SSC CGL Maths · Ratio, Proportion and Variation

Ratios in Income, Savings and Ages

Income is expenditure plus savings for every person, and the gap between two ages never changes, so each of these stories gives one fixed link that turns the ratios into numbers.

All Ratio, Proportion and Variation notes

Why this matters

Fourteen questions in the bank, nine on income and savings and five on ages, with six of the fourteen MODERATE. The savings questions with percentage changes are the slowest here unless you pick easy starting numbers, which is the fast route this page teaches.

Concept 1 of 2: Income, expenditure and savings in a ratio

Every rupee earned is either spent or saved. So if spending and saving are 3 parts and 1 part, income is 4 parts, and a change in any two of the three fixes the third.

Definition

For each person, income = expenditure + savings.
Expenditure : savings = e : s means income is $e + s$ parts.
Two people, two ratios (incomes $a : b$ , expenditures $c : d$ ): write incomes $ak, bk$ and expenditures $cm, dm$ . One more fact (equal savings, or one person's saving rate) links k and m.
Saving a percentage: someone who saves 64% of income spends 36% of it.
Fast route for percentage changes: start from easy numbers that fit the ratio (expenditure 40, savings 10, income 50), apply each change, and recompute savings as income minus expenditure. The percentage change in savings is then a one-line division.

Income identity

I = E + S \qquad S_{\text{new}} = I(1 + r_I) - E(1 + r_E)

I, E, Sincome, expenditure, savings
r_I, r_Ethe fractional rises in income and expenditure

Worked example

A man's income and expenditure are in the ratio 5 : 4. His income rises by 15% and his expenditure by 10%. By what percentage do his savings change?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 25 Sep 2024, Shift 2 · Q69Easy

Example 1 · Ratio, Proportion and Variation · Ratios in Income, Savings and Ages

The ratio of expenditure to savings of a woman is 4 : 1. If her income and expenditure are increased by 20% and 10%, respectively, then find the percentage change in her savings.

Combining the percentages instead of the amounts

Income up 20% and expenditure up 10% does not move savings by 10% or 30%. Savings are a difference, so take numbers, apply each rise, and subtract again.

Confusing the saving rate with the spending rate

Saving 25% of income means spending 75% of it. When the stem gives what is saved, turn it into what is spent before comparing expenditures.

Drill 8 more on income, expenditure and savings in a ratio

Concept 2 of 2: Ages in a ratio

Two people age at the same speed, so the gap between their ages is the same every year while their ratio drifts towards 1. The ratio of ages is therefore never a fixed fact, but the difference is.

Definition

Write the present ages as $ak$ and $bk$ . Then t years ago they were $ak - t$ and $bk - t$ ; in t years they will be $ak + t$ and $bk + t$ .
Set the changed fraction equal to the other ratio and solve for k.
The difference of two ages never changes, so it is the fixed ruler of these questions.
Fast route: scale the two ratios so their DIFFERENCES are equal. Then the age gap is the same number of parts in both, and the change in either term, in parts, equals the years that passed.
Find the ages first, then answer what is asked (an age in five years, a third person's age, or a new ratio).

Ages after t years

\frac{ak + t}{bk + t} = \frac{m}{n} \qquad bk - ak \text{ stays fixed}

tyears ahead (use $-t$ for years ago)

Worked example

The ages of a mother and her son are in the ratio 7 : 2. In 10 years they will be in the ratio 9 : 4. Find their present ages.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 18 Sep 2024, Shift 2 · Q72Easy

Example 2 · Ratio, Proportion and Variation · Ratios in Income, Savings and Ages

The present ages (in years) of A and B are in the ratio of

3 : 4

. Five years back, the ratio of their ages was

2 : 3

. What is the present age (in years) of A?

Adding the years to the ratio terms

Ages in the ratio 6 : 5 do not become 13 : 12 after 7 years. The terms are parts, not years; find the ages first, then add 7 to each.

Letting the age gap change

After any number of years both ages rise by the same amount, so the difference stays the same. If your working changes the gap, a sign or a term has been misread.

Drill 4 more on ages in a ratio

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 (2)

Income, expenditure and savings in a ratio
Income identity
$I = E + S \qquad S_{\text{new}} = I(1 + r_I) - E(1 + r_E)$
Ages in a ratio
Ages after t years
$\frac{ak + t}{bk + t} = \frac{m}{n} \qquad bk - ak \text{ stays fixed}$

Watch out for (4)

Combining the percentages instead of the amounts
→ Income, expenditure and savings in a ratio
Confusing the saving rate with the spending rate
→ Income, expenditure and savings in a ratio
Adding the years to the ratio terms
→ Ages in a ratio
Letting the age gap change
→ Ages in a ratio

Test yourself on Ratio, Proportion and Variation

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.

Take the Ratio, Proportion and Variation chapter test Drill all 14 questions