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Applied arithmetic

Ages: move everyone along the timeline

Translate age statements into equations and use the fact that age differences stay constant while ratios change.

IntermediateAbout 13 min3 worked examples
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01 / THE STARTING POINT

Before you begin

What you need: Ratios and linear equations.

  • Represent present, past and future ages
  • Translate ratio statements at the correct time
  • Check solutions against every age condition
02 / BUILD THE CONCEPT

Understand the idea

Everyone advances by the same time

If a person is x years old now, their age t years later is x + t, and t years ago it was x − t. The same t must be applied to everyone in that statement.

Differences stay fixed

If one person is 12 years older, they remain 12 years older after any shared time shift. Ratios usually change because adding the same amount to both ages does not preserve a ratio.

Ratio parts represent present ages

If present ages are in ratio a:b, write them as ax and bx for a positive scale x. Use a sum, difference or future relation to determine x, then check that past ages are nonnegative.

03 / A DIFFERENT WAY TO SEE IT

Same years added. A different ratio.

WhenYoungerOlderRatio
Now10201:2
In 10 years20302:3
In 20 years30403:4
The 10-year age difference stays fixed in every row.
04 / FROM IDEA TO ANSWER

A method you can reuse

KEEP THIS HANDY

Present ages ax, bx → after t years: ax + t, bx + t

Age difference stays (b − a)x; the ratio generally changes.
  1. Choose a variable for one age or for a shared ratio unit.
  2. Translate the sum, difference or time-shifted ratio into an equation.
  3. Solve and substitute into both the present and future/past statements.
A useful insight

A known age difference is often the fastest way to determine the ratio scale: divide the age difference by the difference of the ratio terms.

05 / WATCH THE METHOD WORK

Worked examples

Read the question first. Try a step yourself, then compare your reasoning.

EXAMPLE 01

Use a sum

Two ages are in ratio 2:3 and sum to 50. Find the ages.

  1. Write the ages as 2x and 3x.
  2. 5x = 50, so x = 10.
  3. Ages are 20 and 30.
Answer20 years and 30 years
EXAMPLE 02

A future ratio

A father is three times his son's age. In 10 years he will be twice the son's age. Find present ages.

  1. Let the son be x and father 3x.
  2. 3x + 10 = 2(x + 10).
  3. 3x + 10 = 2x + 20, so x = 10.
AnswerSon 10; father 30
EXAMPLE 03

Use the fixed difference

An older person is 12 years older than a younger person. Their ages are in ratio 5:3. Find them.

  1. Write older = 5x and younger = 3x.
  2. 2x = 12, so x = 6.
  3. Check the difference: 30 − 18 = 12.
Answer30 years and 18 years
06 / YOUR TURN

Check your understanding

Quick questions, with explanations. These are for self-study and do not affect your account score.

0 of 3 questions checked

QUESTION 1A person is 18 now. Their age 7 years ago?

See the explanation

11. 18 − 7 = 11.

QUESTION 2Ages are 12 and 20 now. Difference after 5 years?

See the explanation

8. The future ages are 17 and 25, still 8 years apart.

QUESTION 3Ages in ratio 1:4 sum to 45. Younger age?

See the explanation

9. Five parts total 45, so one part is 9.

07 / TAKE THE NEXT STEP

Put your understanding to work

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EXPLORE THE NUMBERS

Age Ratio Calculator

Change the inputs and follow the working.

Open calculator →TEST YOUR KNOWLEDGE

Ages Practice Quiz 1

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Ages Practice Quiz 2

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Ages Practice Quiz 3

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