Problems on Ages Cheat Sheet

Problems on Ages Cheat Sheet

Quick Revision

Problems on ages are linear equations in disguise. Name the present age with a variable, then shift it for the past or the future.

1. Past, Present and Future

If the present age is x:

t years ago   →  x − t
t years hence →  x + t

The difference between two people's ages never changes with time.

2. The Ratio Method

When two ages are in a ratio, write them with a common multiplier k:

Ages in ratio a : b  →  ak and bk

Then form one equation from the second condition and solve for k.

3. Worked Example

A father and son are aged in the ratio 4 : 1. After 10 years the father will be twice as old as the son. Find their present ages.

Let son = x, father = 4x
After 10 years:  4x + 10 = 2(x + 10)
                 4x + 10 = 2x + 20
                 2x = 10  →  x = 5

So the son is 5 and the father is 20.

4. Exam Quick Tips

• The age gap is constant — use it to check your answer.
• For "t years ago" subtract t from every person's age, not just one.
• A ratio alone cannot fix the ages; you need a second condition to find k.

Keep revising with the rest of the Maths section:

• All Maths topics
• Average Cheat Sheet
• Ratio and Proportion Cheat Sheet

How this was made: Drafted with AI assistance; spot-checked by the editor on 31 October 2026.