Ratio and Proportion MCQs

1. If the ratio of two numbers is 3 : 4 and their sum is 56, find the two numbers.

a) 21, 35 b) 24, 32 c) 28, 42 d) 30, 40

Correct Answer: b) 24, 32

Explanation:
Let the numbers be $3 x$ and $4 x$ .
Given sum = 56.
Therefore, $3 x + 4 x = 56$
$7 x = 56 \Rightarrow x = 8$
So, the numbers are:
$3 x = 24$ and $4 x = 32$
Shortcut: Ratio sum = 3 + 4 = 7 parts. Value of one part = 56 ÷ 7 = 8.
First number = 3 × 8 = 24, Second number = 4 × 8 = 32.

2. The ratio of the present ages of A and B is 5 : 7. After 6 years, the ratio becomes 3 : 4. Find their present ages.

a) 20, 28 b) 25, 35 c) 30, 42 d) 15, 21

Correct Answer: c) 30, 42

Explanation:
Let the present ages be $5 x$ and $7 x$ .
After 6 years, $\frac{5 x + 6}{7 x + 6} = \frac{3}{4}$
Cross-multiplying gives:
$4 (5 x + 6) = 3 (7 x + 6)$
Expanding and solving:
$20 x + 24 = 21 x + 18$
Rearranging terms:
$20 x - 21 x = 18 - 24$
$-x = -6 \Rightarrow x = 6$
Therefore, ages are:
A = 5 × 6 = 30 B = 7 × 6 = 42
Shortcut: Multiply the terms of the final ratio: 4 × (5x + 6) = 3 × (7x + 6) Solve directly to find x = 6.

3. A sum of ₹ 900 is divided among A, B, and C in the ratio 2 : 3 : 4. How much does B get?

a) ₹ 300 b) ₹ 200 c) ₹ 400 d) ₹ 350

Correct Answer: a) ₹ 300

Explanation:
Let the shares of A, B, and C be 2x, 3x, 4x respectively.
Total sum = ₹ 900
$2 x + 3 x + 4 x = 900$
$9 x = 900$
$x = 100$
B's share = $\frac{3}{9} \times 900 = 300$
Shortcut:
Total ratio = 2 + 3 + 4 = 9 parts.
B's share = ₹ (3 ÷ 9) × 900 = ₹ 300.

4. The ratio of the length and breadth of a rectangle is 5 : 3. If the perimeter is 64 cm, find the length and breadth.

a) 20 cm, 12 cm b) 25 cm, 15 cm c) 16 cm, 10 cm d) 24 cm, 14 cm

Correct Answer: a) 20 cm, 12 cm

Explanation:
Let the length and breadth be $5 x$ and $3 x$ .
Perimeter = 64 cm
$2 (5 x + 3 x) = 64$
$2 (8 x) = 64$
$16 x = 64$
$x = 4$
Therefore,
Length = $5 \times 4 = 20$ cm, Breadth = $3 \times 4 = 12$ cm.
Shortcut:
Total ratio = 5 + 3 = 8 parts.
Length = (5 ÷ 8) × 64 ÷ 2 = 20 cm.
Breadth = (3 ÷ 8) × 64 ÷ 2 = 12 cm.

5. The income of A and B is in the ratio 3 : 4. Their expenses are in the ratio 5 : 7. If they save ₹ 2000 each, find their incomes.

a) ₹ 12000, ₹ 16000 b) ₹ 7000, ₹ 9000 c) ₹ 5000, ₹ 7000 d) ₹ 8000, ₹ 10000

Correct Answer: a) ₹ 12000, ₹ 16000

Explanation:
Let incomes be $3 x$ and $4 x$ . Let expenses be $5 y$ and $7 y$ . According to the savings condition:
$3 x - 5 y = 2000$
$4 x - 7 y = 2000$
Solving these equations:
Multiply the first equation by 4 and the second by 3:
$12 x - 20 y = 8000$
$12 x - 21 y = 6000$
Subtracting gives:
$y = 2000$
From the first equation:
$3 x - 5 (800) = 2000$
$x = 4000$
Therefore, incomes = ₹ 12000 and ₹ 16000.
Shortcut: Let incomes = 3x, 4x. Expenses = 5y, 7y. Solve the two equations to get x = 4000, y = 2000.

6. Two numbers are in the ratio 7 : 9. If their LCM is 252, find the numbers.

a) 21, 27 b) 14, 18 c) 28, 36 d) 35, 45

Correct Answer: c) 28, 36

Explanation:
Let the numbers be $7 x$ and $9 x$ .
LCM = $63 x = 252$
$x = 4$
Therefore, numbers = 28, 36.
Shortcut: LCM = 7 × 9 × x = 252 ⇒ x = 4.

7. The cost of a pen and a pencil are in the ratio 3 : 5. If the pen costs ₹ 45, find the cost of the pencil.

a) ₹ 60 b) ₹ 75 c) ₹ 90 d) ₹ 30

Correct Answer: b) ₹ 75

Explanation:
Let the cost of a pen = 3x and pencil = 5x.
Given 3x = 45 ⇒ x = 15.
Pencil cost = 5 × 15 = ₹ 75.
Shortcut: Pencil = (5 ÷ 3) × 45 = ₹ 75.

8. If 15 workers can complete a task in 12 days, how many days will 20 workers take to complete the same task?

a) 8 days b) 10 days c) 12 days d) 15 days

Correct Answer: a) 8 days

Explanation:
15 × 12 = 20 × x ⇒ x = 9 days.
Shortcut: Inverse proportion: (15 × 12) ÷ 20 = 8 days.

9. If A : B = 4 : 5 and B : C = 2 : 3, find A : C.

a) 8 : 15 b) 4 : 5 c) 6 : 15 d) 5 : 8

Correct Answer: a) 8 : 15

Explanation:
Given: $A : B = 4 : 5$ , $B : C = 2 : 3$
To make B equal, multiply A : B by 2 and B : C by 5:
$A : B = 8 : 10$ , $B : C = 10 : 15$
Therefore, $A : C = 8 : 15$
Shortcut: Multiply the ratios to make B equal, then combine terms.

10. The ratio of milk to water in a mixture is 5 : 3. If 16 liters of water are added, the new ratio becomes 5 : 4. Find the initial quantity of milk.

a) 80 liters b) 50 liters c) 60 liters d) 30 liters

Correct Answer: a) 80 liters

Explanation:
Let milk = $5 x$ and water = $3 x$ After adding 16 liters of water, the new ratio becomes:
$\frac{5x}{3x + 16} = \frac{5}{4}$
Cross-multiplying gives:
$5 x \times 4 = 5 \times (3 x + 16)$
Expanding terms:
$20 x = 15 x + 80$
Solving for x:
$5 x = 80$
$x = 16$
Therefore, initial milk = $5 \times 16 = 80$ liters.
Shortcut: Set up the ratio equation, simplify, and solve directly.

Ratio and Proportion MCQs

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