Comparing Quantities Class 8 Maths NCERT

Understanding the Basics: Ratios, Percentages, and Their Applications

Comparing quantities means figuring out how one amount relates to another. The simplest way to do this is using Ratios and Percentages.

Ratios: A ratio compares two quantities of the same kind by division. For example, if there are 10 boys and 15 girls in a class, the ratio of boys to girls is 10:15, which simplifies to 2:3. This means for every 2 boys, there are 3 girls. Ratios help us understand relative sizes.

Percentages: A percentage means 'out of one hundred'. It's a special type of ratio where the second term is always 100. For example, 75% means 75 out of 100, or 75/100. Percentages are incredibly useful for comparing different quantities, even if their total amounts are different. Imagine scoring 40 out of 50 in Science and 75 out of 100 in Maths. Which is better? Converting to percentages:

• Science: (40/50) * 100% = 80%
• Maths: (75/100) * 100% = 75%

So, 80% in Science is better than 75% in Maths. Percentages make comparisons clear and easy. They are widely used in calculating profit/loss, discounts, interest rates, and even in reporting statistics.

Converting Fractions/Decimals to Percentages:

• To convert a fraction to a percentage, multiply by 100. Example: 3/4 = (3/4) * 100% = 75%.
• To convert a decimal to a percentage, multiply by 100. Example: 0.25 = 0.25 * 100% = 25%.

Converting Percentages to Fractions/Decimals:

• To convert a percentage to a fraction, divide by 100. Example: 60% = 60/100 = 3/5.
• To convert a percentage to a decimal, divide by 100. Example: 60% = 60/100 = 0.60.

Understanding these foundational concepts is crucial before moving on to more complex applications like profit, loss, and interest.

Step-by-Step: Profit, Loss, Discount, and Simple Interest

1. Calculating Profit and Loss — Profit or Loss is determined by comparing the Cost Price (CP), which is the price at which an item is bought, with the Selling Price (SP), the price at which it is sold. Step 1: Identify CP and SP. If SP > CP, there is a Profit. Profit = SP - CP. If CP > SP, there is a Loss. Loss = CP - SP. Step 2: Calculate Profit % or Loss % (always on CP). Profit % = (Profit / CP) × 100 Loss % = (Loss / CP) × 100 Example: A shopkeeper buys a toy for ₹150 (CP) and sells it for ₹180 (SP). Since SP > CP, it's a Profit. Profit = ₹180 - ₹150 = ₹30. Profit % = (₹30 / ₹150) × 100 = (1/5) × 100 = 20%.
2. Calculating Discount and Sales Tax (GST) — When an item is sold, its price can be affected by discounts or additional taxes like GST. Step 1: Understand Marked Price (MP) and Discount. Marked Price (MP) is the price tagged on the item. Discount is the reduction offered on the MP. Discount is always calculated on the MP. Discount = Discount % of MP. Step 2: Find Selling Price (SP) after Discount. SP = MP - Discount. Step 3: Add Sales Tax (GST). Sales Tax/GST is calculated on the selling price after discount. Tax Amount = Tax % of SP. Final Price = SP + Tax Amount. Example: A shirt is marked at ₹800. A discount of 10% is offered, and GST is 12%. Discount = 10% of ₹800 = (10/100) × 800 = ₹80. SP after discount = ₹800 - ₹80 = ₹720. GST Amount = 12% of ₹720 = (12/100) × 720 = ₹86.40. * Final Price = ₹720 + ₹86.40 = ₹806.40.
3. Calculating Simple Interest — Interest is the extra money paid by the borrower to the lender for using their money. Simple Interest (SI) is calculated only on the original principal amount. Step 1: Identify Principal (P), Rate (R), and Time (T). Principal (P): The initial amount of money borrowed or deposited. Rate (R): The interest rate per annum (per year), usually given as a percentage. Time (T): The duration for which the money is borrowed or deposited, in years. Step 2: Use the Simple Interest Formula. Simple Interest (SI) = (P × R × T) / 100 Step 3: Calculate the Amount. Amount (A) = Principal (P) + Simple Interest (SI). Example: Calculate the simple interest on ₹5000 at 8% per annum for 3 years. P = ₹5000, R = 8%, T = 3 years. SI = (5000 × 8 × 3) / 100 = (50 × 8 × 3) = ₹1200. Amount = ₹5000 + ₹1200 = ₹6200.

Exam Tips and Avoiding Common Mistakes

To excel in 'Comparing Quantities', always pay close attention to the details in the problem statement. Here are some key tips and common pitfalls to avoid:

1. Read Carefully: Many mistakes happen because students misinterpret what's asked. Is it profit or loss? Is the rate per annum or per month? Are you finding the percentage increase or the new amount?
2. Units Consistency: Ensure all quantities being compared or used in calculations have the same units. If you're comparing 500 grams to 2 kilograms, convert kilograms to grams (2 kg = 2000 g) before forming a ratio.
3. Base for Percentage: Remember, profit and loss percentages are always calculated on the Cost Price (CP), unless specified otherwise. Discount is always calculated on the Marked Price (MP).
4. Time in Years: For Simple Interest, the 'Time' (T) must always be in years. If given in months, convert to years (e.g., 6 months = 6/12 = 0.5 years).
5. Compound Interest vs. Simple Interest: Understand the fundamental difference. Simple interest is calculated only on the principal, while compound interest is calculated on the principal plus accumulated interest from previous periods. Class 8 mainly focuses on simple interest, but it's good to know the distinction.
6. Step-by-Step Approach: Break down complex problems into smaller, manageable steps. For example, first calculate discount, then SP, then GST. This prevents errors and makes your solution clear.

Practice Questions with Solutions

• Q: A basket contains 30 oranges and 20 apples. Find the ratio of apples to oranges and the percentage of apples in the basket. A: Step 1: Find the ratio of apples to oranges. Number of apples = 20 Number of oranges = 30 Ratio of apples to oranges = 20:30 = 2:3 Step 2: Calculate the total number of fruits. Total fruits = 30 + 20 = 50 Step 3: Calculate the percentage of apples. Percentage of apples = (Number of apples / Total fruits) 100 Percentage of apples = (20 / 50) 100 = (2/5) * 100 = 40% Final answer: The ratio of apples to oranges is 2:3, and the percentage of apples in the basket is 40%.
• Q: A shopkeeper bought a cycle for ₹3500 and spent ₹500 on its repairs. He then sold it for ₹4500. Find his profit or loss percentage. A: Step 1: Calculate the total Cost Price (CP). Initial cost = ₹3500 Repair cost = ₹500 Total CP = ₹3500 + ₹500 = ₹4000 Step 2: Identify the Selling Price (SP). SP = ₹4500 Step 3: Determine if there is a profit or loss. Since SP (₹4500) > CP (₹4000), there is a profit. Profit = SP - CP = ₹4500 - ₹4000 = ₹500 Step 4: Calculate the profit percentage. Profit % = (Profit / CP) 100 Profit % = (₹500 / ₹4000) 100 = (1/8) * 100 = 12.5% Final answer: The shopkeeper made a profit of 12.5%.
• Q: An item is marked at ₹1200. A discount of 20% is offered on it. Calculate the selling price of the item. A: Step 1: Identify the Marked Price (MP) and Discount Percentage. MP = ₹1200 Discount % = 20% Step 2: Calculate the discount amount. Discount = 20% of MP = (20/100) * ₹1200 = ₹240 Step 3: Calculate the Selling Price (SP). SP = MP - Discount = ₹1200 - ₹240 = ₹960 Final answer: The selling price of the item is ₹960.
• Q: Find the simple interest on a principal of ₹8000 at a rate of 5% per annum for 4 years. Also, find the total amount. A: Step 1: Identify the Principal (P), Rate (R), and Time (T). P = ₹8000 R = 5% per annum T = 4 years Step 2: Calculate the Simple Interest (SI) using the formula SI = (P × R × T) / 100. SI = (₹8000 × 5 × 4) / 100 SI = (80 × 5 × 4) = 400 × 4 = ₹1600 Step 3: Calculate the total Amount (A). Amount = Principal + Simple Interest Amount = ₹8000 + ₹1600 = ₹9600 Final answer: The simple interest is ₹1600, and the total amount is ₹9600.

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