Ages

• How age problems use linear relationships, ratios, and time shifts (years ago/hence).
• To translate "in 5 years", "5 years ago" into algebraic expressions cleanly.
• To solve ratio-of-ages questions when totals or differences are given.
• To handle Class 12 age puzzles combining ratios with future/past conditions.

Level 1 — Foundations

Verbal: If present age is x, then in t years age = x+t; t years ago = x−t (assume valid non-negative age).

Standard setup: Define present ages as variables; write equations from each clue.

Ratio ages: If A:B = p:q now, ages pt and qt only if same multiplier k → A=pk, B=qk.

Typical clues:

Consistency: Ages increase equally for all people over same time gap.

Level 2 — Exam depth

Reversal check: Substitute solved ages into every clue including future/past.

Average age: Sum ÷ count — if average changes when one person joins, set sum equations.

Age difference constant: Parent-child age gap unchanged over time — use gap to eliminate variables.

Ratio + sum: a:b=3:5 and sum 48 → parts 8, multiply 6 → 18 and 30.

Exam trick: "Age of father 4 years ago equals age of son 4 years hence" → F−4 = S+4.

Father-son age with future condition

Father is 4 times son's age. In 20 years, father will be twice son's age. Find ages.
Let son = x, father = 4x.
Clue: 4x+20 = 2(x+20) → 4x+20 = 2x+40 → 2x=20 → x=10.
Son **10**, father **40**. Check: in 20 years 60 and 30 → twice ✓.

Ratio ages with given sum

A:B = 5:7, sum 36 → k=36/12=3 → ages **15** and **21**. Gap 6 years preserved in future clues.

• Present age 17 — age 9 years ago and 6 years hence?
• Sum 55; ratio 2:3 — find ages.
• Write equation: "Mother 36, in 4 years daughter half her age."
• Stretch: Ages in AP — three siblings sum 45; middle 15 — find others?

Revision tip: Revisit adjacent topics in Quantitative Reasoning before mixed practice on Ages.

Open the Practice tab for graded questions on Ages.

Define present ages with single letters and write time-shifted ages in parentheses beside them (+5, −3). Ratio problems always use nk form, never raw ratio numbers as ages. After solving, verify every time phrase ("ten years hence," "four years ago") in one pass. Age questions rarely need quadratic equations at Class 12 reasoning level — linear systems suffice.

Age puzzles combine ratio setup with linear equations — master ratio splits before tackling time-shift clauses. Similar logic models work–wage and exam score comparison problems. Graph age timelines on a number line when multiple "years ago/hence" clauses appear — visualisation beats repeated algebra. Verify solutions by plugging back into every temporal phrase, including hidden ones in compound sentences.