Worksheet Solutions: Ratio and Proportion | Mathematics Class 10 ICSE PDF Download

Section A: Very Short Answer Questions (1 mark each)

Q1. Simplify the ratio 48:60.
Solution:
Step 1: Find HCF of 48 and 60 → HCF = 12.
Step 2: Divide both terms by 12 → (48 ÷ 12):(60 ÷ 12) = 4:5.
Answer: 4:5

Q2. If the cost of 7 apples is ₹140, what is the cost of 1 apple?
Solution:
Step 1: Cost of 7 apples = ₹140.
Step 2: Cost of 1 apple = 140 ÷ 7 = ₹20.
Answer: ₹20

Q3. Write the duplicate ratio of 5:7.
Solution:
Duplicate ratio = 5²:7² = 25:49.
Answer: 25:49

Q4. Find the sub-duplicate ratio of 81:256.
Solution:
Sub-duplicate ratio = √81:√256 = 9:16.
Answer: 9:16

Q5. Find the reciprocal ratio of 8:13.
Solution:
Reciprocal ratio = 13:8.
Answer: 13:8

Section B: Short Answer Questions (2–3 marks each)

Q6. The monthly salary of A and B are in the ratio 7:9. If A earns ₹21,000, find B’s salary.
Solution:
Step 1: Let B’s salary = x.
Step 2: Ratio A:B = 7:9 → 21000:x = 7:9.
Step 3: Cross multiply → 7x = 21000 × 9 = 189000.
Step 4: x = 189000 ÷ 7 = ₹27,000.
Answer: B’s salary = ₹27,000

Q7. Two numbers are in the ratio 4:7. If their sum is 132, find the numbers.
Solution:
Step 1: Let the numbers be 4x and 7x.
Step 2: Sum = 4x + 7x = 11x.
Step 3: 11x = 132 → x = 12.
Step 4: Numbers = 4x = 48 and 7x = 84.
Answer: 48 and 84

Q8. If a:b = 3:5 and b:c = 2:7, find a:b:c.
Solution:
Step 1: a:b = 3:5 → multiply by 2 → 6:10.
Step 2: b:c = 2:7 → multiply by 5 → 10:35.
Step 3: Combine → a:b:c = 6:10:35.
Answer: 6:10:35

Q9. The income of two friends is in the ratio 5:8 and their expenses in the ratio 3:5. If each saves ₹4000, find their incomes.
Solution:
Step 1: Let incomes = 5x and 8x.
Step 2: Let expenses = 3y and 5y.
Step 3: Savings = Income – Expense.
So, 5x – 3y = 4000 …(i)
And 8x – 5y = 4000 …(ii)
Step 4: Multiply (i) by 5: 25x – 15y = 20000.
Multiply (ii) by 3: 24x – 15y = 12000.
Step 5: Subtract → x = 8000.
Step 6: Incomes = 5x = 40000, 8x = 64000.
Answer: ₹40,000 and ₹64,000

Q10. If 3a = 2b and 4b = 5c, find the ratio a:b:c.
Solution:
Step 1: From 3a = 2b → a:b = 2:3.
Step 2: From 4b = 5c → b:c = 5:4.
Step 3: a:b = 2:3 → multiply by 5 → 10:15.
Step 4: b:c = 5:4 → multiply by 3 → 15:12.
Step 5: Combine → a:b:c = 10:15:12.
Answer: 10:15:12

Section C: Long Answer Questions (4–5 marks each)

Q11. A sum of ₹3,000 is divided among A, B, and C in the ratio 2:3:5. Find the share of each.
Solution:
Step 1: Total ratio = 2 + 3 + 5 = 10 parts.
Step 2: Value of 1 part = 3000 ÷ 10 = 300.
Step 3: Shares:
A = 2 × 300 = 600
B = 3 × 300 = 900
C = 5 × 300 = 1500
Answer: A = ₹600, B = ₹900, C = ₹1500

Q12. The cost prices of three articles are in the ratio 2 : 3 : 5. Their selling prices are in the ratio 3 : 4 : 6. If the total profit on all three articles is Rs. 720, find the cost price of each article.

Solution:

1. Let the cost prices be 2x, 3x, and 5x.

2. Let the selling prices be 3y, 4y, and 6y.

3. Profit = (sum of selling prices) – (sum of cost prices)
   = (3y + 4y + 6y) – (2x + 3x + 5x)
   = 13y – 10x

4. Given total profit = 720 → 13y – 10x = 720.
   To proceed, assume a common multiplier so that selling price and cost price relate properly.

• Suppose x = 60 and y = 60.
  Then, CP = 2(60) + 3(60) + 5(60) = 600.
  SP = 3(60) + 4(60) + 6(60) = 780.
  Profit = 780 – 600 = 180.

But we want profit = 720.
So we multiply all CP and SP by a scaling factor = 720 ÷ 180 = 4.

Thus:

• Cost prices = 120×4 = 240, 180×4 = 720, 300×4 = 1200.

• Selling prices = 180×4 = 720, 240×4 = 960, 360×4 = 1440.

Check: Total CP = 240 + 360 + 600 = 1200.
Wait correction: Let me carefully scale again.

Correct approach:
Base CP = {120, 180, 300} (from 2x, 3x, 5x with x=60).
Base SP = {180, 240, 360} (from 3y, 4y, 6y with y=60).

Base profit = (180+240+360) – (120+180+300)
= 780 – 600 = 180.

Scale factor = 720 ÷ 180 = 4.

So actual CPs = {120×4, 180×4, 300×4} = {480, 720, 1200}.
Total CP = 2400.
Actual SPs = {180×4, 240×4, 360×4} = {720, 960, 1440}.
Total SP = 3120.

Final Answer:
The cost prices of the three articles are Rs. 480, Rs. 720, and Rs. 1200.

Q13. The fares of a bus increase in the ratio 7:9. If the new fare is ₹675, find the original fare and the increase in fare.
Solution:
Step 1: Original fare : New fare = 7:9.
Step 2: Let original fare = x. Then, 9x/7 = 675.
Step 3: x = (675 × 7) ÷ 9 = 525.
Step 4: Increase = 675 – 525 = 150.
Answer: Original fare = ₹525, Increase = ₹150