Problems on Ages — Timeline Method and Shortcuts

Age problems look like riddles but they all use one fact:

Everyone gets older by the same amount. So the difference between two people's ages never changes.

The method

1. Choose one person's present age as x (or use a ratio like 3k and 5k).
2. Write every other age in terms of x, moving along the timeline (past: subtract, future: add — for both people).
3. Make one equation from the condition and solve.
4. Check your answer against the original sentence.

Example 1

A father is 30 years older than his son. In 5 years, the father will be 3 times as old as the son. Find their present ages.

Son = x, father = x + 30. In 5 years: x + 35 = 3(x + 5) → x + 35 = 3x + 15 → x = 10.

Son = 10, father = 40. Check: in 5 years 15 and 45, and 45 = 3 × 15 ✓

Example 2 — ratios

The ratio of the ages of A and B is 4 : 5. Six years later it will be 5 : 6. Find their present ages.

A = 4k, B = 5k. (4k + 6)/(5k + 6) = 5/6 → 24k + 36 = 25k + 30 → k = 6.

A = 24, B = 30.

Shortcut for ratios: the difference stays the same. Now the difference is 1 part (5 − 4); later it is also 1 part (6 − 5). So "1 part" is the same number of years in both ratios. A goes from 4 parts to 5 parts in 6 years → 1 part = 6 → A = 24, B = 30.

Example 3 — the past

Ten years ago, Meera's age was half of what it will be 15 years from now. What is her present age?

x − 10 = ½(x + 15) → 2x − 20 = x + 15 → x = 35.

Example 4 — sum of ages

The sum of the ages of a mother and daughter is 50. Five years ago the mother was 7 times as old as the daughter. Find their ages.

Daughter = x, mother = 50 − x. Five years ago: (45 − x) = 7(x − 5) → 45 − x = 7x − 35 → 8x = 80 → x = 10.

Daughter = 10, mother = 40.

Example 5 — average of a family

The average age of a husband and wife was 25 when they married. Six years later they have a child. The average age of the three is now 22. How old is the child?

At marriage the couple's total was 2 × 25 = 50. Six years later it is 50 + 6 + 6 = 62. The family's total now is 3 × 22 = 66. Child = 66 − 62 = 4 years.

Use the options (a huge time-saver)

In an exam, you can plug each option into the question. Example 1 options might be (10, 40), (12, 42), (15, 45), (8, 38). Only (10, 40) passes "in 5 years father = 3 × son". This takes 20 seconds.

Common mistakes

• Adding years to one person only. Time passes for everyone.
• Forgetting that the difference between ages stays the same — a free equation.
• Answering with the age at a different time (the question asks "present age" but you give the age "5 years later").

Quick recap

• One variable + timeline.
• Age difference never changes.
• Ratios: use parts; the difference in parts is the same number of years.
• Plug in the options to check fast.