SSC CGL Maths · Linear Equations

Age Problems

Write every age at the time the sentence talks about, remember that everyone ages by the same number of years, and the age problem becomes a linear equation or a pair of them.

All Linear Equations notes

Why this matters

Fourteen questions in the bank, seven EASY and seven MODERATE. Three give today's ages or their gap and ask when a condition holds, eight give conditions at two different times, and three tie three people together. One fact cuts most of them short: the gap between two people's ages never changes.

Concept 1 of 3: The gap between two ages never changes

In ten years both people are ten years older, so the difference between their ages stays exactly the same. Only the ratio of their ages changes. If you know the gap, a condition like "three times as old" tells you the ages at once.

Definition

After $t$ years, add $t$ to every age; $t$ years ago, subtract $t$ from every age. Write each condition with the ages at its own time.
The difference of two ages is the same at every time. The ratio is not.
Fast route, times as old: if the elder is $k$ times the younger at some moment, the gap is $k - 1$ times the younger's age at that moment. So the younger was $\frac{\text{gap}}{k - 1}$ .
Fast route, ratio: if the ages are in the ratio $m : n$ at some moment, the gap is $m - n$ parts, so one part $= \frac{\text{gap}}{m - n}$ .
The full method always works: let the unknown number of years be $x$ and write one equation.

Age-gap shortcut

\text{younger then} = \frac{\text{gap}}{k - 1} \qquad \text{one ratio part} = \frac{\text{gap}}{m - n}

khow many times as old the elder is at that moment
m : nthe ratio of the elder's age to the younger's at that moment

Worked example

A father is 44 and his son is 12. In how many years will the father be three times as old as the son?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2023 · Tier I, 19 Jul 2023, Shift 2 · Q74Easy

Example 1 · Linear Equations · Age Problems

A woman is 28 years older than her son. In three years, her age will be twice the age of her son. The present age of the son is:

Moving only one person in time

"In 5 years, A will be twice B" means

A + 5 = 2(B + 5)

. Adding the 5 years to A alone,

A + 5 = 2B

, is the most common slip in age questions.

Expecting the ratio of ages to stay fixed

If a mother is twice her son's age today, she will be less than twice his age in 5 years. The difference of the ages stays fixed; the ratio moves towards 1 as both grow older.

Drill 2 more on the gap between two ages never changes

Concept 2 of 3: Age conditions at two different times

Two conditions, each about a different moment, give two equations in today's two ages. Write both in terms of today's ages and solve the pair.

Definition

Let today's ages be two letters. Write each condition at its own time: "6 years ago she was 5 times as old" is $M - 6 = 5(D - 6)$ .
Each condition usually gives one age in terms of the other; set the two expressions equal and solve.
A ratio condition such as $\frac{B + 9}{A + 9} = \frac{4}{3}$ becomes $3(B + 9) = 4(A + 9)$ by cross-multiplying.
Fast route, the options: when the options print both ages, test the simpler condition first. It usually leaves one option, and the other condition confirms it.
"Times more than": "5 times more than $a$ " strictly means $a + 5a = 6a$ , but some questions use it to mean $5a$ . Work out both readings and keep the one that reaches a printed option.

Worked example

Five years ago a mother was 5 times as old as her son. Five years from now she will be 3 times as old as him. Find their present ages.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 9 Sep 2024, Shift 2 · Q56Moderate

Example 2 · Linear Equations · Age Problems

Seven years ago, Prachi was four times as old as her daughter was at that time. Four years from now, Prachi will be two-and-a-half times as old as her daughter would then be. Find the sum of the present ages (in years) of Prachi and her daughter.

Using today's ages inside a past condition

"Eight years ago he was 3 times as old as his son" is

F - 8 = 3(S - 8)

. Writing

F - 8 = 3S

moves the father back in time and leaves the son where he is.

Reading "times more than" one way only

"Five times more than" strictly means six times as much, but question setters also use it for five times. Try both readings; usually only one of them lands on a printed option.

Stopping at one age when the question asks for a combination

Age questions often ask for the sum of the present ages, their ratio or their difference. Solving for one person and choosing that number is a common way to pick a printed wrong option.

Drill 7 more on age conditions at two different times

Concept 3 of 3: Three or more people linked by relations

A family question chains people together: the mother through the son, the brother through the friend. Pick the one person everyone else is described by, write every age in terms of that one letter, and a single fact finishes it.

Definition

Choose as $x$ the person the other relations are measured from, and write every other age in terms of $x$ .
Use the fact left over, a sum, a difference or a future condition, as the one equation.
"A and B differ by 7" does not say who is older. Try both orders and keep the one that gives whole, sensible ages.
A fraction in a relation ("a quarter of his brother's age") is easiest cleared by multiplying the whole equation early.

Worked example

A is twice as old as B, and C is 4 years younger than B. The three ages add up to 56. How old is C?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 19 Sep 2024, Shift 1 · Q53Moderate

Example 3 · Linear Equations · Age Problems

In a family, the combined age of husband, wife and a son is 90 years. The husband is the oldest. The difference in the ages of the husband and wife is

\frac{1}{3}

rd the age of their son. After 10 years, the age of the son would be half the age of his father. What is the present age of his mother (in years)?

Deciding who is older when the question does not say

"Their ages differ by 7 years" allows either person to be the elder. Settle the order from the other facts, or try both and keep the one that gives whole, sensible ages.

Drill 2 more on three or more people linked by relations

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

The gap between two ages never changes
Age-gap shortcut
$\text{younger then} = \frac{\text{gap}}{k - 1} \qquad \text{one ratio part} = \frac{\text{gap}}{m - n}$

Watch out for (6)

Moving only one person in time
→ The gap between two ages never changes
Expecting the ratio of ages to stay fixed
→ The gap between two ages never changes
Using today's ages inside a past condition
→ Age conditions at two different times
Reading "times more than" one way only
→ Age conditions at two different times
Stopping at one age when the question asks for a combination
→ Age conditions at two different times
Deciding who is older when the question does not say
→ Three or more people linked by relations

Test yourself on Linear Equations

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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