Revision and important questions: Fractions in Disguise

6.1 Formula box

Keep these relationships together. Every question in the chapter is a combination of them.

1. Percentage to fraction: x% = x/100.
2. Fraction to percentage: multiply the fraction by 100.
3. Percentage of a quantity: y% of a = (y/100) × a.
4. Percentage change: (change/original) × 100, always measured against the original value.
5. Profit percentage: (SP - CP)/CP × 100. Loss percentage: (CP - SP)/CP × 100.
6. Discount: discount = MP - SP, and the discount percentage is taken on the marked price.
7. Simple interest amount: p(1 + rt), where r is the annual rate written as a decimal.
8. Compound interest amount: p(1 + r)ᵗ.
9. Repeated change: multiply the multipliers. A rise of r1 followed by a rise of r2 gives the factor (1 + r1)(1 + r2).
10. Depreciation: value after t years = p(1 - r)ᵗ.

6.2 Common mistakes

1. Adding successive percentages. A 30% discount followed by a 20% discount is not a 50% discount. The multipliers give 0.7 × 0.8 = 0.56, which is a 44% discount.
2. Using the selling price as the base. A profit of ₹250 on a cost of ₹500 is 50%, not 250/750 = 33.33%. The second figure measures profit against revenue, which answers a different question.
3. Forgetting the principal. The simple interest on ₹50,000 at 7% for 3 years is ₹10,500, but the amount is ₹60,500. Read whether the question asks for interest or amount.
4. Reading a fall as a final percentage. A fall of 85% leaves 15%, so the new value is 15% of the old one, not 85%.
5. Dropping the whole-number check. A percentage of a count of people or objects must give a sensible whole number; 38.4 students means the data and the question do not match exactly.
6. Changing the base in a comparison. The two buffaloes sold at the same price gave different cost prices, so a 5% profit and a 10% loss did not cancel.

6.3 Important exam-style questions

Question 1. Convert 7/8 to a percentage, and convert 0.35 to a percentage.

Answer. 7/8 × 100 = 700/8 = 87.5%. Also 0.35 = 35/100 = 35%. The decimal shift is two places to the right.

Question 2. Find 15% of 240 and 125% of 80.

Answer. 15% of 240 = 0.15 × 240 = 36. 125% of 80 = 1.25 × 80 = 100. The second answer is larger than the whole 80 because the percentage is above 100%.

Question 3. One number is 60% of another. The smaller number is 72. Find the larger number.

Answer. Let the larger number be x. Then 0.6x = 72, so x = 72/0.6 = 720/6 = 120. Check: 60% of 120 = 72.

Question 4. A shirt's price rises from ₹250 to ₹300. Find the percentage increase.

Answer. Increase = 300 - 250 = 50. Base = 250. Percentage increase = 50/250 × 100 = 20%.

Question 5. A shopkeeper buys 40 pens at ₹15 each and sells all of them for ₹720. Find the profit percentage.

Answer. Total cost = 40 × 15 = 600. Total selling price = 720. Profit = 120. Profit percentage = 120/600 × 100 = 20%.

Question 6. ₹12,000 is invested at 8% p.a. for 2 years. Find the amount with simple interest and with compound interest, and the difference.

Answer. Simple amount = 12,000(1 + 0.08 × 2) = 12,000 × 1.16 = ₹13,920. Compound amount = 12,000 × 1.08² = 12,000 × 1.1664 = ₹13,996.80. The difference is ₹76.80, which is 8% of the first year's interest of ₹960.

Question 7. A shop offers two schemes on a ₹500 item: two successive discounts of 20% and 10%, or a single 30% discount. Which is cheaper for the customer?

Answer. Successive: 500 × 0.8 × 0.9 = 500 × 0.72 = ₹360. Single 30%: 500 × 0.7 = ₹350. The single 30% discount is cheaper by ₹10, because the successive discounts give only 28% off.

Question 8. A town's population is 80,000 and falls by 5% each year. Find the population after 2 years.

Answer. After one year: 80,000 × 0.95 = 76,000. After two years: 76,000 × 0.95 = 72,200. In one line, 80,000 × 0.95² = 80,000 × 0.9025 = 72,200.

Question 9. A shirt is marked at ₹800. A 20% discount is given, and then 5% GST is charged on the discounted price. What does the customer pay?

Answer. After the discount: 800 × 0.8 = ₹640. GST is 5% of 640 = ₹32. Final price = 640 + 32 = ₹672. In one line, 800 × 0.8 × 1.05 = 672.

Question 10. An article sold at a 10% loss would bring ₹60 less than if it were sold at a 5% profit. Find the cost price.

Answer. Let the cost price be c. The two selling prices are 0.9c and 1.05c. Their difference is 1.05c - 0.9c = 0.15c. The question says this difference is ₹60, so 0.15c = 60 and c = 400. Check: 10% loss gives ₹360 and 5% profit gives ₹420, and 420 - 360 = 60.