CBSE Notes Class 7 Maths Chapter 7 - Comparing Quantities

Comparing Quantities involves using ratios, percentages, and proportions to compare two or more values. A ratio shows how many times one value contains another, while percentages express a value as a fraction of 100. Proportion compares two equal ratios. Concepts like profit, loss, and simple interest help in comparing financial quantities effectively.

1.0Comparing Quantities: An Introduction

To compare two quantities, ensure both quantities are expressed in the same units. Here are some examples: 

1. Joe’s height is 150 cm, while Tom’s height is 100 cm. The ratio of Joe's height to Tom's is:

$\text{ Ratio }=\frac{150}{100}=\frac{3}{2}\text{ or }3:2$

2. The ratio of 3 km to 30 m is:

$\frac{3\mathrm{km}}{30\mathrm{m}}=\frac{3000\mathrm{m}}{30\mathrm{m}}=100:1$

2.0Ratios

A ratio compares two quantities by showing how many times one value contains or is contained within the other. For instance, if there are 4 girls and 7 boys in a class, the ratio of girls to boys is 4:7.

Equivalent Ratios:

Multiplying both the numerator and denominator of a ratio by the same non-zero number gives an equivalent ratio. For example:

$\frac{2}{3}\text{ and }\frac{4}{6}$ are equivalent ratios.

3.0Proportions

When two ratios are equal, they are said to be in proportion. This can be represented as either "::" or "=". For example:

$2:3::6:9\text{ or }\frac{2}{3}=\frac{6}{9}\text{. }$

4.0Calculating Percentage Increase or Decrease

To find the percentage change, use the following formula:

$\text{ Percentage Change }=\frac{\text{ Change in Value }}{\text{ Original Value }}\times 100$

For example, if the price of a book increases from ₹20 to ₹25, the percentage increase is:

Change = 25 – 20 = 5, 

$\text{ Percentage Increase }=\frac{5}{20}\times 100=25\%$

5.0Percentages

Percentages represent a ratio as a fraction of 100. For example:

$40\%=\frac{40}{100},25\%=\frac{25}{100}$

If the denominator is not 100, convert the fraction to one that has a denominator of 100. For example:

$\frac{3}{5}=\frac{60}{100}=60\%$

6.0Converting Between Decimals, Fractions, and Percentages 

• To change a decimal into a percentage, multiply the by 100. For example, 0.44 becomes 44%.
• To convert a fraction to a percentage, first make its denominator 100 and then express it as a percentage.

7.0Estimation and Interpretation Using Percentages

Percentages help interpret data more effectively. For example, if 60% of 200 chocolates were given to Joe and 40% to Tom:

$\text{ Joe’s share }=\frac{60}{100}\times 200=120\text{ chocolates, Tom’s share }=\frac{40}{100}\times 200=80\text{ chocolates. }$

Ratios can also be expressed as percentages to enhance understanding of certain situations.

1. Profit Percentage:

Profit Percentage is the percentage of profit made on the cost price of an item. It is calculated using the formula:

$\text{ Profit Percentage }=\left(\frac{\text{ Profit }}{\text{ Cost Price }}\right)\times 100$

Since Profit = Selling Price – Cost Price, we can express the formula as:

$\text{ Profit Percentage }=\left(\frac{\mathrm{SP}-\mathrm{CP}}{\mathrm{CP}}\right)\times 100$

This formula helps to determine the percentage of gain in relation to the original cost price.

2. Loss Percentage

Loss Percentage is the percentage of loss incurred on the cost price of an item. It is calculated using the formula:

$\text{ Loss Percentage }=\left(\frac{\text{ Loss }}{\text{ Cost Price }}\right)\times 100$

Since Loss = Cost Price – Selling Price, the formula becomes:

$\text{ Loss Percentage }=\left(\frac{CP-SP}{CP}\right)\times 100$

This gives the percentage of loss relative to the cost price.

8.0Prices Related to Buying and Selling

When discussing prices of items, there are two key terms:

1. Selling Price (SP): This is the price at which an item is sold to a buyer.
2. Cost Price (CP): This represents the original price at which an item was purchased.

To calculate the profit or loss made from a sale:

• Profit = Selling Price – Cost Price (when SP > CP).
• Loss = Cost Price – Selling Price (when CP > SP).

If SP is equal to CP, there is neither a profit nor a loss.

9.0Simple and Compound Interest

• The principal (P) is the initial sum borrowed.
• Interest is the extra amount paid to the lender.
• The total amount to be repaid (A) is: Amount = Principal + Interest.
• Simple Interest (SI) is calculated using:

$SI=\frac{P\times R\times T}{100}$

where P is the principal, R is the rate of interest, and T is the time in years.

For example, if P = ₹200, R = 10%, and T = 3 years:

S I $=\frac{200\times 10\times 3}{100}$ = ₹60.

Thus, the total amount to be paid is:

Amount = Principal + Simple Interest

   = 200 + 60 = ₹260.

10.0Solved Examples on Comparing Quantities

Example 1: Compare 3 kilometers and 200 meters using ratios.

Solution: 

First, convert kilometers to meters.

3 km = 3000 meters.

Now, the ratio is:

$\frac{3000}{200}=15:1$

Thus, the ratio of 3 km to 200 m is 15:1.

Example 2: Convert the fraction $\frac{3}{4}$ into a percentage.

Solution: 

To convert a fraction to a percentage, multiply it by 100:

$\frac{3}{4}\times 100=75\%$

Thus, $\frac{3}{4}$ as a percentage is 75%.

Example 3: Find 20% of 250.

Solution: 

To find 20% of 250, use the formula:

$\text{ Percentage of a number }=\frac{\text{ percentage }}{100}\times \text{ number }$

$20\%\text{ of }250=\frac{20}{100}\times 250=50$

Thus, 20% of 250 is 50.

Example 4: A shopkeeper buys a toy for ₹400 and sells it for ₹500. What is the profit percentage?

Solution: 

First, calculate the profit:

Profit = Selling Price – Cost Price

= ₹ 500 – ₹ 400 = ₹100.

Now, calculate the profit percentage:

$\text{ Profit Percentage }=\left(\frac{\text{ Profit }}{\text{ Cost Price }}\right)\times 100=\left(\frac{100}{400}\right)\times 100=25\%\text{. }$

Thus, the profit percentage is 25%.

Example 5: A person buys a book for ₹250 and sells it for ₹200. What is the loss percentage?

Solution: 

First, calculate the loss:

Loss = Cost Price – Selling Price

 = ₹ 250 – ₹200 = ₹50.

Now, calculate the loss percentage:

$\text{ Loss Percentage }=\left(\frac{\text{ Loss }}{\text{ Cost Price }}\right)\times 100=\left(\frac{50}{250}\right)\times 100=20\%\text{. }$

Thus, the loss percentage is 20%.

Example 6: The ratio of girls to boys in a class is 3:5. Find the percentage of girls and boys in the class.

Solution: 

Total parts = 3 + 5 = 8.

Percentage of girls:

$\frac{3}{8}\times 100=37.5\%$

Percentage of boys:

$\frac{5}{8}\times 100=62.5\%$

Thus, 37.5% of the class are girls and 62.5% are boys.

Example 7: If 6 : 8 :: x : 12, find the value of x.

Solution: 

Set up the proportion:

$\frac{6}{8}=\frac{x}{12}$

Cross-multiply:

$6\times 12=8\times x\text{. }$

Simplifying:

$72=8x\Rightarrow x=\frac{72}{8}=9$

Thus, x = 9.

Example 8: Convert 0.75 into a percentage.

Solution: 

To convert a decimal to a percentage, multiply it by 100:

$0.75\times 100=75\%$

Thus, 0.75 as a percentage is 75%.

11.0Practice Questions on Comparing Quantities

1. Convert the Following Fractions into Percentages:

• $\frac{2}{5}$
• $\frac{5}{6}$

2. Find the Ratio of the Following Quantities in Same Units:

• 5 kg to 200 g  
• 6 hours to 90 minutes  
• ₹500 to ₹2000  

3. Percentage of a Number:

• Find 15% of 300.  
• Find 45% of 800.  

4. A shopkeeper buys a mobile phone for ₹15,000 and sells it for ₹18,000. Find the profit percentage.  
5. If a car is bought for ₹3,00,000 and sold for ₹2,70,000, find the loss percentage.  
6. In a school, 65% of the students are boys. If the total number of students is 600, how many boys are there in the school?  Convert it into percentage.
7. The ratio of green balls to blue balls in a box is 4:6. What percentage of balls are green and what percentage are blue?

12.0Sample Questions on Comparing Quantities

1. How Do We Find the Percentage of a Given Quantity?

Ans: To find the percentage of a quantity, multiply the given quantity by the percentage value and divide it by 100.

Formula:

$\text{ Percentage of a quantity }=\frac{\text{ Percentage value }}{100}\times \text{ Quantity }$

2. How Do We Convert a Fraction to a Percentage?

Ans: To convert a fraction into a percentage, multiply the fraction by 100.

Formula:

$\text{ Percentage }=\frac{\text{ Numerator }}{\text{ Denominator }}\times 100$

3. What is the Formula for Simple Interest?

Ans: The formula for Simple Interest (SI) is:

$SI=\frac{P\times R\times T}{100}$

where:

P = Principal (initial amount)

R = Rate of interest

T = Time (in years)