Problems on Ages: Ratio Method

Every year, every person gets exactly one year older. So the difference between two people's ages never changes. A mother who is 24 years older than her daughter today was 24 years older ten years ago and will be 24 years older in twenty years' time.

Ratios of ages, on the other hand, do change: a father may be four times his son's age now but only twice it later. Age problems give you ratios at different times and ask for actual ages. The constant difference is the bridge between those ratios, and it turns most questions into a one-line calculation.

Worked example 1: difference in units

A mother is three times as old as her daughter, and the difference in their ages is 24 years. The ratio 3 : 1 has a difference of 2 units, and 2 units = 24 years, so 1 unit is 12. The daughter is 12 and the mother is 36.

When a question gives two ratios at two different times, follow three steps.

Find the difference between the terms of each ratio.
Scale one ratio so both differences are the same number of units. This is allowed because the real age difference is the same.
Compare the same person's units in both ratios. The change in units equals the number of years that passed.

Worked example 2: 3 : 5 now, 2 : 3 in 6 years

The differences are 2 units and 1 unit. Double the second ratio to 4 : 6 so both differences are 2. A's units go from 3 to 4 in 6 years, so 1 unit = 6 years. A is 3 × 6 = 18.

B is 5 × 6 = 30. Check: in 6 years they will be 24 and 36, and 24 : 36 = 2 : 3.

	A	B	Difference
Now	3	5	2
In 6 years (scaled)	4	6	2
Change	+1 unit	+1 unit	same

Notice that both people gain the same number of units, as they must, because both gain the same number of years.

"A father is four times as old as his son" is simply the ratio 4 : 1. The ratio-unit method works exactly as before.

Worked example 3: four times now, twice in 20 years

Now 4 : 1 has a difference of 3 units. Later 2 : 1 has a difference of 1 unit, so scale it to 6 : 3. The son goes from 1 unit to 3 units, a change of 2 units, in 20 years. So 1 unit = 10 years and the son is 10. The father is 40; in 20 years they will be 60 and 30.

If you prefer algebra, the same question is 4x + 20 = 2(x + 20), which gives x = 10. Use whichever you find faster, but always check by substituting back.

When the question gives the sum of the present ages and a ratio from the past, move the sum to that past time first. Each person was k years younger, so the sum was smaller by k times the number of people.

Worked example 4: sum 50, ratio 3 : 2 five years ago

Five years ago the sum was 50 − 10 = 40. A's share was 3/5 × 40 = 24, so A is now 24 + 5 = 29, and B is 21.

Worked example 5: a ratio now, a half before

The present ages of A and B are in the ratio 3 : 4. Ten years ago A was half B's age. With ages 3x and 4x, 3x − 10 = (4x − 10) ÷ 2, so 6x − 20 = 4x − 10 and x = 5. The ages are 15 and 20, a sum of 35.

Average questions about ages are really questions about totals. Convert each average into a total by multiplying by the number of people, and adjust each total for the years that have passed: a group of n people gains n years of total age every year.

Worked example 6: husband, wife and child

Three years ago the average age of a husband, wife and child was 27. Five years ago the average age of the wife and child was 20. Today the three have a total of 3 × (27 + 3) = 90, and the wife and child have 2 × (20 + 5) = 50. The husband is 90 − 50 = 40.

✘ Adding the years to only one person. Both get older by the same amount.
✘ Adjusting a sum of two ages by k instead of 2k.
✘ Comparing ratios without first matching their differences.
✘ Answering with the past or future age when the question asks for the present one.

Age problems share the ratio skills used in partnership problems and alligation and mixture. For more quick methods, see the Quant cheat code.

Q1. The ages of two brothers are in the ratio 4 : 7, and the elder is 15 years older than the younger. What is the elder brother's age? easy

(a) 28 years (b) 21 years (c) 35 years (d) 42 years

✅ Answer: (c) 35 years

💡 Explanation: The difference is 7 − 4 = 3 units, and 3 units = 15 years, so 1 unit = 5 years. The elder brother is 7 × 5 = 35 years old.

Q2. The present ages of A and B are in the ratio 5 : 3. After 4 years the ratio will be 3 : 2. What is A's present age? easy

(a) 20 years (b) 15 years (c) 25 years (d) 12 years

✅ Answer: (a) 20 years

💡 Explanation: Write the ages as 5x and 3x. Then (5x + 4) ÷ (3x + 4) = 3 ÷ 2, so 10x + 8 = 9x + 12 and x = 4. A is 5 × 4 = 20 years old.

Q3. A father is three times as old as his son. After 15 years he will be twice as old as his son. What is the father's present age? medium

(a) 30 years (b) 60 years (c) 40 years (d) 45 years

✅ Answer: (d) 45 years

💡 Explanation: Let the son be x. Then 3x + 15 = 2(x + 15), so x = 15. The father is 3 × 15 = 45 years old.

Q4. The sum of the ages of two sisters is 60 years. Six years ago their ages were in the ratio 5 : 3. What is the elder sister's present age? medium

(a) 30 years (b) 36 years (c) 40 years (d) 33 years

✅ Answer: (b) 36 years

💡 Explanation: Six years ago the sum was 60 − 12 = 48. The elder's share then was 5/8 × 48 = 30, so she is now 30 + 6 = 36 years old.

Q5. The average age of three children is 12 years. Including their mother, the average becomes 18 years. How old is the mother? easy

(a) 30 years (b) 42 years (c) 36 years (d) 24 years

✅ Answer: (c) 36 years

💡 Explanation: The children's total is 3 × 12 = 36. The total with the mother is 4 × 18 = 72. So the mother is 72 − 36 = 36 years old.

Q6. Eight years ago, A was three times as old as B. Four years from now, A will be twice as old as B. What is A's present age? hard

(a) 44 years (b) 40 years (c) 36 years (d) 48 years

✅ Answer: (a) 44 years

💡 Explanation: A − 8 = 3(B − 8) gives A = 3B − 16. A + 4 = 2(B + 4) gives 3B − 12 = 2B + 8, so B = 20 and A = 44. Check: 36 = 3 × 12 and 48 = 2 × 24.

When should I use equations instead of the ratio-unit method?

Use ratio units when the question gives two ratios for the same two people. Use a simple equation when it gives a multiple such as 'three times as old' at two different times, or when three or more people are involved.

How can I check my answer quickly?

Put the ages back into every statement in the question. Both the ratio now and the ratio at the other time must come out exactly, and the ages must be sensible, for example a parent older than the child by a realistic margin.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

How this page was prepared. Drafted with AI assistance (Claude (Anthropic AI model)) from the sources listed above. 13 calculations on this page were recomputed by software and matched. Automated checks also covered structure, internal links and duplicate text. Software checks cannot confirm facts. Reviewed by praveenkumar on 2026-10-20 against the sources listed above.

Found an error? Report it and we will correct the page.