Percentage of it is Given
Sometimes, we are given the value of a certain percentage of a quantity and need to find the total (whole) quantity. We can solve such problems by setting up a simple equation.
Let the whole quantity be $Q$. If $p\%$ of $Q$ is equal to a value $V$, we can write the equation:
$p\%$ of $Q = V$
Convert the percentage to a fraction:
$\frac{p}{100} \times Q = V$
... (iii)
Now, we can solve for $Q$ by isolating it:
$Q = V \times \frac{100}{p}$
... (iv)
Example 3. 10% of a number is 60. Find the number.
Answer:
Given: 10% of a number is 60.
To Find: The number.
Solution:
Let the unknown number be $x$.
According to the problem statement, 10% of $x$ is equal to 60.
$10\%$ of $x = 60$
Using the formula $\frac{p}{100} \times Q = V$, where $p=10$, $Q=x$, and $V=60$:
$\frac{10}{100} \times x = 60$
Simplify the fraction $\frac{10}{100} = \frac{1}{10}$:
$\frac{1}{10} \times x = 60$
To isolate $x$, multiply both sides of the equation by 10 (or transpose $\frac{1}{10}$ to the RHS):
$x = 60 \times 10$
(Multiplying both sides by 10)
$x = 600$
The number is 600.
Check: Is 10% of 600 equal to 60? 10% of $600 = \frac{10}{100} \times 600 = \frac{1}{10} \times 600 = 60$. Yes. The answer is correct.
Alternatively, using the formula $Q = V \times \frac{100}{p}$ directly:
$x = 60 \times \frac{100}{10}$
$x = 60 \times 10$
$x = 600$
This gives the same result.
Finding Increase or Decrease Percentage
Percentages are widely used to describe changes in quantities, whether it's an increase or a decrease. For example, prices of goods might increase or decrease, populations might grow or shrink, or the value of an investment might go up or down. Expressing these changes as a percentage gives a standardised way to understand the relative size of the change compared to the original amount.
Percentage Increase
When a quantity increases from an original value to a new value, the percentage increase tells us the increase amount as a percentage of the original quantity. This is always calculated with respect to the initial (original) quantity.
First, calculate the actual increase in the quantity:
Increase in Quantity = New Quantity $-$ Original Quantity
... (i)
Then, express this increase as a fraction of the original quantity and multiply by 100%:
Percentage Increase $= \frac{\text{Increase in Quantity}}{\text{Original Quantity}} \times 100\%$
... (ii)
Example 1. The price of petrol in a city increased from $\textsf{₹}90$ per litre to $\textsf{₹}99$ per litre. Find the percentage increase in the price.
Answer:
Given:
- Original Price of petrol = $\textsf{₹}90$ per litre.
- New Price of petrol = $\textsf{₹}99$ per litre.
To Find:
- The percentage increase in the price.
Solution:
Step 1: Calculate the increase in price.
Increase in Price = New Price $-$ Original Price
[Using Formula (i)]
Increase in Price $= \textsf{₹}99 - \textsf{₹}90 = \textsf{₹}9$
Step 2: Calculate the percentage increase using the original price as the base.
Percentage Increase $= \frac{\text{Increase in Price}}{\text{Original Price}} \times 100\%$
[Using Formula (ii)]
Percentage Increase $= \frac{\textsf{₹}9}{\textsf{₹}90} \times 100\%$
Simplify the fraction and calculate the percentage:
$= \frac{\cancel{9}^{\normalsize 1}}{\cancel{90}_{\normalsize 10}} \times 100\%$
$= \frac{1}{10} \times 100\% = \frac{100}{10}\% = 10\%$
The percentage increase in the price of petrol is 10%.
Percentage Decrease
When a quantity decreases from an original value to a new value, the percentage decrease tells us the decrease amount as a percentage of the original quantity. Similar to percentage increase, this is always calculated with respect to the initial (original) quantity.
First, calculate the actual decrease in the quantity:
Decrease in Quantity = Original Quantity $-$ New Quantity
... (iii)
Then, express this decrease as a fraction of the original quantity and multiply by 100%:
Percentage Decrease $= \frac{\text{Decrease in Quantity}}{\text{Original Quantity}} \times 100\%$
... (iv)
Example 2. The number of students in a school decreased from 500 to 480 in a year. Find the percentage decrease in the number of students.
Answer:
Given:
- Original Number of Students = 500.
- New Number of Students = 480.
To Find:
- The percentage decrease in the number of students.
Solution:
Step 1: Calculate the decrease in the number of students.
Decrease in Number of Students = Original Number $-$ New Number
[Using Formula (iii)]
Decrease in Students $= 500 - 480 = 20$ students
Step 2: Calculate the percentage decrease using the original number of students as the base.
Percentage Decrease $= \frac{\text{Decrease in Number}}{\text{Original Number}} \times 100\%$
[Using Formula (iv)]
Percentage Decrease $= \frac{20}{500} \times 100\%$
Simplify the fraction and calculate the percentage:
$= \frac{\cancel{20}^{\normalsize 1}}{\cancel{500}_{\normalsize 25}} \times 100\%$
$= \frac{1}{25} \times 100\% = \frac{100}{25}\% = 4\%$
The percentage decrease in the number of students is 4%.
Profit & Loss and its Related Terms
In business and trade, buying and selling items is common. When we buy an item at a certain price and sell it at another price, we either make a profit (gain) or incur a loss. Understanding these concepts and how to calculate them, often in terms of percentages, is an important application of comparing quantities.
Basic Terms in Profit and Loss
Let's define the fundamental terms used in buying and selling:
- Cost Price (CP): The price at which an article or item is purchased or bought. This is the expense incurred by the buyer or seller.
- Selling Price (SP): The price at which an article or item is sold. This is the revenue received by the seller.
The comparison between the Selling Price (SP) and the Cost Price (CP) determines whether there is a profit or a loss.
- Profit (or Gain): If the Selling Price (SP) is greater than the Cost Price (CP) (SP > CP), then there is a profit. The buyer pays more than the seller spent.
- Loss: If the Selling Price (SP) is less than the Cost Price (CP) (SP < CP), then there is a loss. The buyer pays less than the seller spent.
- Neither Profit nor Loss: If the Selling Price (SP) is equal to the Cost Price (CP) (SP = CP), then there is neither profit nor loss.
The amount of profit or loss is calculated as the difference between the Selling Price and the Cost Price:
Profit = SP $-$ CP
... (i)
Loss = CP $-$ SP
... (ii)
Profit Percentage and Loss Percentage
Simply knowing the amount of profit or loss ($\textsf{₹}10$ profit) might not be as informative as knowing the profit or loss relative to the original cost. For example, a $\textsf{₹}10$ profit on an item bought for $\textsf{₹}100$ is much better than a $\textsf{₹}10$ profit on an item bought for $\textsf{₹}1000$. To provide a standard comparison, profit and loss are often expressed as a percentage of the Cost Price (CP).
Profit or loss percentage is always calculated on the Cost Price (CP), unless otherwise specified.
Profit Percentage $= \frac{\text{Profit}}{\text{CP}} \times 100\%$
... (iii)
Loss Percentage $= \frac{\text{Loss}}{\text{CP}} \times 100\%$
... (iv)
Example 1. A shopkeeper bought a chair for $\textsf{₹}500$ and sold it for $\textsf{₹}600$. Find the profit amount and the profit percentage.
Answer:
Given:
- Cost Price (CP) of the chair = $\textsf{₹}500$.
- Selling Price (SP) of the chair = $\textsf{₹}600$.
To Find:
- Profit amount and Profit percentage.
Solution:
Since SP ($\textsf{₹}600$) > CP ($\textsf{₹}500$), there is a profit.
Calculate the profit amount:
Profit = SP $-$ CP
[Using Formula (i)]
Profit $= \textsf{₹}600 - \textsf{₹}500 = \textsf{₹}100$
Calculate the profit percentage:
Profit Percentage $= \frac{\text{Profit}}{\text{CP}} \times 100\%$
[Using Formula (iii)]
Profit Percentage $= \frac{\textsf{₹}100}{\textsf{₹}500} \times 100\%$
Simplify the fraction and calculate the percentage:
$= \frac{\cancel{100}^{\normalsize 1}}{\cancel{500}_{\normalsize 5}} \times 100\%$
$= \frac{1}{5} \times 100\% = \frac{100}{5}\% = 20\%$
The profit is $\textsf{₹}100$ and the profit percentage is 20%.
Example 2. A vendor bought a bicycle for $\textsf{₹}1200$ and sold it for $\textsf{₹}1000$. Find the loss amount and the loss percentage.
Answer:
Given:
- Cost Price (CP) of the bicycle = $\textsf{₹}1200$.
- Selling Price (SP) of the bicycle = $\textsf{₹}1000$.
To Find:
- Loss amount and Loss percentage.
Solution:
Since SP ($\textsf{₹}1000$) < CP ($\textsf{₹}1200$), there is a loss.
Calculate the loss amount:
Loss = CP $-$ SP
[Using Formula (ii)]
Loss $= \textsf{₹}1200 - \textsf{₹}1000 = \textsf{₹}200$
Calculate the loss percentage:
Loss Percentage $= \frac{\text{Loss}}{\text{CP}} \times 100\%$
[Using Formula (iv)]
Loss Percentage $= \frac{\textsf{₹}200}{\textsf{₹}1200} \times 100\%$
Simplify the fraction and calculate the percentage:
$= \frac{\cancel{200}^{\normalsize 1}}{\cancel{1200}_{\normalsize 6}} \times 100\%$
$= \frac{1}{6} \times 100\% = \frac{100}{6}\% = \frac{50}{3}\%$
We can express this as a mixed number or decimal:
$\frac{50}{3} \% = 16\frac{2}{3}\%$ or $16.67\%$ (approximately)
The loss is $\textsf{₹}200$ and the loss percentage is $16\frac{2}{3}\%$.
Finding SP or CP When Profit/Loss Percentage is Given
We can also find the Selling Price (SP) or Cost Price (CP) if we know one of them and the profit or loss percentage.
Consider a Profit of $P\%$. This means the profit amount is $P\%$ of CP. Profit $= \frac{P}{100} \times \text{CP}$. Since SP = CP + Profit, SP $= \text{CP} + \frac{P}{100} \times \text{CP}$ SP $= \text{CP} \times (1 + \frac{P}{100})$
SP $= \text{CP} \times (\frac{100+P}{100})$
... (v)
CP $= \text{SP} \times (\frac{100}{100+P})$
... (vi)
Consider a Loss of $L\%$. This means the loss amount is $L\%$ of CP. Loss $= \frac{L}{100} \times \text{CP}$. Since SP = CP - Loss, SP $= \text{CP} - \frac{L}{100} \times \text{CP}$ SP $= \text{CP} \times (1 - \frac{L}{100})$
SP $= \text{CP} \times (\frac{100-L}{100})$
... (vii)
CP $= \text{SP} \times (\frac{100}{100-L})$
... (viii)
Example 3. A shopkeeper sold a table for $\textsf{₹}880$, making a profit of 10%. Find the cost price of the table.
Answer:
Given:
- Selling Price (SP) = $\textsf{₹}880$.
- Profit percentage (P%) = 10%.
To Find:
- Cost Price (CP).
Solution:
We are given SP and Profit %. We need to find CP. We can use the formula that relates CP, SP, and Profit %:
CP $= \text{SP} \times (\frac{100}{100+P})$
[Using Formula (vi)]
Substitute the given values SP = 880 and P = 10:
CP $= 880 \times (\frac{100}{100+10})$
CP $= 880 \times \frac{100}{110}$
Simplify the fraction $\frac{100}{110} = \frac{10}{11}$:
CP $= 880 \times \frac{10}{11}$
Calculate the product:
CP $= \cancel{880}^{\normalsize 80} \times \frac{10}{\cancel{11}_{\normalsize 1}}$
CP $= 80 \times 10 = 800$
The cost price of the table is $\textsf{₹}800$.
Check: If CP is $\textsf{₹}800$ and Profit is 10%, Profit amount = 10% of $\textsf{₹}800 = \frac{10}{100} \times 800 = \textsf{₹}80$. Selling Price = CP + Profit $= \textsf{₹}800 + \textsf{₹}80 = \textsf{₹}880$. This matches the given SP, so the answer is correct.
Discount and its Related Terms
In retail, it is common practice to offer a reduction on the price of an item to attract customers or clear stock. This reduction is known as a discount. Understanding discount involves relating it to the price at which the item is listed and the price at which it is finally sold.
Basic Terms in Discount
Let's define the key terms related to discount:
- Marked Price (MP): This is the price that is written or printed on the label or tag of an article. It is the price at which the item is intended to be sold before any discount is applied. Marked Price is also sometimes called the List Price.
- Discount: This is the reduction in the price offered on the Marked Price. It is the amount by which the original price is lowered.
- Selling Price (SP): This is the price at which the article is actually sold to the customer after the discount has been deducted from the Marked Price.
The relationship between Marked Price, Discount, and Selling Price is:
Selling Price (SP) = Marked Price (MP) $-$ Discount
... (i)
From this, we can also find the Discount if we know the MP and SP:
Discount = Marked Price (MP) $-$ Selling Price (SP)
... (ii)
Discount Percentage
Discount is typically offered and expressed as a percentage of the Marked Price. Calculating the discount percentage provides a standard measure of the reduction offered, regardless of the actual price.
Discount percentage is always calculated on the Marked Price (MP).
Discount Percentage $= \frac{\text{Discount Amount}}{\text{Marked Price (MP)}} \times 100\%$
... (iii)
Example 1. A shirt is marked at $\textsf{₹}1200$. A customer gets a discount of $\textsf{₹}240$. Find the discount percentage.
Answer:
Given:
- Marked Price (MP) of the shirt = $\textsf{₹}1200$.
- Discount Amount = $\textsf{₹}240$.
To Find:
- Discount Percentage.
Solution:
We have the Discount Amount and the Marked Price. We can directly use the formula for Discount Percentage:
Discount Percentage $= \frac{\text{Discount}}{\text{MP}} \times 100\%$
[Using Formula (iii)]
Substitute the given values:
Discount Percentage $= \frac{\textsf{₹}240}{\textsf{₹}1200} \times 100\%$
Simplify the fraction and calculate the percentage:
$= \frac{\cancel{240}^{\normalsize 1}}{\cancel{1200}_{\normalsize 5}} \times 100\%$
$= \frac{1}{5} \times 100\% = \frac{100}{5}\% = 20\%$
The discount percentage is 20%.
From this, we can also find the Selling Price (SP) if needed: SP = MP - Discount Amount $= \textsf{₹}1200 - \textsf{₹}240 = \textsf{₹}960$.
Finding SP or MP When Discount Percentage is Given
If the discount percentage ($D\%$) is given, we can find the Selling Price (SP) directly from the Marked Price (MP), or vice versa, using the concept that the SP is the remaining percentage of the MP after the discount is applied.
If a discount of $D\%$ is offered on MP, the customer pays $(100 - D)\%$ of the MP.
So, Selling Price $= (100 - D)\%$ of Marked Price.
Converting the percentage to a fraction:
SP $= \frac{100-D}{100} \times \text{MP}$
... (iv)
From this formula, we can also find the Marked Price if we know the Selling Price and the Discount Percentage:
MP $= \text{SP} \times \frac{100}{100-D}$
... (v)
Example 2. An article is marked at $\textsf{₹}1500$. A discount of 12% is offered on it. Find the selling price of the article.
Answer:
Given:
- Marked Price (MP) = $\textsf{₹}1500$.
- Discount Percentage (D%) = 12%.
To Find:
- Selling Price (SP).
Solution:
The discount is 12%. This means the selling price is $(100 - 12)\%$ of the Marked Price, which is 88% of the MP.
SP $= (100 - 12)\%$ of $\textsf{₹}1500$
SP $= 88\%$ of $\textsf{₹}1500$
Using the formula SP $= \frac{100-D}{100} \times \text{MP}$:
SP $= \frac{88}{100} \times 1500$
[Using Formula (iv)]
Simplify and calculate:
SP $= \frac{88}{\cancel{100}_{\normalsize 1}} \times \cancel{1500}^{\normalsize 15}$
SP $= 88 \times 15$
Perform the multiplication:
$88 \times 15 = 88 \times (10 + 5) = 88 \times 10 + 88 \times 5 = 880 + 440 = 1320$
The selling price of the article is $\textsf{₹}1320$.
Check: Discount amount = 12% of $\textsf{₹}1500 = \frac{12}{100} \times 1500 = 12 \times 15 = \textsf{₹}180$. Selling Price = Marked Price - Discount Amount $= \textsf{₹}1500 - \textsf{₹}180 = \textsf{₹}1320$. This matches the calculated SP.
Goods and Services Tax (GST)
In our daily transactions, especially when buying goods or services, we often see an amount added to the price as tax. In India, a significant tax on the supply of goods and services is the Goods and Services Tax (GST). GST is a single, comprehensive tax levied on the supply of goods and services from the manufacturer/seller to the final consumer. It has replaced many indirect taxes previously levied by the Central and State governments, such as Value Added Tax (VAT), Sales Tax, Excise Duty, etc.
Understanding GST
GST is a value-added tax, meaning it is levied at each stage of production and distribution, but the tax is calculated only on the 'value added' at each stage. However, for the final consumer, GST is simply added to the final selling price of the goods or services.
The amount of GST is calculated as a percentage of the taxable value (which is often the price of the item before tax). This tax amount is then added to the price to arrive at the final amount paid by the customer.
GST rates are decided by the government and vary for different categories of goods and services. Common GST rates in India include 5%, 12%, 18%, and 28%.
Calculation of GST Amount and Bill Amount:
If the price of an article before adding tax is the Taxable Value (or sometimes referred to as Selling Price in contexts where discount is already applied), and the GST Rate is $G\%$, then:
Amount of GST $= \text{GST Rate Percentage} \times \text{Taxable Value}$
Using the percentage as a fraction:
Amount of GST $= \frac{G}{100} \times \text{Taxable Value}$
... (i)
The final amount paid by the customer, which is the Bill Amount (Price including GST), is the sum of the Taxable Value and the Amount of GST.
Bill Amount = Taxable Value + Amount of GST
... (ii)
Substituting the Amount of GST from Formula (i) into Formula (ii):
Bill Amount = Taxable Value $+ \frac{G}{100} \times \text{Taxable Value}$
Bill Amount = Taxable Value $\times (1 + \frac{G}{100})$
Bill Amount = Taxable Value $\times (\frac{100+G}{100})$
... (iii)
This formula directly calculates the Bill Amount if you know the Taxable Value and the GST Rate.
Example 1. A television was bought at a price of $\textsf{₹}18000$ before GST was added. If a GST of 12% is charged, find the final bill amount.
Answer:
Given:
- Price before GST (Taxable Value) = $\textsf{₹}18000$.
- GST Rate (G%) = 12%.
To Find:
- The final bill amount.
Solution:
Method 1: Calculate the GST amount first.
Amount of GST = GST Rate \% $\times$ Taxable Value
Amount of GST $= 12\%$ of $\textsf{₹}18000$
$= \frac{12}{100} \times 18000$
Simplify and calculate:
$= \frac{12}{\cancel{100}_{\normalsize 1}} \times \cancel{18000}^{\normalsize 180}$
$= 12 \times 180$
$= 2160$
[Using multiplication: $12 \times 18 = 216$, then $\times 10$]
Amount of GST = $\textsf{₹}2160$.
Now, calculate the Bill Amount:
Bill Amount = Taxable Value + Amount of GST
[Using Formula (ii)]
Bill Amount $= \textsf{₹}18000 + \textsf{₹}2160 = \textsf{₹}20160$
The final bill amount is $\textsf{₹}20160$.
Alternate Method:
Using the formula Bill Amount = Taxable Value $\times (\frac{100+G}{100})$:
Bill Amount $= 18000 \times (\frac{100+12}{100})$
[Using Formula (iii)]
Bill Amount $= 18000 \times \frac{112}{100}$
Simplify and calculate:
Bill Amount $= \cancel{18000}^{\normalsize 180} \times \frac{112}{\cancel{100}_{\normalsize 1}}$
Bill Amount $= 180 \times 112$
Perform the multiplication $180 \times 112$:
$180 \times 112 = 18 \times 10 \times 112 = 18 \times 1120$
Multiply $18 \times 1120$:
Bill Amount $= 20160$
The final bill amount is $\textsf{₹}20160$. Both methods yield the same result.
Finding Price Before GST When Bill Amount is Given
Sometimes, the final bill amount (including GST) is known, and you need to find the original price before GST was added (the Taxable Value). We can use the formula (iii) from above and rearrange it to solve for the Taxable Value.
Bill Amount = Taxable Value $\times (\frac{100+G}{100})$
Rearranging to find Taxable Value:
Taxable Value = Bill Amount $\times (\frac{100}{100+G})$
... (iv)
Example 2. Rahul bought an article for $\textsf{₹}1344$, which included 12% GST. Find the price of the article before GST was added.
Answer:
Given:
- Bill Amount (Price including GST) = $\textsf{₹}1344$.
- GST Rate (G%) = 12%.
To Find:
- Price before GST (Taxable Value).
Solution:
We are given the Bill Amount and the GST Rate. We need to find the price before GST. We can use the formula that relates Bill Amount, Taxable Value, and GST Rate:
Taxable Value = Bill Amount $\times (\frac{100}{100+G})$
[Using Formula (iv)]
Substitute the given values Bill Amount = 1344 and G = 12:
Taxable Value $= 1344 \times (\frac{100}{100+12})$
Taxable Value $= 1344 \times \frac{100}{112}$
Calculate the value by simplifying and multiplying:
Taxable Value $= \frac{134400}{112}$
Let's perform the division $1344 \div 112$. We can do long division or notice common factors. $112 \times 10 = 1120$. $1344 - 1120 = 224$. $112 \times 2 = 224$. So, $112 \times 12 = 1344$.
Taxable Value $= 12 \times 100 = 1200$
The price of the article before GST was added is $\textsf{₹}1200$.
Check: If the price before GST is $\textsf{₹}1200$ and GST Rate is 12%, Amount of GST = 12% of $\textsf{₹}1200 = \frac{12}{100} \times 1200 = 12 \times 12 = \textsf{₹}144$. Bill Amount = Price before GST + Amount of GST $= \textsf{₹}1200 + \textsf{₹}144 = \textsf{₹}1344$. This matches the given Bill Amount, so the answer is correct.
Simple Interest and Compound Interest
Money is a resource that can grow over time, especially when it is lent or invested. When you borrow money from a bank or a person, you usually have to pay back more than the amount you borrowed. The extra money paid is called interest. Similarly, if you deposit money in a bank or invest it, you may earn extra money over time, which is also called interest. This section introduces two main types of interest: Simple Interest and Compound Interest.
Basic Terms Related to Interest
Before we discuss the types of interest, let's understand some basic terms:
- Principal (P): This is the initial sum of money borrowed or lent or invested. It is the original amount on which interest is calculated.
- Interest: This is the extra money paid by the borrower for using the principal amount, or earned by the lender/investor. It is typically calculated as a percentage of the principal amount per unit of time.
- Time Period (T): This is the duration for which the principal is borrowed, lent, or invested. It is usually specified in years, but can also be in months or days, in which case it needs to be converted to years for calculations using standard formulas.
- Rate of Interest (R): This is the percentage at which the interest is calculated on the principal. It is usually given "per annum" (p.a.) or "per year", meaning the percentage is for one full year. For example, 10% p.a. means 10% of the principal is charged or earned as interest each year.
- Amount (A): This is the total sum paid back by the borrower to the lender, or received by the investor at the end of the time period. It includes the original principal plus the total interest accumulated over the time period.
Amount = Principal + Interest
... (i)
Simple Interest (SI)
Simple Interest is the interest calculated only on the original principal amount for the entire duration of the loan or investment. In simple interest, the interest earned in previous periods is not added to the principal for calculating interest in the current period. The interest amount remains constant for each unit of time (e.g., each year).
The formula for calculating Simple Interest is:
SI $= \frac{P \times R \times T}{100}$
... (ii)
Where:
- P = Principal amount
- R = Rate of Interest per annum (as a percentage, so R is the number, not R/100)
- T = Time Period (in years)
The Total Amount (A) to be paid back or received at the end of T years is the Principal plus the Simple Interest:
Amount (A) = P + SI
Substituting the formula for SI:
A $= P + \frac{P \times R \times T}{100}$
We can also factor out P:
A $= P (1 + \frac{RT}{100})$
... (iii)
Example 1. Find the simple interest on $\textsf{₹}10000$ at 12% per annum for 3 years. Also, find the total amount to be paid back.
Answer:
Given:
- Principal (P) = $\textsf{₹}10000$.
- Rate of Interest (R) = 12% p.a. (So, R = 12 in the formula).
- Time Period (T) = 3 years.
To Find:
- Simple Interest (SI) and Total Amount (A).
Solution:
Calculate the Simple Interest using the formula SI $= \frac{P \times R \times T}{100}$:
SI $= \frac{10000 \times 12 \times 3}{100}$
Simplify and calculate:
SI $= \frac{\cancel{10000}^{\normalsize 100} \times 12 \times 3}{\cancel{100}_{\normalsize 1}}$
SI $= 100 \times 12 \times 3$
SI $= 100 \times 36 = 3600$
The simple interest is $\textsf{₹}3600$.
Calculate the Total Amount using the formula A = P + SI:
Amount (A) = $\textsf{₹}10000 + \textsf{₹}3600 = \textsf{₹}13600$
The total amount to be paid back after 3 years is $\textsf{₹}13600$.
Compound Interest (CI)
Compound Interest is different from simple interest because the interest earned in each period is added to the principal for the next period. This means that the interest for the subsequent periods is calculated on a larger amount (original principal + accumulated interest), resulting in interest earning interest. This effect is called compounding.
In compound interest, the principal does not remain constant throughout the time period. It increases at the end of each compounding period (e.g., annually, half-yearly, quarterly) by the interest earned during that period.
The calculation of compound interest depends on the compounding period, which is how frequently the interest is added to the principal. If the interest is compounded annually, interest is calculated and added once a year. If compounded half-yearly, it's calculated and added every six months, and so on.
The formula for calculating the total amount and compound interest will be discussed in the next section.
Formula for Compound Interest
We have learned that compound interest involves adding the earned interest to the principal at the end of each compounding period. This growing principal then earns interest in the next period, leading to faster growth of the investment or debt compared to simple interest. Let's derive and use the formula for calculating the Amount and Compound Interest.
When Interest is Compounded Annually
Let P be the Principal amount, R be the rate of interest per annum (p.a.), and n be the time period in years. If the interest is compounded annually, it means interest is calculated and added to the principal at the end of each year.
Let's see how the amount grows year by year:
- Amount at the end of 1st year: Interest for 1st year $= P \times \frac{R}{100}$ Amount after 1st year $= P + P \frac{R}{100} = P (1 + \frac{R}{100})$
- Amount at the end of 2nd year: The principal for the 2nd year is the amount at the end of the 1st year, i.e., $P (1 + \frac{R}{100})$. Interest for 2nd year $= [P (1 + \frac{R}{100})] \times \frac{R}{100}$ Amount after 2nd year $= P (1 + \frac{R}{100}) + [P (1 + \frac{R}{100})] \frac{R}{100}$ $= P (1 + \frac{R}{100}) (1 + \frac{R}{100}) = P (1 + \frac{R}{100})^2$
- Amount at the end of 3rd year: The principal for the 3rd year is the amount at the end of the 2nd year, $P (1 + \frac{R}{100})^2$. Amount after 3rd year $= [P (1 + \frac{R}{100})^2] (1 + \frac{R}{100}) = P (1 + \frac{R}{100})^3$
Following this pattern, the formula for the Amount (A) after $n$ years, when interest is compounded annually, is:
A $= P (1 + \frac{R}{100})^n$
... (i)
Where:
- P = Principal
- R = Annual Rate of Interest (as a number, not a fraction or decimal)
- n = Time Period (in years)
The Compound Interest (CI) is the total interest earned over the $n$ years, which is the difference between the final Amount and the original Principal.
CI = Amount $-$ Principal
CI $= P (1 + \frac{R}{100})^n - P$
... (ii)
Example 1. Find the compound interest on $\textsf{₹}10000$ at 10% per annum for 2 years, compounded annually. Also find the amount.
Answer:
Given:
- Principal (P) = $\textsf{₹}10000$.
- Rate of Interest (R) = 10% p.a. (So, R = 10).
- Time Period (n) = 2 years.
- Compounding Frequency: Annually.
To Find:
- Amount (A) and Compound Interest (CI).
Solution:
Calculate the Amount after 2 years using the formula A $= P (1 + \frac{R}{100})^n$:
A $= 10000 (1 + \frac{10}{100})^2$
[Using Formula (i)]
Simplify the term inside the bracket:
A $= 10000 (1 + \frac{1}{10})^2$
A $= 10000 (\frac{10+1}{10})^2 = 10000 (\frac{11}{10})^2$
Calculate the square of the fraction:
A $= 10000 \times \frac{11^2}{10^2} = 10000 \times \frac{121}{100}$
Simplify and calculate:
A $= \cancel{10000}^{\normalsize 100} \times \frac{121}{\cancel{100}_{\normalsize 1}}$
A $= 100 \times 121 = 12100$
The amount after 2 years is $\textsf{₹}12100$.
Calculate the Compound Interest using the formula CI = A - P:
CI $= \textsf{₹}12100 - \textsf{₹}10000 = \textsf{₹}2100$
The compound interest is $\textsf{₹}2100$.
When Interest is Compounded Half-Yearly
If the interest is compounded half-yearly (meaning twice a year), the interest is calculated and added to the principal every six months. In this case, the time period needs to be expressed in terms of half-years, and the annual rate needs to be converted to a half-yearly rate.
If the annual rate is R% p.a., the rate per half-year is $\frac{R}{2}\%$.
If the total time period is $n$ years, there are $n \times 2 = 2n$ half-yearly periods.
We use the same compound interest formula structure, but substitute the half-yearly rate and the number of half-yearly periods:
Amount (A) $= P (1 + \frac{\text{Rate per period}}{100})^{\text{Number of periods}}$
A $= P (1 + \frac{R/2}{100})^{2n}$
... (iii)
Example 2. Find the amount and compound interest on $\textsf{₹}8000$ at 10% per annum for 1 year, compounded half-yearly.
Answer:
Given:
- Principal (P) = $\textsf{₹}8000$.
- Annual Rate (R) = 10% p.a.
- Time Period = 1 year.
- Compounding Frequency: Half-yearly.
To Find:
- Amount (A) and Compound Interest (CI).
Solution:
Since interest is compounded half-yearly:
- Rate per half-year $= \frac{\text{Annual Rate}}{2} = \frac{10\%}{2} = 5\%$. (So, use 5 as R in the formula with compounding periods).
- Number of half-years in 1 year $= 1 \text{ year} \times 2 \text{ half-years/year} = 2$ half-years. (So, use 2 as n in the formula).
Calculate the Amount using the formula A $= P (1 + \frac{\text{Rate per period}}{100})^{\text{Number of periods}}$:
A $= 8000 (1 + \frac{5}{100})^2$
[Using Formula (iii) with R=5, n=2]
Simplify the term inside the bracket:
A $= 8000 (1 + \frac{1}{20})^2$
A $= 8000 (\frac{20+1}{20})^2 = 8000 (\frac{21}{20})^2$
Calculate the square of the fraction:
A $= 8000 \times \frac{21 \times 21}{20 \times 20} = 8000 \times \frac{441}{400}$
Simplify and calculate:
A $= \cancel{8000}^{\normalsize 20} \times \frac{441}{\cancel{400}_{\normalsize 1}}$
A $= 20 \times 441$
A $= 8820$
The amount after 1 year (compounded half-yearly) is $\textsf{₹}8820$.
Calculate the Compound Interest using the formula CI = A - P:
CI $= \textsf{₹}8820 - \textsf{₹}8000 = \textsf{₹}820$
The compound interest is $\textsf{₹}820$.
When Interest is Compounded Quarterly
If the interest is compounded quarterly (meaning four times a year), the interest is calculated and added to the principal every three months. The time period needs to be expressed in terms of quarters, and the annual rate needs to be converted to a quarterly rate.
If the annual rate is R% p.a., the rate per quarter is $\frac{R}{4}\%$.
If the total time period is $n$ years, there are $n \times 4 = 4n$ quarterly periods.
The formula for the Amount (A) after $n$ years, when interest is compounded quarterly, is:
A $= P (1 + \frac{R/4}{100})^{4n}$
... (iv)
Example 3. Find the amount and compound interest on $\textsf{₹}4000$ at 8% per annum for 9 months, compounded quarterly.
Answer:
Given:
- Principal (P) = $\textsf{₹}4000$.
- Annual Rate (R) = 8% p.a.
- Time Period = 9 months.
- Compounding Frequency: Quarterly.
To Find:
- Amount (A) and Compound Interest (CI).
Solution:
Since interest is compounded quarterly:
- Rate per quarter $= \frac{\text{Annual Rate}}{4} = \frac{8\%}{4} = 2\%$. (So, use 2 as the rate per period).
- Number of quarters in 9 months. There are 3 months in a quarter. Number of quarters $= \frac{9 \text{ months}}{3 \text{ months/quarter}} = 3$ quarters. (So, use 3 as the number of periods).
Calculate the Amount using the formula A $= P (1 + \frac{\text{Rate per period}}{100})^{\text{Number of periods}}$:
A $= 4000 (1 + \frac{2}{100})^3$
[Using Formula (iv) structure with R=2, n=3]
Simplify the term inside the bracket:
A $= 4000 (1 + \frac{1}{50})^3$
A $= 4000 (\frac{50+1}{50})^3 = 4000 (\frac{51}{50})^3$
Calculate the cube of the fraction:
A $= 4000 \times \frac{51 \times 51 \times 51}{50 \times 50 \times 50} = 4000 \times \frac{51^3}{125000}$
Simplify and calculate. We can simplify the fraction $\frac{4000}{125000}$. Divide both by 1000: $\frac{4}{125}$.
A $= \cancel{4000}^{\normalsize 4} \times \frac{51^3}{\cancel{125000}_{\normalsize 125}}$
A $= \frac{4 \times 51^3}{125} = \frac{4 \times 132651}{125} = \frac{530604}{125}$
Perform the division:
$\frac{530604}{125} = 4244.832$
The amount after 9 months (compounded quarterly) is $\textsf{₹}4244.83$ (rounding to two decimal places). For currency, we usually round to two decimal places unless specified otherwise.
Calculate the Compound Interest using the formula CI = A - P:
CI $= \textsf{₹}4244.83 - \textsf{₹}4000 = \textsf{₹}244.83$
The compound interest is $\textsf{₹}244.83$.
When Time Period is Not a Whole Number
If the time period is a fraction, like $n \frac{p}{q}$ years, where $n$ is a whole number and $\frac{p}{q}$ is a fraction (and compounding is annual), the interest is compounded for the whole number of years $n$, and then simple interest is calculated on the amount obtained after $n$ years for the remaining fractional part $\frac{p}{q}$ years.
Amount (A) after $n \frac{p}{q}$ years (compounded annually):
A $= P (1 + \frac{R}{100})^n \times (1 + \frac{\frac{p}{q} \times R}{100})$
... (v)
Here, $(1 + \frac{R}{100})^n$ calculates the amount after $n$ full years compounding. Then, $(1 + \frac{\frac{p}{q} \times R}{100})$ calculates the simple interest for the fractional part of the year on this accumulated amount.
Example 4. Find the compound interest on $\textsf{₹}10000$ at 10% per annum for $2\frac{1}{2}$ years, compounded annually. Also find the amount.
Answer:
Given:
- Principal (P) = $\textsf{₹}10000$.
- Annual Rate (R) = 10% p.a.
- Time Period = $2\frac{1}{2}$ years. Here, $n=2$ (whole years) and $\frac{p}{q} = \frac{1}{2}$ (fractional part).
- Compounding Frequency: Annually.
To Find:
- Amount (A) and Compound Interest (CI).
Solution:
We calculate the amount for the whole years first, and then simple interest on that amount for the fractional part.
Step 1: Calculate the Amount after 2 full years compounded annually.
Amount after 2 years $= P (1 + \frac{R}{100})^2$
[Using Formula (i) for n=2]
Amount after 2 years $= 10000 (1 + \frac{10}{100})^2 = 10000 (\frac{110}{100})^2 = 10000 (\frac{11}{10})^2$
$= 10000 \times \frac{121}{100} = \cancel{10000}^{\normalsize 100} \times \frac{121}{\cancel{100}_{\normalsize 1}} = 100 \times 121 = \textsf{₹}12100$
Step 2: Calculate Simple Interest on the amount from Step 1 for the remaining fractional time period ($\frac{1}{2}$ year) at the annual rate R.
Principal for SI calculation = $\textsf{₹}12100$
Rate (R) = 10% p.a.
Time (T) $= \frac{1}{2}$ year.
Simple Interest for last $\frac{1}{2}$ year $= \frac{P \times R \times T}{100}$
[Using SI Formula]
$= \frac{12100 \times 10 \times \frac{1}{2}}{100} = \frac{12100 \times 10}{100 \times 2}$
$= \frac{\cancel{12100}^{\normalsize 121} \times \cancel{10}^{\normalsize 5}}{\cancel{100}_{\normalsize 1} \times \cancel{2}_{\normalsize 1}} = 121 \times 5 = 605$
Interest for the last $\frac{1}{2}$ year is $\textsf{₹}605$.
Step 3: Calculate the Total Amount and Total Compound Interest.
Total Amount after $2\frac{1}{2}$ years = Amount after 2 years + SI for last $\frac{1}{2}$ year
$= \textsf{₹}12100 + \textsf{₹}605 = \textsf{₹}12705$
Total Compound Interest = Total Amount $-$ Original Principal
CI $= \textsf{₹}12705 - \textsf{₹}10000 = \textsf{₹}2705$
The amount is $\textsf{₹}12705$ and the compound interest is $\textsf{₹}2705$.
Using Formula (v):
A $= P (1 + \frac{R}{100})^n (1 + \frac{\frac{p}{q} \times R}{100})$
A $= 10000 (1 + \frac{10}{100})^2 (1 + \frac{\frac{1}{2} \times 10}{100})$
A $= 10000 (1 + \frac{1}{10})^2 (1 + \frac{5}{100})$
A $= 10000 (\frac{11}{10})^2 (1 + \frac{1}{20})$
A $= 10000 \times (\frac{121}{100}) \times (\frac{21}{20})$
A $= \cancel{10000}^{\normalsize 100} \times \frac{121}{\cancel{100}_{\normalsize 1}} \times \frac{21}{20} = 100 \times 121 \times \frac{21}{20}$
A $= \cancel{100}^{\normalsize 5} \times 121 \times \frac{21}{\cancel{20}_{\normalsize 1}} = 5 \times 121 \times 21$
A $= 5 \times (121 \times 21)$
A $= 5 \times 2541 = 12705$
The amount is $\textsf{₹}12705$.
CI $= \textsf{₹}12705 - \textsf{₹}10000 = \textsf{₹}2705$
This confirms the previous calculation.
When Rates are Different for Different Years
If the annual rate of interest changes from year to year, the compound interest formula is modified to reflect the different rates for each period. If the principal is P, and the rates for the 1st, 2nd, 3rd, ..., $n$-th year are $R_1\%, R_2\%, R_3\%, ..., R_n\%$ respectively, and compounding is annual, the Amount (A) after $n$ years is the product of the principal and the growth factors for each year:
A $= P (1 + \frac{R_1}{100}) (1 + \frac{R_2}{100}) \times \dots \times (1 + \frac{R_n}{100})$
... (vi)
Note that each rate is divided by 100 before adding to 1.
Example 5. Find the amount on $\textsf{₹}20000$ for 2 years if the rate of interest is 10% p.a. for the first year and 12% p.a. for the second year, compounded annually.
Answer:
Given:
- Principal (P) = $\textsf{₹}20000$.
- Rate for 1st year ($R_1$) = 10% p.a.
- Rate for 2nd year ($R_2$) = 12% p.a.
- Time = 2 years.
- Compounding Frequency: Annually.
To Find:
- Amount (A) after 2 years.
Solution:
Since the rates are different for each year, use the formula for different rates:
A $= P (1 + \frac{R_1}{100}) (1 + \frac{R_2}{100})$
[Using Formula (vi) for n=2]
Substitute the given values P=20000, R1=10, R2=12:
A $= 20000 (1 + \frac{10}{100}) (1 + \frac{12}{100})$
Simplify the terms inside the brackets:
A $= 20000 (\frac{110}{100}) (\frac{112}{100})$
Simplify the fractions:
A $= 20000 \times \frac{11}{10} \times \frac{112}{100}$
Perform the multiplication:
A $= \cancel{20000}^{\normalsize 200} \times \frac{11}{\cancel{10}_{\normalsize 1}} \times \frac{112}{\cancel{100}_{\normalsize 10}}$
A $= 200 \times 11 \times \frac{112}{10}$
A $= \cancel{200}^{\normalsize 20} \times 11 \times \frac{112}{\cancel{10}_{\normalsize 1}}$
A $= 20 \times 11 \times 112 = 220 \times 112$
The amount after 2 years is $\textsf{₹}24640$.
Compound Interest = Amount - Principal = $\textsf{₹}24640 - \textsf{₹}20000 = \textsf{₹}4640$.
Application of Compound Interest in Growth and Depreciation
The concept of compound interest is not limited to financial calculations involving loans and investments. It is a powerful mathematical model that can be applied to any situation where a quantity increases or decreases at a constant rate over successive periods. This includes phenomena like population growth, the increase or decrease in the value of assets (like vehicles or property), the growth of bacteria, etc.
General Formula for Growth and Depreciation
The formula for compound interest, $A = P (1 + \frac{R}{100})^n$, can be adapted to model growth and depreciation:
In general, if an original quantity $P_0$ changes at a fixed rate of $R\%$ per period for $n$ periods, the final quantity $P_n$ after $n$ periods is given by:
Final Quantity $= \text{Original Quantity} \times (1 + \frac{\text{Rate}}{100})^{\text{Number of Periods}}$
Symbolically:
$P_n = P_0 (1 + \frac{R}{100})^n$
... (i)
Here:
- $P_0$ = Original quantity (similar to Principal)
- $P_n$ = Quantity after $n$ periods (similar to Amount)
- $R$ = Rate of change per period (as a percentage)
- $n$ = Number of periods
The term $(1 + \frac{R}{100})$ is the growth factor per period.
If the quantity is increasing (Growth), $R$ is a positive value, and the factor is $(1 + \frac{R}{100})$.
If the quantity is decreasing (Depreciation or Decay), $R$ is the rate of decrease, and the factor is $(1 - \frac{R}{100})$. In this case, the formula becomes:
$P_n = P_0 (1 - \frac{R}{100})^n$
... (ii)
Applications
1. Population Growth:
If the population of a place increases at a rate of R% per annum, this is a case of growth compounded annually (assuming the rate is applied to the population each year). The population after $n$ years can be found using the growth formula.
Population after $n$ years $= \text{Present Population} \times (1 + \frac{\text{Rate of Increase}}{100})^n$
... (iii)
If you are asked to find the population $n$ years ago, given the present population and the annual growth rate, the present population is the "Amount" and the population $n$ years ago is the "Principal".
Present Population $= \text{Population } n \text{ years ago} \times (1 + \frac{\text{Rate}}{100})^n$
You can then rearrange this to find the Population $n$ years ago.
Example 1. The population of a town was 20000 in the year 2018. It increased at the rate of 5% per annum. Find the population at the end of the year 2020.
Answer:
Given:
- Original Population (in 2018) ($P_0$) = 20000.
- Rate of increase (R) = 5% per annum.
- Time period (n) = From the end of 2018 to the end of 2020 is 2 years (2019, 2020).
To Find:
- Population at the end of the year 2020 ($P_n$).
Solution:
Use the population growth formula, which is based on compound interest:
Population after $n$ years $= \text{Present Population} \times (1 + \frac{R}{100})^n$
[Using Formula (iii)]
Substitute the values P0 = 20000, R = 5, and n = 2:
Population at the end of 2020 $= 20000 \times (1 + \frac{5}{100})^2$
Simplify the term inside the bracket:
$= 20000 \times (1 + \frac{1}{20})^2 = 20000 \times (\frac{20+1}{20})^2 = 20000 \times (\frac{21}{20})^2$
Calculate the square of the fraction:
$= 20000 \times \frac{21 \times 21}{20 \times 20} = 20000 \times \frac{441}{400}$
Simplify and calculate:
$= \cancel{20000}^{\normalsize 50} \times \frac{441}{\cancel{400}_{\normalsize 1}}$
$= 50 \times 441$
Multiply $50 \times 441$:
$50 \times 441 = 5 \times 10 \times 441 = 5 \times 4410 = 22050$
The population at the end of the year 2020 is 22050.
2. Depreciation (Decrease in Value):
The value of items like machines, vehicles, or electronics often decreases over time due to wear and tear, age, or technological advancements. This decrease in value is called depreciation. If the value depreciates at a fixed percentage rate per period (e.g., per year), we can use a formula similar to the compound interest formula, but with a negative sign in the factor, as the quantity is decreasing.
Value after $n$ years $= \text{Original Value} \times (1 - \frac{\text{Rate of Depreciation}}{100})^n$
... (iv)
Example 2. The value of a machine depreciates at the rate of 10% per annum. If its present value is $\textsf{₹}50000$, find its value after 2 years.
Answer:
Given:
- Original Value (Present Value) ($P_0$) = $\textsf{₹}50000$.
- Rate of depreciation (R) = 10% per annum.
- Time period (n) = 2 years.
To Find:
- Value of the machine after 2 years ($P_n$).
Solution:
Since the value is depreciating (decreasing), we use the depreciation formula:
Value after $n$ years $= \text{Original Value} \times (1 - \frac{R}{100})^n$
[Using Formula (iv)]
Substitute the values P0 = 50000, R = 10, and n = 2:
Value after 2 years $= 50000 \times (1 - \frac{10}{100})^2$
Simplify the term inside the bracket:
$= 50000 \times (1 - \frac{1}{10})^2 = 50000 \times (\frac{10-1}{10})^2 = 50000 \times (\frac{9}{10})^2$
Calculate the square of the fraction:
$= 50000 \times \frac{9 \times 9}{10 \times 10} = 50000 \times \frac{81}{100}$
Simplify and calculate:
$= \cancel{50000}^{\normalsize 500} \times \frac{81}{\cancel{100}_{\normalsize 1}}$
$= 500 \times 81$
Multiply $500 \times 81$:
$500 \times 81 = 5 \times 100 \times 81 = 5 \times 8100 = 40500$
The value of the machine after 2 years will be $\textsf{₹}40500$.
The compound interest formula provides a versatile model for understanding exponential growth and decay in various real-world situations.