NCERT Class 7 Maths Chapter 8: Comparing Quantities Exercise 8.2

Welcome, young mathematicians! In CBSE Class 7 Maths, Chapter 8, 'Comparing Quantities', helps us understand how to compare different things in real life. Exercise 8.2 is a crucial part of this chapter, where we dive deeper into percentages, profit, loss, and simple interest. These aren't just textbook topics; they are skills you'll use every day, from calculating discounts while shopping to understanding bank interest.

This page will guide you through all the concepts and types of problems found in NCERT Exercise 8.2. We'll break down complex ideas into simple steps, provide clear examples, and give you practice questions to solidify your understanding. By the end, you'll be confident in converting fractions to percentages, calculating profit or loss, and solving simple interest problems like a pro! Let's embark on this exciting journey of numbers and comparisons.

Understanding Percentages, Profit, Loss, and Simple Interest

Exercise 8.2 of Comparing Quantities for Class 7 builds upon your knowledge of ratios and introduces you to percentages, profit and loss calculations, and the basics of simple interest. A percentage means 'per hundred' or 'out of every hundred'. For example, 25% means 25 out of 100. Percentages are super useful for comparing quantities easily, like exam scores, discounts, or ingredient proportions. To convert a fraction or decimal to a percentage, you simply multiply by 100.

Next, we have Profit and Loss, which are essential concepts in business and everyday transactions. When you buy something and sell it for more than you bought it, you make a profit. If you sell it for less, you incur a loss. These are calculated based on the Cost Price (CP) and Selling Price (SP). We also learn to calculate profit percentage or loss percentage, which tells us how much profit or loss was made relative to the cost price.

Finally, the exercise introduces Simple Interest. This is the extra money paid by an institution (like a bank) for the use of money deposited by us, or paid by us to the bank for money borrowed. It's calculated based on the Principal amount (P), Rate of Interest (R), and Time (T). Understanding these concepts will help you make smart financial decisions in the future.

Key Terms and Formulas

Percentage (%): A way of expressing a number as a fraction of 100. For example, 20% means 20/100.
Cost Price (CP): The price at which an article is bought. This is the initial cost for the buyer.
Selling Price (SP): The price at which an article is sold. This is the price the seller receives.
Profit: When Selling Price (SP) is greater than Cost Price (CP). Formula: Profit = SP - CP.
Loss: When Cost Price (CP) is greater than Selling Price (SP). Formula: Loss = CP - SP.
Profit Percentage: The profit expressed as a percentage of the Cost Price. Formula: (Profit / CP) × 100%.
Loss Percentage: The loss expressed as a percentage of the Cost Price. Formula: (Loss / CP) × 100%.
Principal (P): The initial amount of money deposited or borrowed.
Rate of Interest (R): The percentage at which interest is charged per annum (per year).
Time (T): The duration for which the money is deposited or borrowed, usually in years.
Simple Interest (SI): The interest calculated only on the principal amount. Formula: SI = (P × R × T) / 100.
Amount (A): The total money repaid or received, which includes the principal and the interest. Formula: A = P + SI.

Step-by-Step: Calculating Profit/Loss and Percentages

Step 1: Identify Cost Price (CP) and Selling Price (SP) — Read the problem carefully to find out the price at which an item was bought (CP) and the price at which it was sold (SP). Sometimes, overhead expenses (like transportation or repair costs) are added to the original CP to find the 'total CP'.
Step 2: Determine if it's a Profit or Loss — Compare CP and SP: If SP > CP, there is a Profit. If CP > SP, there is a Loss. * If SP = CP, there is no profit and no loss.
Step 3: Calculate the Profit or Loss Amount — Use the respective formula: Profit = SP - CP Loss = CP - SP
Step 4: Calculate Profit Percentage or Loss Percentage — Apply the percentage formulas. Remember, both profit percentage and loss percentage are always calculated on the Cost Price (CP): Profit % = (Profit / CP) × 100 Loss % = (Loss / CP) × 100 Make sure to express your final answer with the '%' symbol.

Worked Examples from Ex 8.2

Example 1: Converting a Ratio to Percentage Question: Convert the ratio 3:4 to percentage. Solution: Step 1: Write the ratio as a fraction. 3:4 can be written as 3/4. Step 2: Multiply the fraction by 100 to convert it to a percentage. (3/4) × 100% = (3 × 25)% = 75% Final Answer: 3:4 is 75%.
Example 2: Finding a Percentage of a Quantity Question: Find 20% of ₹150. Solution: Step 1: Convert the percentage to a fraction or decimal. 20% = 20/100. Step 2: Multiply the fraction by the given quantity. (20/100) × ₹150 = (1/5) × ₹150 = ₹30 Final Answer: 20% of ₹150 is ₹30.
Example 3: Profit and Loss Calculation Question: A shopkeeper buys an article for ₹500 and sells it for ₹575. Find the profit percentage. Solution: Step 1: Identify CP and SP. CP = ₹500, SP = ₹575. Step 2: Determine if it's a profit or loss. Since SP > CP, it's a profit. Step 3: Calculate the profit amount. Profit = SP - CP = ₹575 - ₹500 = ₹75. Step 4: Calculate the profit percentage. Profit % = (Profit / CP) × 100 = (₹75 / ₹500) × 100 = (75 / 5) = 15% Final Answer: The profit percentage is 15%.
Example 4: Simple Interest Calculation Question: Calculate the simple interest on a principal of ₹8000 at a rate of 5% per annum for 3 years. Solution: Step 1: Identify the given values. Principal (P) = ₹8000 Rate (R) = 5% p.a. Time (T) = 3 years Step 2: Apply the simple interest formula: SI = (P × R × T) / 100. SI = (₹8000 × 5 × 3) / 100 SI = (80 × 5 × 3) SI = 400 × 3 = ₹1200 Final Answer: The simple interest is ₹1200.

Exam Tip: Avoiding Common Mistakes

When solving problems from Exercise 8.2, students often make a few common errors. Firstly, always remember that profit percentage and loss percentage are always calculated on the Cost Price (CP), not the Selling Price (SP). Many students mistakenly use SP in the denominator. Secondly, ensure your time (T) for simple interest calculations is always in years. If given in months, convert it to years (e.g., 6 months = 6/12 = 0.5 years). Lastly, pay close attention to units and ensure your final answer includes the correct unit (e.g., %, ₹). Double-check your calculations, especially when dealing with percentages, to avoid small arithmetic errors that can lead to incorrect final answers.

Practice Questions with Solutions

Q: A football team won 10 matches out of the total number of matches they played. If their win percentage was 40, how many matches did they play in all? A: Step 1: Let the total number of matches played be 'x'. Step 2: We are given that 40% of the total matches is 10. Step 3: Write this as an equation: (40/100) x = 10. Step 4: Solve for x: x = (10 100) / 40 = 1000 / 40 = 25. Final answer: The team played 25 matches in all.
Q: Rahul bought a refrigerator for ₹9000 and sold it for ₹8100. Find his loss percentage. A: Step 1: Identify CP and SP. CP = ₹9000, SP = ₹8100. Step 2: Calculate the loss amount (since CP > SP). Loss = CP - SP = ₹9000 - ₹8100 = ₹900. Step 3: Calculate the loss percentage. Loss % = (Loss / CP) × 100 = (₹900 / ₹9000) × 100 = (1/10) × 100 = 10%. Final answer: Rahul's loss percentage is 10%.
Q: Find the amount to be paid at the end of 2 years for a principal of ₹1500 at 6% p.a. simple interest. A: Step 1: Identify P, R, and T. P = ₹1500, R = 6%, T = 2 years. Step 2: Calculate Simple Interest (SI). SI = (P × R × T) / 100 = (1500 × 6 × 2) / 100 = 15 × 6 × 2 = 180. Step 3: Calculate the Total Amount (A). A = P + SI = ₹1500 + ₹180 = ₹1680. Final answer: The amount to be paid is ₹1680.
Q: Convert 0.65 to a percentage. A: Step 1: To convert a decimal to a percentage, multiply by 100. Step 2: 0.65 × 100% = 65%. Final answer: 0.65 is 65%.

Frequently Asked Questions

What is the main focus of Class 7 Maths Chapter 8 Exercise 8.2?

Exercise 8.2 primarily focuses on practical applications of percentages, including converting ratios and decimals to percentages, calculating profit and loss percentages in commercial transactions, and understanding the basics of simple interest calculations for deposits and loans.

How do I calculate profit or loss percentage?

To calculate profit or loss percentage, first find the actual profit (SP - CP) or loss (CP - SP). Then, divide this amount by the Cost Price (CP) and multiply the result by 100. Remember, both profit and loss percentages are always calculated with respect to the Cost Price.

What is the formula for Simple Interest?

The formula for Simple Interest is SI = (P × R × T) / 100, where 'P' is the Principal amount, 'R' is the Rate of Interest per annum (in percentage), and 'T' is the Time period in years. After calculating SI, you can find the Total Amount (A) by adding Principal and Simple Interest: A = P + SI.