4.3 Age & Relationship Problems
Key Takeaways
- Translate each verbal statement into an equation immediately: "X is twice as old as Y" → X = 2Y; "sum of ages is 34" → X + Y = 34.
- Two unknowns need two equations — pair a sum/difference or sum/ratio to solve by elimination or substitution.
- For sum S and difference D: older = (S + D)/2, younger = (S − D)/2. Sum 34, difference 8 → ages 21 and 13.
- Decode family-tree chains from the speaker outward: "my mother's husband" = my father; "my father's sister" = my aunt; combine step by step.
- Gender-neutral terms (cousin, sibling) are traps — read whether the relationship is maternal (mother's side) or paternal (father's side) before picking an option.
4.3 Age & Relationship Problems
Quick Answer: Turn each sentence into an equation on the spot. Two unknowns require two equations; solve by elimination. For relationships, decode the chain one link at a time starting from the speaker.
Two sub-types appear under this heading on the PAF GD Pilot Initial Test: age word problems (algebraic translation) and relationship-decoding puzzles (kinship logic). Both reward disciplined translation rather than mental gymnastics.
Age Problems — The Translation Method
Read each clause and write an equation. Do not try to solve in your head.
Common translations:
| Verbal phrase | Equation |
|---|---|
| "X is twice as old as Y" | X = 2Y |
| "X is 5 years older than Y" | X = Y + 5 |
| "Sum of their ages is 34" | X + Y = 34 |
| "X's age 4 years ago" | X − 4 |
| "In 3 years, X will be…" | X + 3 |
| "X was twice as old as Y 5 years ago" | X − 5 = 2(Y − 5) |
The Sum-and-Difference Shortcut
If you are given the sum S and difference D of two ages, the ages are:
- Older = (S + D) / 2
- Younger = (S − D) / 2
Worked Example (candidate-reported style)
The sum of Ali's and Ahmed's ages is 34, and Ali is 8 years older than Ahmed. Find their ages.
Using the shortcut:
- Older (Ali) = (34 + 8) / 2 = 42 / 2 = 21.
- Younger (Ahmed) = (34 − 8) / 2 = 26 / 2 = 13.
Check: 21 + 13 = 34 ✓; 21 − 13 = 8 ✓.
Algebraic alternative: let Ahmed = x, Ali = x + 8. Then x + (x + 8) = 34 → 2x = 26 → x = 13, Ali = 21.
Ratio Age Problems
"X and Y are in the ratio 3:2 and X is 6 years older than Y." Let the common multiplier be k: X = 3k, Y = 2k. Then 3k − 2k = k = 6, so X = 18 and Y = 12.
Future/Past Age Problems
A father is 4 times as old as his son. In 20 years, he will be twice as old. Find their present ages.
Let son = s, father = 4s. In 20 years: 4s + 20 = 2(s + 20) → 4s + 20 = 2s + 40 → 2s = 20 → s = 10. Son = 10, father = 40.
Relationship-Decoding — Chain Method
Decode the chain one link at a time, starting from the speaker and moving outward.
Worked Example
A man points to a photograph and says: "Her mother's husband is my mother's only brother." How is the woman in the photograph related to him?
Step 1 — "her mother's husband" = her father. Step 2 — "my mother's only brother" = my maternal uncle. Step 3 — So her father = my maternal uncle, meaning she is my uncle's daughter → cousin.
Relationship Translation Table
| Phrase | Decoded relation |
|---|---|
| My mother's husband | My father |
| My father's sister | My aunt (paternal) |
| My mother's brother | My maternal uncle |
| My father's brother | My paternal uncle |
| My uncle's son/daughter | My cousin |
| My brother's wife | My sister-in-law |
| My wife's brother | My brother-in-law |
| My wife's sister | My sister-in-law |
| My sister's husband | My brother-in-law |
| My mother's sister | My maternal aunt |
| My father's mother | My grandmother |
| My son's wife | My daughter-in-law |
Common Traps
- Maternal vs paternal. The PAF exam may distinguish "maternal uncle" (mother's brother) from "paternal uncle" (father's brother) — read carefully.
- Gender of the photograph subject. "He is the son of my father's only son" — if the speaker is male, "my father's only son" could be the speaker himself, making the boy his son, not his cousin.
- "Only"/"sole" qualifiers. "My mother's only brother" rules out multiple uncles — the chain has exactly one resolution.
- Mixed time references in age problems. "Was" vs "will be" flips the sign of the offset: 5 years ago subtracts 5, in 5 years adds 5.
Worked Example: Three-Variable Age System
The sum of Aamina's, Bilal's, and Chaudhry's ages is 80. Aamina is twice as old as Bilal. Chaudhry is 10 years older than Bilal. Find each age.
Step 1 — Assign variables: let Bilal = b. Then Aamina = 2b, Chaudhry = b + 10. Step 2 — Translate the sum: 2b + b + (b + 10) = 80 → 4b + 10 = 80 → 4b = 70 → b = 17.5. Step 3 — Solve: Bilal = 17.5, Aamina = 35, Chaudhry = 27.5. Step 4 — Check: 17.5 + 35 + 27.5 = 80 ✓; 35 = 2 × 17.5 ✓; 27.5 = 17.5 + 10 ✓.
Three-variable systems are rarer but appear in the harder third of the verbal battery. The discipline is the same: one variable per unknown, one equation per clause, then eliminate.
Worked Example: In-Law Chain Decode
A woman says, "He is the son of my husband's sister." How is he related to her?
Step 1 — "my husband's sister" = my sister-in-law. Step 2 — "son of my sister-in-law" = my nephew (by marriage). Step 3 — He is her nephew (sister-in-law's son).
The trap: candidates stop at "sister-in-law" and pick "brother-in-law." Decode one link further — the son of the sister-in-law is a nephew, not an in-law.
Procedure
For age items: (1) assign variables, (2) translate every clause, (3) solve the system, (4) check against the original wording. For relationship items: (1) start from the speaker, (2) decode each link, (3) combine, (4) read all options before locking in.
The sum of two brothers' ages is 50 and the elder is 4 years older than the younger. What are their ages?
A man says, "She is the daughter of my grandfather's only son." How is she related to him?
A father is 3 times as old as his son. In 12 years, he will be twice as old. What is the son's present age?
Pointing to a woman, a man says, "Her brother's father is my father." How is the woman related to him?