Technique

The age difference between two people never changes

Arithmetic • Age Problems

Statement

If two people are now aged

a

and

b

, then

t

years earlier their ages were

a - t

and

b - t

, and

t

years later they will be

a + t

and

b + t

; in every case the difference is

a - b

. The gap is therefore the same at every date, so naming it

d

and carrying it across the dates in the question removes one unknown.

How to use it

1When facts are given at two different dates, write the gap once as $d$ and translate each date into an equation in $d$ and one person's age.
2When the question asks for the difference itself, make $d$ the unknown and no further step is needed at the end.
3Use it to check an answer: compute the gap at each date mentioned and confirm it comes out the same.

Worked example

Three years ago a father was five times as old as his son. He is now three times as old as his son. What is the difference between their ages?

Step-by-step

1
Son $s$ now, father $s + d$
Why: The gap $d$ is the same at both dates, so it is named once and reused.
2
Now: $s + d = 3 s \Rightarrow d = 2 s$
Why: Translate the present-day fact.
3
Three years ago: $s + d - 3 = 5 (s - 3) \Rightarrow d = 4 s - 12$
Why: Both ages drop by $3$ , so the gap is untouched.
4
$2 s = 4 s - 12 \Rightarrow s = 6$ , so $d = 12$

Answer

12

Watch out

Only the difference is invariant. The RATIO of two ages changes every year, so a ratio given at one date may never be carried to another — that changing ratio is exactly what a two-date age problem is built on.

Builds on

Turn a worded condition into a linear equation in one unknown

Also known as

age difference is constant, age gap invariant, difference of ages never changes, ages at two different dates, age problem with two dates

Appears in past papers

2020 (II)

Q27 Q28