Lessons / Quantitative aptitude
Quantitative aptitude · Day 6

Ages

The gap between two people's ages never changes. Anchor on the gap and every ratio question becomes counting units.

Practise (11 questions) Present as slides

Why this topic matters

Everyone gets older at the same speed. So the DIFFERENCE between two people's ages never changes.

Age questions are ratio and linear equations dressed up. Use the fact that the age gap is fixed, or just try the options.

The fixed gap, and the bar model

If a father is 30 years older than his son now, he was 30 years older 10 years ago and will be 30 years older in 10 years.

Ratios change with time; the gap does not. Put the gap in the bar model and one box's value is found.

SonFathergap
Now 2 : 5. The 3-box gap is fixed forever
Example 1The ratio of a father's age to his son's is 5 : 2. The father is 30 years older. Their ages?
  1. The gap is 5 - 2 = 3 parts.Bar model: 3 boxes.
  2. 3 parts = 30, so 1 part = 10.
  3. Father 50, son 20.
Example 2Four years ago, A was 3 times as old as B. In 4 years A will be twice as old as B. A's present age?
  1. Let B four years ago = x, A = 3x. Now: A = 3x + 4, B = x + 4.Start from the time the question describes first.
  2. In 4 years: 3x + 8 = 2(x + 8).
  3. x = 8.
  4. A now = 28.

Try the options

Age questions always have whole-number answers, and the options test directly: put the option in, check every sentence.

Example 1The sum of the ages of a mother and daughter is 50. Five years ago the mother was 7 times as old as the daughter. Mother's age? (a) 35 (b) 40 (c) 42 (d) 45
  1. Try 40: daughter 10. Five years ago: 35 and 5. 35 = 7 × 5. Fits.Check both sentences.
  2. 40.

Formula summary

Age difference is constant
If ratio changes from a:b to c:d, the difference in units must match: scale both ratios to the same difference.
n years ago / hence
Subtract / add n to EVERY person.
Sum after n years
Sum + n × (number of people).
Children born at intervals
Ages x, x + k, x + 2k ... sum = n × + k(0 + 1 + ... + n - 1).

Shortcuts and tricks

Ratio-units trick (no equations)

Use it when: 'Ratio now 4:5, 5 years ago 7:9'.

  1. Differences: 5 - 4 = 1 unit, 9 - 7 = 2 units. Make them equal: now 8:10.
  2. Now 8:10, then 7:9. Each person dropped 1 unit = 5 years.
  3. So 1 unit = 5, ages 40 and 50.
ExampleA:B = 4:5 now, 7:9 five years ago. B's age?
  1. Scale to 8:10 vs 7:9.
  2. 1 unit = 5 years.
  3. B = 10 × 5.

Times-older problems

Use it when: 'Father is 3 times son now; in 15 years, twice'.

  1. Now: 3:1, gap 2 units. Later 2:1, gap 1 unit → scale to 4:2.
  2. Now 3:1, later 4:2: son went 1 → 2 units = 15 years.
  3. Son 15, father 45.
ExampleFather's age?
  1. 45.

Before you move on, you should be able to...

  • use the fixed age gap
  • turn ratio statements into parts
  • check options against every sentence

Practice questions (11)

Try each one before opening the solution. Or practise them one by one so your score is saved.

easy The sum of ages of 5 children born 3 years apart is 50. Youngest?
  1. 5x + 3(0+1+2+3+4) = 50 → 5x = 20.

Answer: 4

medium Ten years ago A was half B's age. Their present ages are 3:4. Sum of present ages?
  1. Now 3:4 (gap 1 unit). Then 1:2 (gap 1 unit) - gaps already match.
  2. Each dropped 2 units (3 → 1, 4 → 2) = 10 years, so 1 unit = 5.
  3. Ages 15 and 20, sum 35.

Answer: 35

medium A is 4 years more than twice B's age; the product of their ages is 240. A's age?
  1. b(2b + 4) = 240 → b = 10.

Answer: 24

medium A couple's average age at marriage was 23. Five years later, with one child, the family's average age is 20. Child's age?
  1. At marriage total 46; 5 years later 56.
  2. Family of 3 total 60 → child 4.

Answer: 4

hard A father's age is twice the sum of his two children's ages. In 20 years his age will equal the sum of theirs. Father's age now?
  1. F = 2S. F + 20 = S + 40.
  2. 2S + 20 = S + 40 → S = 20.

Answer: 40

hard Six years ago Ravi was 3 times as old as Sita. Six years from now he will be twice her age. Ravi's age now?
  1. r - 6 = 3(s - 6), r + 6 = 2(s + 6).
  2. r = 3s - 12 and r = 2s + 6, so s = 18.
  3. r = 42.

Answer: 42

hard The ratio of A's and B's ages is 5:7. After 8 years it will be 7:9. The age difference?
  1. Ratio gap 2 units in both.
  2. 5 → 7 = 2 units = 8 years → 1 unit = 4.
  3. Gap 2 units = 8.

Answer: 8

hard A and B are 6:5 and their ages add to 44. Ratio after 8 years?
  1. 24 and 20 → 32 : 28.

Answer: 8:7

hard Three years ago a family of 5 averaged 17 years. A baby has since been born, and the average is still 17. Baby's age?
  1. Now the 5 total 85 + 15 = 100.
  2. 6 people × 17 = 102.

Answer: 2

hard A mother is 4 times her daughter's age. In 5 years she will be 3 times. Mother's age now?
  1. m = 4d, m + 5 = 3(d + 5) → d = 10.

Answer: 40

hard Two brothers differ by 10 years. 15 years ago the elder was twice the younger. Elder's age now?
  1. e = y + 10, e - 15 = 2(y - 15) → y = 25.

Answer: 35

Class plan (2 hours, for trainers)

A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.

10 minWarm-up.
25 minFixed gap; bar model.
30 minPast and future equations.
20 minTry the options.
30 minPractice set.
5 minRecap.
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