Ratio and Proportion
CSAT · Quantitative Aptitude
Introduction
Ratio questions test whether you can think in 'parts'. Once quantities are written as multiples of one unknown part, sharing money, mixing, ages and coins all become one-line problems.
Key concepts
Ratios as parts
A : B = 3 : 5 means A = 3k and B = 5k for some k. If the total is known, k = total ÷ (3 + 5).
Combining ratios
To merge A : B and B : C, make B the same in both. A : B = 2 : 3 and B : C = 4 : 5 → multiply the first by 4 and the second by 3: A : B : C = 8 : 12 : 15.
Proportion
a : b = c : d means ad = bc. The mean proportional of a and b is √(ab); the third proportional to a and b is b²/a; the fourth proportional to a, b, c is bc/a.
Changing a ratio
If x is added to both terms of a : b to make p : q, solve q(a + x) = p(b + x). Adding the same number to both terms moves the ratio towards 1.
Variation
- Direct: y = kx (more workers → more output).
- Inverse: y = k/x (more workers → fewer days).
Coins and notes
Write the numbers of each coin as ratio parts, convert each to rupees (50 paise = ½, 25 paise = ¼), add, and solve for k.
- Share of A in a : b : c = a/(a + b + c) × total
- a/b = c/d ⇒ ad = bc
- Mean proportional = √(ab)
- Compounded ratio of a:b and c:d = ac : bd
Solved examples
Example 1
A bag has ₹1, 50-paise and 25-paise coins in the ratio 4 : 2 : 4. The total value is ₹126. How many 50-paise coins are there?
(a) 46 (b) 42 (c) 84 (d) 63
- Let the numbers be 4x, 2x and 4x.
- Value = 4x + 2x/2 + 4x/4 = 6x.
- 6x = 126 → x = 21.
- 50-paise coins = 2 × 21 = 42.
Answer: (b) 42
Example 2
If A : B = 4 : 7 and B : C = 5 : 4, then A : B : C is:
(a) 20 : 35 : 28 (b) 4 : 7 : 4 (c) 28 : 35 : 20 (d) 4 : 35 : 4
- Make B equal in both ratios: multiply the first by 5 and the second by 7.
- A : B = 20 : 35, B : C = 35 : 28.
- A : B : C = 20 : 35 : 28 = 20 : 35 : 28.
Answer: (a) 20 : 35 : 28
Example 3
If A : B = 4 : 2 and B : C = 2 : 4, then A : B : C is:
(a) 3 : 1 : 2 (b) 4 : 2 : 4 (c) 4 : 4 : 4 (d) 2 : 1 : 2
- Make B equal in both ratios: multiply the first by 2 and the second by 2.
- A : B = 8 : 4, B : C = 4 : 8.
- A : B : C = 8 : 4 : 8 = 2 : 1 : 2.
Answer: (d) 2 : 1 : 2
Example 4
What number must be added to both 6 and 13 so that the ratio becomes 5 : 6?
(a) 31 (b) 29 (c) 30 (d) 28
- (6 + x) / (13 + x) = 5/6.
- 6(6 + x) = 5(13 + x) → 36 + 6x = 65 + 5x.
- x = (65 − 36) ÷ (6 − 5) = 29.
Answer: (b) 29
Example 5
What number must be added to both 12 and 2 so that the ratio becomes 6 : 3?
(a) 8 (b) 7 (c) 9 (d) 10
- (12 + x) / (2 + x) = 6/3.
- 3(12 + x) = 6(2 + x) → 36 + 3x = 12 + 6x.
- x = (12 − 36) ÷ (3 − 6) = 8.
Answer: (a) 8
Example 6
If A : B = 7 : 7 and B : C = 7 : 3, then A : B : C is:
(a) 3 : 7 : 7 (b) 8 : 7 : 3 (c) 7 : 7 : 3 (d) 7 : 49 : 3
- Make B equal in both ratios: multiply the first by 7 and the second by 7.
- A : B = 49 : 49, B : C = 49 : 21.
- A : B : C = 49 : 49 : 21 = 7 : 7 : 3.
Answer: (c) 7 : 7 : 3
Example 7
A bag has ₹1, 50-paise and 25-paise coins in the ratio 1 : 1 : 5. The total value is ₹33. How many 50-paise coins are there?
(a) 12 (b) 13 (c) 60 (d) 24
- Let the numbers be 1x, 1x and 5x.
- Value = 1x + 1x/2 + 5x/4 = 2 3/4x.
- 2 3/4x = 33 → x = 12.
- 50-paise coins = 1 × 12 = 12.
Answer: (a) 12
Example 8
₹1,400 is divided among P, Q and R in the ratio 5 : 3 : 6. What is Q's share?
(a) ₹600 (b) ₹466 (c) ₹400 (d) ₹300
- Total parts = 5 + 3 + 6 = 14.
- One part = 1,400 ÷ 14 = 100.
- Q gets 3 × 100 = ₹300.
Answer: (d) ₹300
- Check answers for divisibility: a share must be a multiple of its ratio part.
- If a ratio and the difference are given, one part = difference ÷ (difference of ratio terms).
- Invert ratios for inverse proportion (time vs speed, days vs workers).
- Combining ratios without equalising the common term.
- Treating the number of coins as their value.
- Reading 'A : B = 2 : 3' as A = 2, B = 3 rather than 2k and 3k.
Practice set
Attempt all 12 questions (about 90 seconds each), then check the worked solutions below.
Q1. If A : B = 7 : 4 and B : C = 4 : 6, then A : B : C is:
(a) 6 : 4 : 7 (b) 7 : 4 : 6 (c) 7 : 16 : 6 (d) 8 : 4 : 6
Q2. ₹2,700 is divided among P, Q and R in the ratio 3 : 7 : 8. What is Q's share?
(a) ₹1155 (b) ₹1050 (c) ₹1200 (d) ₹900
Q3. If A : B = 5 : 2 and B : C = 4 : 6, then A : B : C is:
(a) 5 : 2 : 6 (b) 5 : 8 : 6 (c) 5 : 2 : 3 (d) 3 : 2 : 5
Q4. ₹1,800 is divided among P, Q and R in the ratio 2 : 3 : 4. What is Q's share?
(a) ₹720 (b) ₹660 (c) ₹800 (d) ₹600
Q5. If A : B = 3 : 3 and B : C = 7 : 4, then A : B : C is:
(a) 4 : 7 : 7 (b) 7 : 7 : 4 (c) 3 : 21 : 4 (d) 3 : 3 : 4
Q6. ₹1,920 is divided among P, Q and R in the ratio 7 : 5 : 4. What is P's share?
(a) ₹840 (b) ₹640 (c) ₹960 (d) ₹600
Q7. A bag has ₹1, 50-paise and 25-paise coins in the ratio 4 : 4 : 2. The total value is ₹143. How many 50-paise coins are there?
(a) 88 (b) 110 (c) 44 (d) 97
Q8. A bag has ₹1, 50-paise and 25-paise coins in the ratio 4 : 5 : 4. The total value is ₹45. How many 50-paise coins are there?
(a) 24 (b) 36 (c) 33 (d) 30
Q9. If A : B = 6 : 2 and B : C = 2 : 7, then A : B : C is:
(a) 6 : 2 : 7 (b) 7 : 2 : 6 (c) 7 : 2 : 7 (d) 6 : 4 : 7
Q10. What number must be added to both 2 and 4 so that the ratio becomes 4 : 5?
(a) 8 (b) 7 (c) 6 (d) 5
Q11. ₹2,400 is divided among P, Q and R in the ratio 8 : 5 : 7. What is P's share?
(a) ₹800 (b) ₹600 (c) ₹960 (d) ₹1080
Q12. A bag has ₹1, 50-paise and 25-paise coins in the ratio 4 : 2 : 3. The total value is ₹138. How many 50-paise coins are there?
(a) 53 (b) 96 (c) 72 (d) 48
Q1 – (b) 7 : 4 : 6. Make B equal in both ratios: multiply the first by 4 and the second by 4. A : B = 28 : 16, B : C = 16 : 24. A : B : C = 28 : 16 : 24 = 7 : 4 : 6.
Q2 – (b) ₹1050. Total parts = 3 + 7 + 8 = 18. One part = 2,700 ÷ 18 = 150. Q gets 7 × 150 = ₹1,050.
Q3 – (c) 5 : 2 : 3. Make B equal in both ratios: multiply the first by 4 and the second by 2. A : B = 20 : 8, B : C = 8 : 12. A : B : C = 20 : 8 : 12 = 5 : 2 : 3.
Q4 – (d) ₹600. Total parts = 2 + 3 + 4 = 9. One part = 1,800 ÷ 9 = 200. Q gets 3 × 200 = ₹600.
Q5 – (b) 7 : 7 : 4. Make B equal in both ratios: multiply the first by 7 and the second by 3. A : B = 21 : 21, B : C = 21 : 12. A : B : C = 21 : 21 : 12 = 7 : 7 : 4.
Q6 – (a) ₹840. Total parts = 7 + 5 + 4 = 16. One part = 1,920 ÷ 16 = 120. P gets 7 × 120 = ₹840.
Q7 – (a) 88. Let the numbers be 4x, 4x and 2x. Value = 4x + 4x/2 + 2x/4 = 6 1/2x. 6 1/2x = 143 → x = 22. 50-paise coins = 4 × 22 = 88.
Q8 – (d) 30. Let the numbers be 4x, 5x and 4x. Value = 4x + 5x/2 + 4x/4 = 7 1/2x. 7 1/2x = 45 → x = 6. 50-paise coins = 5 × 6 = 30.
Q9 – (a) 6 : 2 : 7. Make B equal in both ratios: multiply the first by 2 and the second by 2. A : B = 12 : 4, B : C = 4 : 14. A : B : C = 12 : 4 : 14 = 6 : 2 : 7.
Q10 – (c) 6. (2 + x) / (4 + x) = 4/5. 5(2 + x) = 4(4 + x) → 10 + 5x = 16 + 4x. x = (16 − 10) ÷ (5 − 4) = 6.
Q11 – (c) ₹960. Total parts = 8 + 5 + 7 = 20. One part = 2,400 ÷ 20 = 120. P gets 8 × 120 = ₹960.
Q12 – (d) 48. Let the numbers be 4x, 2x and 3x. Value = 4x + 2x/2 + 3x/4 = 5 3/4x. 5 3/4x = 138 → x = 24. 50-paise coins = 2 × 24 = 48.
- Think in parts: 3k, 5k
- Equalise the common term before combining
- ad = bc
- Coins: count ratio → value ratio
