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Ratio & Proportion

Ratios, proportions, dividing amounts and variation.

45 solved questions · 15 easy · 20 medium · 10 hard

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Shortcuts & formulas

  • a:b = c:d means ad = bc (product of extremes = product of means).
  • Fourth proportional to a, b, c is bc/a; third proportional to a, b is b²/a; mean proportional between a and b is √(ab).
  • To divide N in the ratio a:b:c, one part = N/(a+b+c); shares are a, b, c times one part.
  • To combine a:b and b:c, make the common term b equal, then write a:b:c.
  • Compound ratio of a:b and c:d is ac:bd; duplicate ratio of a:b is a²:b²; sub-duplicate is √a:√b.
  • Direct variation: y/x is constant. Inverse variation: xy is constant (more men, fewer days).
  • Ratio of ages: write ages as ax and bx, then apply 'after n years' as (ax+n) and (bx+n).
  • Fractions as a ratio: 1/2:1/3:1/4 becomes 6:4:3 after multiplying by the LCM (12).

Solved examples

Try each one first, then open the answer and the step-by-step solution.

Easy questions

Q1.

Express the ratio 48 : 72 in its simplest form.

  1. A2:3
  2. B3:4
  3. C2:5
  4. D1:2
Show answer & solution

Answer: A. 2:3

Solution

HCF of 48 and 72 is 24. 48 ÷ 24 = 2 and 72 ÷ 24 = 3. Answer: 2:3

Q2.

Divide ₹2,400 between Asha and Ravi in the ratio 5 : 3. How much does Asha get?

  1. A₹900
  2. B₹1,200
  3. C₹1,440
  4. D₹1,500
Show answer & solution

Answer: D. ₹1,500

Solution

Total parts = 5 + 3 = 8 One part = 2,400 ÷ 8 = ₹300 Asha's share = 5 × 300 = ₹1,500 Answer: ₹1,500

Q3.

Find the fourth proportional to 4, 9 and 12.

  1. A16
  2. B24
  3. C27
  4. D36
Show answer & solution

Answer: C. 27

Solution

Fourth proportional d satisfies 4 : 9 = 12 : d. d = (9 × 12) ÷ 4 = 108 ÷ 4 = 27 Answer: 27

Q4.

Find the third proportional to 9 and 12.

  1. A15
  2. B16
  3. C18
  4. D14
Show answer & solution

Answer: B. 16

Solution

Third proportional c satisfies 9 : 12 = 12 : c. c = (12 × 12) ÷ 9 = 144 ÷ 9 = 16 Answer: 16

Q5.

What is the mean proportional between 4 and 25?

  1. A14.5
  2. B10
  3. C12
  4. D100
Show answer & solution

Answer: B. 10

Solution

Mean proportional = √(4 × 25) = √100 = 10 Answer: 10

Q6.

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

  1. A8:15
  2. B2:5
  3. C6:5
  4. D10:12
Show answer & solution

Answer: A. 8:15

Solution

Make B equal: A:B = 2:3 = 8:12 (multiply by 4), B:C = 4:5 = 12:15 (multiply by 3). So A:B:C = 8:12:15. A:C = 8:15 Answer: 8:15

Q7.

Find x if 3 : x = 12 : 20.

  1. A4
  2. B6
  3. C8
  4. D5
Show answer & solution

Answer: D. 5

Solution

Cross-multiply: 12x = 3 × 20 = 60 x = 60 ÷ 12 = 5 Answer: 5

Q8.

What is the ratio of 40 minutes to 2 hours?

  1. A1:2
  2. B2:1
  3. C1:3
  4. D1:5
Show answer & solution

Answer: C. 1:3

Solution

Convert to the same unit: 2 hours = 120 minutes. Ratio = 40 : 120 = 1 : 3 Answer: 1:3

Q9.

The ages of two brothers are in the ratio 3 : 5 and their sum is 64 years. What is the age of the younger brother?

  1. A20 years
  2. B32 years
  3. C24 years
  4. D40 years
Show answer & solution

Answer: C. 24 years

Solution

Total parts = 3 + 5 = 8 One part = 64 ÷ 8 = 8 years Younger = 3 × 8 = 24 years Answer: 24 years

Q10.

In a class of 360 students the ratio of boys to girls is 5 : 4. How many girls are there?

  1. A200
  2. B160
  3. C180
  4. D144
Show answer & solution

Answer: B. 160

Solution

Total parts = 5 + 4 = 9 One part = 360 ÷ 9 = 40 Girls = 4 × 40 = 160 Answer: 160

Q11.

What number must be added to both terms of the ratio 7 : 11 to make it 3 : 4?

  1. A5
  2. B3
  3. C4
  4. D6
Show answer & solution

Answer: A. 5

Solution

Let the number be x: (7 + x)/(11 + x) = 3/4 4(7 + x) = 3(11 + x) 28 + 4x = 33 + 3x x = 5 Check: 12/16 = 3/4. Answer: 5

Q12.

Which of the following ratios is the greatest?

  1. A3:4
  2. B5:7
  3. C2:3
  4. D7:9
Show answer & solution

Answer: D. 7:9

Solution

Convert to decimals: 3/4 = 0.750, 5/7 ≈ 0.714, 7/9 ≈ 0.778, 2/3 ≈ 0.667. The largest is 7/9. Answer: 7:9

Q13.

What is the duplicate ratio of 2 : 3?

  1. A2:9
  2. B4:6
  3. C8:27
  4. D4:9
Show answer & solution

Answer: D. 4:9

Solution

Duplicate ratio of a:b is a² : b². 2² : 3² = 4 : 9 Answer: 4:9

Q14.

What is the sub-duplicate ratio of 16 : 25?

  1. A8:5
  2. B256:625
  3. C4:5
  4. D2:5
Show answer & solution

Answer: C. 4:5

Solution

Sub-duplicate ratio of a:b is √a : √b. √16 : √25 = 4 : 5 Answer: 4:5

Q15.

If 8 pens cost ₹96, what is the cost of 15 pens of the same kind?

  1. A₹150
  2. B₹180
  3. C₹120
  4. D₹192
Show answer & solution

Answer: B. ₹180

Solution

Cost of 1 pen = 96 ÷ 8 = ₹12 Cost of 15 pens = 15 × 12 = ₹180 Answer: ₹180

Medium questions

Q16.

₹60,000 is divided among A, B and C in the ratio 2 : 3 : 5. How much more does C get than B?

  1. A₹12,000
  2. B₹18,000
  3. C₹6,000
  4. D₹30,000
Show answer & solution

Answer: A. ₹12,000

Solution

Total parts = 2 + 3 + 5 = 10 One part = 60,000 ÷ 10 = ₹6,000 B = 3 × 6,000 = ₹18,000, C = 5 × 6,000 = ₹30,000 Difference = 30,000 − 18,000 = ₹12,000 Answer: ₹12,000

Shortcut: Difference is (5 − 3) = 2 parts = 2 × 6,000.

Q17.

A bag has ₹1, 50-paise and 25-paise coins in the ratio 2 : 3 : 4 (by number). If the total value is ₹54, how many 50-paise coins are there?

  1. A36
  2. B24
  3. C48
  4. D72
Show answer & solution

Answer: A. 36

Solution

Let the numbers be 2x, 3x and 4x. Value = 2x × 1 + 3x × 0.50 + 4x × 0.25 = 2x + 1.5x + x = 4.5x 4.5x = 54, so x = 12 50-paise coins = 3 × 12 = 36 Answer: 36

Q18.

Two numbers are in the ratio 3 : 4 and their LCM is 240. What is their sum?

  1. A120
  2. B180
  3. C70
  4. D140
Show answer & solution

Answer: D. 140

Solution

Let the numbers be 3x and 4x. Their LCM = 12x (3 and 4 are co-prime). 12x = 240, so x = 20 Numbers are 60 and 80, sum = 140 Answer: 140

Q19.

The incomes of A and B are in the ratio 5 : 4 and their expenditures are in the ratio 3 : 2. If each saves ₹1,600, what is A's income?

  1. A₹3,200
  2. B₹5,000
  3. C₹4,000
  4. D₹2,400
Show answer & solution

Answer: C. ₹4,000

Solution

Let incomes be 5x and 4x, expenditures 3y and 2y. 5x − 3y = 1,600 ... (1) 4x − 2y = 1,600 ... (2) From (2): y = 2x − 800. Put in (1): 5x − 6x + 2,400 = 1,600, so x = 800 A's income = 5 × 800 = ₹4,000 (Check: y = 800, expenditure of A = 2,400, saving 1,600.) Answer: ₹4,000

Q20.

Two numbers are in the ratio 5 : 7. If 10 is added to each, the ratio becomes 15 : 19. What is the sum of the original numbers?

  1. A36
  2. B48
  3. C60
  4. D24
Show answer & solution

Answer: B. 48

Solution

Let the numbers be 5x and 7x. (5x + 10)/(7x + 10) = 15/19 95x + 190 = 105x + 150 10x = 40, x = 4 Numbers: 20 and 28, sum = 48 Answer: 48

Q21.

The present ages of Rohit and Sunil are in the ratio 4 : 5. After 6 years the ratio will be 5 : 6. What is Sunil's present age?

  1. A24 years
  2. B30 years
  3. C36 years
  4. D25 years
Show answer & solution

Answer: B. 30 years

Solution

Let the ages be 4x and 5x. (4x + 6)/(5x + 6) = 5/6 24x + 36 = 25x + 30 x = 6 Sunil's age = 5 × 6 = 30 years Answer: 30 years

Q22.

The sum of two numbers is 120 and their difference is 24. What is the ratio of the larger to the smaller?

  1. A3:2
  2. B5:3
  3. C4:3
  4. D5:2
Show answer & solution

Answer: A. 3:2

Solution

Larger = (120 + 24)/2 = 72, smaller = (120 − 24)/2 = 48. Ratio = 72 : 48 = 3 : 2 Answer: 3:2

Q23.

₹1,300 is divided among A, B and C in the ratio 1/2 : 1/3 : 1/4. What is B's share?

  1. A₹600
  2. B₹300
  3. C₹433
  4. D₹400
Show answer & solution

Answer: D. ₹400

Solution

Multiply by the LCM 12: 1/2 : 1/3 : 1/4 = 6 : 4 : 3 Total parts = 13, one part = 1,300 ÷ 13 = ₹100 B's share = 4 × 100 = ₹400 Answer: ₹400

Q24.

Find the compound ratio of 2 : 3, 6 : 7 and 7 : 8.

  1. A2:3
  2. B1:3
  3. C1:2
  4. D12:7
Show answer & solution

Answer: C. 1:2

Solution

Compound ratio = (2 × 6 × 7) : (3 × 7 × 8) = 84 : 168 Divide by 84: 1 : 2 Answer: 1:2

Q25.

If a : b = 2 : 3, find (3a + 2b) : (a + 4b).

  1. A12:11
  2. B5:7
  3. C6:7
  4. D3:4
Show answer & solution

Answer: C. 6:7

Solution

Take a = 2k, b = 3k. 3a + 2b = 6k + 6k = 12k a + 4b = 2k + 12k = 14k Ratio = 12 : 14 = 6 : 7 Answer: 6:7

Q26.

x is inversely proportional to y. If x = 12 when y = 5, find x when y = 15.

  1. A36
  2. B4
  3. C9
  4. D60
Show answer & solution

Answer: B. 4

Solution

Inverse variation: x × y = constant. 12 × 5 = 60 = x × 15 x = 60 ÷ 15 = 4 Answer: 4

Q27.

12 men can finish a piece of work in 15 days. How many days will 20 men take to finish it?

  1. A9 days
  2. B25 days
  3. C10 days
  4. D12 days
Show answer & solution

Answer: A. 9 days

Solution

Men and days are inversely proportional: men × days = constant. 12 × 15 = 180 man-days Days for 20 men = 180 ÷ 20 = 9 Answer: 9 days

Q28.

Two numbers are in the ratio 8 : 11. If 4 is subtracted from each, the ratio becomes 2 : 3. What is the larger number?

  1. A16
  2. B33
  3. C18
  4. D22
Show answer & solution

Answer: D. 22

Solution

Let the numbers be 8x and 11x. (8x − 4)/(11x − 4) = 2/3 24x − 12 = 22x − 8 2x = 4, x = 2 Larger number = 11 × 2 = 22 Answer: 22

Q29.

A : B = 2 : 3 and B : C = 4 : 5. If ₹1,050 is divided among A, B and C in the ratio A : B : C, what does C get?

  1. A₹360
  2. B₹525
  3. C₹420
  4. D₹450
Show answer & solution

Answer: D. ₹450

Solution

A:B = 2:3 = 8:12 and B:C = 4:5 = 12:15, so A:B:C = 8:12:15. Total parts = 35, one part = 1,050 ÷ 35 = ₹30 C = 15 × 30 = ₹450 Answer: ₹450

Q30.

If x : y = 3 : 2, find (x² + y²) : (x² − y²).

  1. A5:1
  2. B13:1
  3. C13:5
  4. D9:4
Show answer & solution

Answer: C. 13:5

Solution

Take x = 3, y = 2. x² + y² = 9 + 4 = 13 x² − y² = 9 − 4 = 5 Ratio = 13 : 5 Answer: 13:5

Q31.

If 2A = 3B = 4C, what is A : B : C?

  1. A2:3:4
  2. B6:4:3
  3. C4:3:2
  4. D3:4:6
Show answer & solution

Answer: B. 6:4:3

Solution

Let 2A = 3B = 4C = 12 (the LCM). A = 6, B = 4, C = 3 A : B : C = 6 : 4 : 3 Answer: 6:4:3

Shortcut: A is the largest since it is multiplied by the smallest coefficient: A:B:C = 1/2 : 1/3 : 1/4.

Q32.

The ages of A and B are in the ratio 7 : 9. Five years ago the sum of their ages was 54 years. What is B's present age?

  1. A36 years
  2. B28 years
  3. C32 years
  4. D40 years
Show answer & solution

Answer: A. 36 years

Solution

Let the ages be 7x and 9x. Five years ago: (7x − 5) + (9x − 5) = 54 16x = 64, x = 4 B = 9 × 4 = 36 years Answer: 36 years

Q33.

₹4,500 is divided among A, B and C. A gets 2/3 of B's share and B gets 3/4 of C's share. How much does C get?

  1. A₹2,000
  2. B₹1,000
  3. C₹1,500
  4. D₹1,800
Show answer & solution

Answer: A. ₹2,000

Solution

Let C = 4 parts. B = 3/4 of C = 3 parts. A = 2/3 of B = 2 parts. A:B:C = 2:3:4, total 9 parts. One part = 4,500 ÷ 9 = ₹500 C = 4 × 500 = ₹2,000 Answer: ₹2,000

Q34.

On a map, 1 cm represents 25 km. A road is 450 km long. How long is it on the map?

  1. A11.25 cm
  2. B16 cm
  3. C20 cm
  4. D18 cm
Show answer & solution

Answer: D. 18 cm

Solution

Map length = 450 ÷ 25 = 18 cm Answer: 18 cm

Q35.

If 6, 10, x and 25 are in proportion, find x.

  1. A12
  2. B20
  3. C15
  4. D16.67
Show answer & solution

Answer: C. 15

Solution

6 : 10 = x : 25 10x = 6 × 25 = 150 x = 15 Answer: 15

Hard questions

Q36.

In a school the ratio of boys to girls is 7 : 5. When 12 boys leave and 8 girls join, the numbers become equal. How many students were there originally?

  1. A60
  2. B120
  3. C100
  4. D144
Show answer & solution

Answer: B. 120

Solution

Let boys = 7x and girls = 5x. 7x − 12 = 5x + 8 2x = 20, x = 10 Original total = 12x = 120 Answer: 120

Q37.

A : B = 2 : 3 and B : C = 5 : 4. If A and C together have ₹2,200, how much does B have?

  1. A₹1,000
  2. B₹1,500
  3. C₹1,200
  4. D₹1,650
Show answer & solution

Answer: B. ₹1,500

Solution

A:B = 2:3 = 10:15 and B:C = 5:4 = 15:12, so A:B:C = 10:15:12. A + C = 22 parts = 2,200, so one part = ₹100 B = 15 × 100 = ₹1,500 Answer: ₹1,500

Q38.

The salaries of A, B and C are in the ratio 2 : 3 : 5. They get raises of 15%, 10% and 20% respectively. What is the new ratio of their salaries?

  1. A23:33:60
  2. B2:3:5
  3. C23:30:50
  4. D46:33:60
Show answer & solution

Answer: A. 23:33:60

Solution

Take salaries 2, 3, 5 (in thousands). New A = 2 × 1.15 = 2.3, new B = 3 × 1.10 = 3.3, new C = 5 × 1.20 = 6.0 Ratio = 2.3 : 3.3 : 6 = 23 : 33 : 60 Answer: 23:33:60

Q39.

The present ages of a father and his son are in the ratio 7 : 3. After 6 years the father will be twice as old as the son. What was the father's age 10 years ago?

  1. A42 years
  2. B36 years
  3. C28 years
  4. D32 years
Show answer & solution

Answer: D. 32 years

Solution

Let the ages be 7x and 3x. 7x + 6 = 2(3x + 6) 7x + 6 = 6x + 12, so x = 6 Father now = 42, so 10 years ago he was 32 Answer: 32 years

Q40.

A : B = 2 : 3 and B : C = 4 : 5. If ₹8,750 is divided among A, B and C in the ratio A : B : C, how much more does C get than A?

  1. A₹2,250
  2. B₹1,500
  3. C₹1,750
  4. D₹1,250
Show answer & solution

Answer: C. ₹1,750

Solution

A:B:C = 8:12:15 (make B equal to 12). Total = 35 parts, one part = 8,750 ÷ 35 = ₹250 C − A = (15 − 8) × 250 = 7 × 250 = ₹1,750 Answer: ₹1,750

Q41.

y varies directly as the square of x. If y = 18 when x = 3, find y when x = 5.

  1. A30
  2. B90
  3. C50
  4. D75
Show answer & solution

Answer: C. 50

Solution

y = kx², so k = 18 ÷ 9 = 2. When x = 5: y = 2 × 25 = 50 Answer: 50

Shortcut: y2/y1 = (x2/x1)² = 25/9, so y = 18 × 25/9 = 50.

Q42.

A box has ₹100, ₹50 and ₹10 notes in the ratio 3 : 5 : 8 (by number). If the total amount is ₹12,600, how many notes are there altogether?

  1. A100
  2. B320
  3. C280
  4. D160
Show answer & solution

Answer: B. 320

Solution

Let the numbers be 3x, 5x and 8x. Value = 3x × 100 + 5x × 50 + 8x × 10 = 300x + 250x + 80x = 630x 630x = 12,600, so x = 20 Total notes = (3 + 5 + 8) × 20 = 16 × 20 = 320 Answer: 320

Q43.

If a/b = 2/3, b/c = 4/5 and c/d = 6/7, find a : d.

  1. A16:35
  2. B48:35
  3. C8:21
  4. D2:7
Show answer & solution

Answer: A. 16:35

Solution

a/d = (a/b) × (b/c) × (c/d) = (2/3) × (4/5) × (6/7) = 48/105 = 16/35 Answer: 16:35

Q44.

The sums (a + b), (b + c) and (c + a) are in the ratio 6 : 7 : 8 and a + b + c = 21. What is the largest of a, b and c?

  1. A7
  2. B8
  3. C10
  4. D9
Show answer & solution

Answer: D. 9

Solution

Let a + b = 6k, b + c = 7k, c + a = 8k. Adding: 2(a + b + c) = 21k, so 42 = 21k and k = 2. a + b = 12, b + c = 14, c + a = 16 c = 21 − 12 = 9, a = 21 − 14 = 7, b = 21 − 16 = 5 Largest = 9 Answer: 9

Q45.

₹2,400 is divided among A, B and C so that 1/3 of A's share = 1/4 of B's share = 1/5 of C's share. What does C get?

  1. A₹600
  2. B₹800
  3. C₹720
  4. D₹1,000
Show answer & solution

Answer: D. ₹1,000

Solution

Let 1/3 of A = 1/4 of B = 1/5 of C = t. Then A = 3t, B = 4t, C = 5t, total 12t = 2,400. t = 200, so C = 5 × 200 = ₹1,000 Answer: ₹1,000

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