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Ages

    • Theory
    • Set 1
    • Set 2
    • Set 3
    • Set 4
    • Set 1
    • Set 2
  1. Aptitude
  3. Quantitative
  5. Ages
  7. Theory

Problems on Ages: Concepts, Formulas, and Clear Examples

Age questions look like family stories, but they are linear equations with a time shift. Present ages are unknowns; phrases like “5 years ago” or “after 8 years” add or subtract the same ttt to every person in that clause.

Campus tests (TCS NQT, Infosys, Wipro) often give today’s ratio and ask for the ratio after ttt years—or the present ages from sum and difference. One sign error produces a neat trap option.

Use a multiplier for ratios (3:5⇒3k,5k3:5 \Rightarrow 3k, 5k3:5⇒3k,5k) and remember the age gap is constant between two people at any date.

Timeline: present, past, future

If A’s present age is aaa, then ttt years ago A was (a−t)(a - t)(a−t) and after ttt years will be (a+t)(a + t)(a+t). Every person in the same sentence uses the same ttt.

Shift by t years

future age=a+t,past age=a−t\text{future age} = a + t,\quad \text{past age} = a - tfuture age=a+t,past age=a−t
  • Difference A−BA - BA−B never changes with time.
  • Reject negative ages after solving.

Ratio ages and multiplier k

From ratio

A:B=m:n  ⇒  A=mk,  B=nkA : B = m : n \;\Rightarrow\; A = mk,\; B = nkA:B=m:n⇒A=mk,B=nk

Ratio after t years

mk+tnk+t=pq\frac{mk + t}{nk + t} = \frac{p}{q}nk+tmk+t​=qp​

Example — Present ages from future ratio

Ratio now 4 : 5. After 6 years, ratio 5 : 6.

4k+65k+6=56\frac{4k+6}{5k+6}=\frac{5}{6}5k+64k+6​=65​. Cross-multiply: 6(4k+6)=5(5k+6)6(4k+6)=5(5k+6)6(4k+6)=5(5k+6) → 24k+36=25k+3024k+36=25k+3024k+36=25k+30 → k=6k=6k=6. Ages: 24 and 30.

Sum and difference

Two unknowns

A+B=S, A−B=D  ⇒  A=S+D2, B=S−D2A+B=S,\ A-B=D \;\Rightarrow\; A=\frac{S+D}{2},\ B=\frac{S-D}{2}A+B=S, A−B=D⇒A=2S+D​, B=2S−D​

Was / will be multiples

“Father was thrice son’s age 12 years ago” → (F−12)=3(S−12)(F-12)=3(S-12)(F−12)=3(S−12). “Will be twice after 8 years” → (F+8)=2(S+8)(F+8)=2(S+8)(F+8)=2(S+8). Solve the pair.

Past

F−t=k(S−t)F - t = k(S - t)F−t=k(S−t)

Future

F+t=k′(S+t)F + t = k'(S + t)F+t=k′(S+t)

Average age and replacements

  • Sum of ages = (average) ×\times× (count).
  • Removing age aaa and adding bbb changes sum by (b−a)(b-a)(b−a); average shifts by (b−a)/n(b-a)/n(b−a)/n.

Verification checklist

  • Plug into every time clause in the question.
  • Check ratio direction (A:B not B:A).
  • Integer ages when options are integers.

Step-by-step method for exam questions

  1. Mark Present / t years ago / t years hence for each person.
  2. Convert ratios to mk,nkmk, nkmk,nk.
  3. Write equations from sum, difference, or ratio conditions.
  4. Solve for kkk or ages; discard invalid values.
  5. Substitute back into all time phrases.
  6. Pick the matching option.

Formula cheat sheet

SituationFormulaWhen to use
Ratio m:n todayA=mk, B=nkA=mk,\ B=nkA=mk, B=nkBefore other conditions
Time shiftage±t\text{age} \pm tage±tAgo / after wording
Fixed gapA−B=constantA-B=\text{constant}A−B=constantFather–son type
Sum S, diff DA=S+D2A=\frac{S+D}{2}A=2S+D​Two-person MCQs
Average changeΔxˉ=b−an\Delta\bar{x}=\frac{b-a}{n}Δxˉ=nb−a​Member replaced

Frequently asked questions

Why do I get negative k?

Usually swapped ratio order or used +t instead of −t for ‘years ago’. Flip ratio or change sign of t.

Can I use one variable for father and son?

Yes: son = S, father = S + d with constant d, or use ratio multiplier.

Do ages have to be integers?

Often yes in MCQs; fractional k suggests a setup error.

Fast verification?

Check both past and future statements in the stem with your final ages.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.

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