CDS Mathematics · Ratio, Proportion and Variation

Ratios in Income, Savings and Ages

Write each quantity as a multiple of an unknown, apply the change the story describes, and solve the ratio that results.

All Ratio, Proportion and Variation notes CDS Mathematics strategy

Why this matters

Ten PYQs. Incomes and expenditures, ages then and now, fares before and after a rise: each is two ratios linked by a fixed difference or a fixed change. The data-sufficiency versions test whether the ratios alone can give an absolute amount — they cannot.

Concept 1 of 2: Income, expenditure and savings

Savings = income − expenditure. Write the incomes as

pk, qk

and the expenditures as

rm, sm

; a condition on savings then links

k

and

m

Definition

Income $-$ expenditure $=$ saving, for each person.
Equal savings: $pk - rm = qk - sm$ . Given savings $S$ : substitute and solve.
Ratios alone fix only proportions; an absolute amount needs at least one rupee figure.
Percentage changes: multiply each term of the ratio by its own factor ( $1.2$ , $1.3$ ).
Combining groups: find the actual counts in each group before adding.

Savings

\text{saving} = \text{income} - \text{expenditure}

Worked example

The incomes of A and B are in the ratio

5 : 4

and their expenditures

3 : 2

. Each saves Rs. 1,600. Find A's income.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (I) 2019 — Elementary Mathematics · Q22Easy

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

The monthly incomes of

A

and

B

are in the ratio 4 : 3. Each saves Rs. 600. If their expenditures are in the ratio 3 : 2, then what is the monthly income of

A

Two ratios do not decide who saves more

Incomes

1 : 2

and expenses

1 : 3

are consistent with either person saving more, depending on the actual amounts. Try two cases before choosing.

Drill 5 more on income, expenditure and savings

Concept 2 of 2: Ages in a ratio

The difference between two people's ages never changes. So a ratio now and a ratio

t

years ago (or later) give two equations in the two present ages.

Definition

Present ages $M$ , $D$ ; $t$ years ago: $M - t$ , $D - t$ ; $t$ years later: $M + t$ , $D + t$ .
The age difference is constant: if present ages are $4t$ and $5t$ , the difference is $t$ .
A ratio of sum to difference: $\dfrac{F + S}{F - S} = \dfrac pq$ fixes $F : S$ .
Whole-number ages restrict the possible values.

Ages then and now

\dfrac{M - t}{D - t} = \dfrac pq

Worked example

Five years ago a father was four times as old as his son; five years from now he will be twice as old. Find their present ages.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (II) 2018 — Elementary Mathematics · Q37Moderate

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

Ten (10) years before, the ages of a mother and her daughter were in the ratio 3 : 1. In another 10 years from now, the ratio of their ages will be 13 : 7. What are their present ages ?

Shift both ages

Ten years ago BOTH were ten years younger. Subtracting from only one age turns a correct setup into a wrong answer that is often printed.

Drill 3 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
Savings
$\text{saving} = \text{income} - \text{expenditure}$
Ages in a ratio
Ages then and now
$\dfrac{M - t}{D - t} = \dfrac pq$

Watch out for (2)

Two ratios do not decide who saves more
→ Income, expenditure and savings
Shift both ages
→ Ages in a ratio

Test yourself on Ratio, Proportion and Variation

20 past CDS questions from this chapter, timed at 24 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 10 questions