Age

Module 13 — Lesson 0013 · In Depth · 35 min

Age problems are linear equations in disguise — past, present, future with one constant: time moves equally for everyone. Place x at present, shift by ±n, and the ratio does the rest.

1. The Golden Rules

Rule	Statement	Example
1. Time travel is universal	If `n` years pass, everyone ages `n` years	Father `x`, Son `y` → after 5y: `x+5`, `y+5`
2. Difference is constant	Age gap never changes	Brother 5y older today → still 5y older in 50 years
3. Ratio changes	Multiples/ratios are snapshot-specific	Father 3× Son today → not 3× after 10y

Why Rule 3 matters: ratio 3:1 today means gap = 2 units. Adding 10y to both keeps gap same but total grows, so ratio shrinks. Never carry a ratio across time without shifting both ages.

2. Setting Up Variables — The Timeline Method

Place x at present — then past/future are offsets:

Timeline	Keywords	Expression
Past	ago, before, back then, was	`x − n`
Present	now, current, at present, is	`x`
Future	hence, after, from now, will be	`x + n`

Mathematical Relationships

Type A — Ratios: ratio 4:5 → A=4x, B=5x
Type B — Multipliers: "Father is 3× Son" → Son=x, Father=3x
Type C — Linear: "A is 20y older than B" → B=x, A=x+20

Setup tip: If ratio is given, use ax, bx immediately — not x, y. One variable is enough. For "20 years older", one variable is also enough (x, x+20), not two.

3. Solving Methodologies

Method 1: The Equation Approach (Standard)

Define current ages with one variable (e.g., 4x, 5x).
Shift to the time mentioned (add/subtract n from both).
Write equation from new condition, solve for x.

Worked: Ratio now 4:5, after 5y ratio 5:6.
Now: 4x, 5x → after 5y: 4x+5, 5x+5
→ (4x+5)/(5x+5)=5/6 → cross: 6(4x+5)=5(5x+5) → 24x+30=25x+25 → x=5
Ages now: 20 and 25. Check: after 5y → 25 and 30 → 5:6 ✓

Method 2: The Constant Difference Shortcut

Since age difference is constant, the gap in ratio units must be same at both times — compare the change in units directly.

Same example, faster:
Ratio now 4:5 (diff =1 unit), later 5:6 (diff =1 unit) — gaps match, so 1 unit of change corresponds to 5 years.
4→5 is +1 unit → 1 unit =5y → now: 4×5=20, 5×5=25 instantly.
When gaps don't match, scale one ratio to equalize the difference before comparing. E.g., 2:3 (diff 1) vs 3:5 (diff 2) → scale first to 4:6 (diff 2), then compare.

4. Averages in Ages

Total age = Average × N
Group aging: average of N people is A today → after Y years, average = A+Y (if no one leaves/joins) — everyone ages equally.
New baby: N increases, baby age =0 → total increases by 0, but divisor grows → average drops sharply. Example: average of 2 parents is 32, after baby average of 3 is (32×2+0)/3≈21.3 immediately.
Replacement: if a person leaves and is replaced, change in total = difference in their ages.

5. Complex Logic — "Was as old as…"

High-level template: "A is twice as old as B was when A was as old as B is now."

Decode step-by-step:
Let present ages be A, B, gap D = A−B.
"When A was as old as B is now" was D years ago (because A has aged D since then to reach current A).
At that time: A's age was A−D = B, B's age was B−D.
Statement "A is twice as old as B was then": A = 2×(B−D) = 2×(B − (A−B)) = 2×(2B−A) → 3A = 4B.
Key move: the gap D is the time shift between "now" and "then".

6. Summary of Variables

Symbol	Meaning
`x, y`	Present ages (usually one variable suffices)
`n, t`	Years ago/hence
`a:b`	Ratio → `ax, bx`
`Average`	Sum / count

Time travel rule: after n years, ages become?

Age difference over time?

Ratio now 4:5 → present ages?

Ratio 4:5 now, 5:6 after 5y — using equation, x=?

Constant-difference shortcut for 4:5 → 5:6 in 5y: 1 unit = ?

Average of N people is A today. After Y years (same group) average = ?

"A is twice as old as B was when A was as old as B is now" — first step?

Notes

Golden Rules :-

Universal ⇒ +n to everyone ; Gap ⇒ constant ; Ratio ⇒ changes (snapshot only)

Timeline :-

Present = x ; Past ⇒ x−n ; Future ⇒ x+n

Ratio a:b ⇒ ax,bx ; Times ⇒ Son=x, Father=3x ; Older ⇒ B=x, A=x+20

Methods :-

a) Equation ⇒ define now → shift → new ratio → solve (eg 4:5→5:6 in 5y →x=5)

b) Constant diff ⇒ gap in units same → 1 unit = years changed (scale to equalize diff if needed)

Averages :-

Total = Avg×N ; group +Y years → avg +Y ; baby (0) → avg drops sharply

Complex ⇒ gap D = A−B :-

"When A was as old as B now" ⇒ D years ago → A−D=B, B−D ; then statement → A=2(B−D)

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