Mixed practice: Fractions in Disguise

5.1 Mixed practice: putting the ideas together

These nine problems are new: they are not taken from the textbook. Each one mixes two or more ideas from the chapter, so begin by naming the base and the quantity being compared. Work with fractions and percentages side by side, keep exact values until the final step, and check that the answer makes sense in the situation before moving on.

Question 1. A library has 1,800 books. Of these, 35% are fiction, and 2/5 of the fiction books are mystery stories. How many mystery books does the library have?

Answer. Step 1. Find the fiction books: 35% of 1,800 = (35/100) × 1,800 = 630. Step 2. Find 2/5 of the fiction books: (2/5) × 630 = 252. Step 3. Check: 630 + 1,170 = 1,800, so the fiction count fits the total, and 252 is less than 630. The library has 252 mystery books.

Question 2. Two students took different tests. Ananya scored 18 out of 25 in one test and 29 out of 40 in another. Which score is better, and by how many percentage points?

Answer. Convert each score to a percentage. Step 1. 18/25 = (18 × 4)/(25 × 4) = 72/100 = 72%. Step 2. 29/40 = (29 × 2.5)/(40 × 2.5) = 72.5/100 = 72.5%. Step 3. The second score is higher by 72.5% − 72% = 0.5 percentage points. Percentage points compare the two percentages directly; they are not the same as a 0.5% increase.

Question 3. A bakery's daily sale rises from ₹4,500 to ₹5,400. Find the percentage increase.

Answer. The increase is ₹5,400 − ₹4,500 = ₹900. The base is the earlier sale, ₹4,500. Percentage increase = (900/4,500) × 100 = 20%. Check: 20% of 4,500 is 900, and 4,500 + 900 = 5,400.

Question 4. A jacket's price falls from ₹2,400 to ₹1,800. Find the percentage decrease. The shop then raises the price by 10% from ₹1,800. What is the final price, and what is the overall percentage change from ₹2,400?

Answer. Step 1. Decrease = 2,400 − 1,800 = ₹600. Step 2. Percentage decrease = (600/2,400) × 100 = 25%. Step 3. A 10% rise on ₹1,800 gives 1,800 × 1.10 = ₹1,980. Step 4. Overall change = 1,980 − 2,400 = −₹420, so the overall percentage change = (−420/2,400) × 100 = −17.5%. The final price is lower than the original because the rise was taken on the reduced price.

Question 5. A cycle is marked at ₹12,500. A shop offers a 16% discount and sells it for ₹10,500. The shop had bought the cycle for ₹9,000. Find the discount, the profit, and the profit percentage.

Answer. Step 1. Discount = 16% of ₹12,500 = 0.16 × 12,500 = ₹2,000, so the selling price is 12,500 − 2,000 = ₹10,500. Step 2. Profit = SP − CP = 10,500 − 9,000 = ₹1,500. Step 3. Profit percentage is calculated on the cost price: (1,500/9,000) × 100 = 16.67% (to two decimal places). The discount percentage uses the marked price as its base, while the profit percentage uses the cost price.

Question 6. A school bag is marked at ₹2,000. A shop first gives a 25% discount and then a further 12% discount on the reduced price. A second shop offers a single 35% discount. Which shop gives the lower price, and by how much?

Answer. Step 1. After 25% off: 2,000 × 0.75 = ₹1,500. Step 2. After a further 12% off: 1,500 × 0.88 = ₹1,320. Step 3. The two successive multipliers are 0.75 × 0.88 = 0.66, which is a 34% discount overall, not 37%. Step 4. The single 35% discount gives 2,000 × 0.65 = ₹1,300. The second shop is cheaper by ₹20.

Question 7. A trader sells two lamps for ₹1,200 each. One sale gives a 20% profit and the other gives a 20% loss. Find the cost price of each lamp, the total cost, the total selling price, and the overall profit or loss percentage.

Answer. Step 1. For the profitable lamp, SP = 1.20 × CP, so CP = 1,200/1.20 = ₹1,000. Step 2. For the loss-making lamp, SP = 0.80 × CP, so CP = 1,200/0.80 = ₹1,500. Step 3. Total CP = 1,000 + 1,500 = ₹2,500; total SP = 1,200 + 1,200 = ₹2,400. Step 4. Overall loss = 2,500 − 2,400 = ₹100, so loss percentage = (100/2,500) × 100 = 4%. Equal percentage changes on different bases do not cancel.

Question 8. ₹15,000 is deposited at 8% per annum for 2 years. Find the amount with simple interest and with compound interest, and the difference between them.

Answer. Step 1. Simple interest amount = 15,000(1 + 0.08 × 2) = 15,000 × 1.16 = ₹17,400. Step 2. Compound interest amount = 15,000(1.08)² = 15,000 × 1.1664 = ₹17,496. Step 3. Difference = 17,496 − 17,400 = ₹96. The extra ₹96 is the interest earned in the second year on the first year's interest.

Question 9. A test is out of 150 marks. A student scores 72%. How many marks did the student get, and how many more marks are needed to reach 80%?

Answer. Step 1. 72% of 150 = 0.72 × 150 = 108 marks. Step 2. 80% of 150 = 0.80 × 150 = 120 marks. Step 3. The student needs 120 − 108 = 12 more marks. Check: 12 marks out of 150 is 8 percentage points, and 72% + 8% = 80%.