AFCAT Numerical Ability · Ratio and Proportion

When a Ratio Changes: Additions, Ages and Incomes

Write the old quantities as pk and qk, apply the change, and set the new ratio equal to the one given: for added numbers, ages, percentage rises and savings.

All Ratio and Proportion notes

Why this matters

This is where AFCAT asks most: seven of the chapter's twelve past-year questions are on this page, from 2015 to 2017. They are ages, a school roll that grows, and two people's incomes. Each one is a single equation in k, or two equations when incomes and expenditures each have their own ratio.

Concept 1 of 4: Adding or subtracting the same number from both terms

Adding the same number to both terms changes the ratio, because each term grows by a different fraction of itself. Write the old quantities as

pk

and

qk

, apply the change, and set the new ratio equal to the one given.

Definition

Old numbers $pk$ and $qk$ . After adding $x$ to each: $\dfrac{pk + x}{qk + x} = \dfrac rs$ . Cross-multiply and solve for $k$ .
When only one term changes (more boys join, the girls stay), the unchanged term fixes $k$ quickly.
'What must be added to each term of $a : b$ to make it $r : s$ ?' is the same equation with the numbers known and $x$ unknown.
Check by putting the answer back: the new ratio must come out exactly.

A changed ratio

\dfrac{pk + x}{qk + x} = \dfrac rs

Worked example

Two numbers are in the ratio

4 : 7

. If 6 is added to each, the ratio becomes

2 : 3

. Find the numbers.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2017 · AFCAT (II) 2017 — Numerical Ability · 04-305Easy

Example 1 · Ratio and Proportion · Change in Ratio, Ages and Income

The ratio between boys and girls in a school is 4 : 6 respectively. If the number of boys is increased by 200 the ratio becomes 5 : 6 respectively. How many girls are there in the school?

Add to the quantities, not to the ratio

If 5 is added to two numbers in the ratio

2 : 3

, the new numbers are

2k + 5

and

3k + 5

. Writing

7 : 8

assumes

k = 1

Drill 9 more on adding or subtracting the same number from both terms

Concept 2 of 4: Age problems with ratios

Everyone ages by the same number of years. So a ratio of ages now becomes a different ratio

n

years later, and the equation is the one from the last concept: add

n

to both terms.

Definition

Present ages $pk$ and $qk$ . After $n$ years they are $pk + n$ and $qk + n$ ; $n$ years ago, $pk - n$ and $qk - n$ .
The difference between two ages never changes.
A ratio together with a sum, an average or a product of ages: a sum splits into parts, an average times 2 is the sum, and a product gives $k^2$ .
'Three times as old' is the ratio $3 : 1$ .

Ages after n years

\dfrac{pk + n}{qk + n} = \dfrac rs

Worked example

The ages of A and B are in the ratio

5 : 3

. Six years from now the ratio will be

3 : 2

. Find their present ages.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2015 · AFCAT (I) 2015 — Numerical Ability · 04-313Easy

Example 2 · Ratio and Proportion · Change in Ratio, Ages and Income

The present ages of

X

and

Y

are in the ratio 7 : 5, respectively. Four years hence, the ratio of their ages will become 11 : 9 respectively. What is the present age of

Y

Both people age

Six years later both ages rise by 6. Adding 6 to one age only is the usual slip.

'Ago' subtracts

'Four years ago' means

pk - 4

and

qk - 4

. Writing

+4

by habit still gives a neat number, so check it in the stem's ratio.

Drill 17 more on age problems with ratios

Concept 3 of 4: When each term changes by a percentage

A percentage change multiplies a quantity. If the two terms are multiplied by different factors, the new ratio is the old one multiplied by the ratio of the factors.

Definition

$p : q$ raised by $a\%$ and $b\%$ becomes $p\left(1 + \tfrac{a}{100}\right) : q\left(1 + \tfrac{b}{100}\right)$ .
A ratio stays the same only if both terms change by the same percentage.
For savings, take an income of 100: work out the expenditure and the savings, apply each change, and compare.

Terms scaled by percentages

p : q \;\to\; p\left(1 + \tfrac{a}{100}\right) : q\left(1 + \tfrac{b}{100}\right)

Worked example

Seats in two courses are in the ratio

3 : 4

. They are increased by 20% and 50%. Find the new ratio.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Ratio and Proportion · Change in Ratio, Ages and Income

Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. There is a proposal to increase these seats by 40%, 50% and 75%, respectively. What will be the ratio of increased seats?

A percentage multiplies the term

Raising

3 : 4

by 20% and 50% gives

3.6 : 6

. Adding the percentages to the terms,

23 : 54

, has no meaning.

Drill 4 more on when each term changes by a percentage

Concept 4 of 4: Income, expenditure and savings

Savings are income minus expenditure. When incomes are given as one ratio and expenditures as another, write the incomes as

4x, 3x

and the expenditures as

5y, 4y

. The two savings give two equations in

x

and

y

Definition

Incomes $ax$ and $bx$ , expenditures $cy$ and $dy$ : the savings are $ax - cy$ and $bx - dy$ .
Two savings figures give two equations. Multiply so that one letter has the same coefficient, then subtract.
If both save the same amount $S$ , the equations are $ax - cy = S$ and $bx - dy = S$ .
A single ratio of income to expenditure, $p : q$ , makes the savings $p - q$ parts.

Savings

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

Worked example

The incomes of A and B are in the ratio

5 : 4

and their expenditures in the ratio

3 : 2

. Each saves ₹2,000. Find their incomes.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2016 · AFCAT (I) 2016 — Numerical Ability · 04-309Moderate

Example 4 · Ratio and Proportion · Change in Ratio, Ages and Income

The ratio of incomes of two persons is 5 : 3 and that of their expenditures is 9 : 5. The income of each person, if they save ₹1300 and ₹900 respectively, is

Two ratios need two letters

Incomes

4 : 3

and expenditures

5 : 4

need

4x, 3x

and

5y, 4y

. Using one

k

for both makes spending a fixed share of income, which the question never says.

Drill 10 more on income, expenditure and savings

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 (4)

Adding or subtracting the same number from both terms
A changed ratio
$\dfrac{pk + x}{qk + x} = \dfrac rs$
Age problems with ratios
Ages after n years
$\dfrac{pk + n}{qk + n} = \dfrac rs$
When each term changes by a percentage
Terms scaled by percentages
$p : q \;\to\; p\left(1 + \tfrac{a}{100}\right) : q\left(1 + \tfrac{b}{100}\right)$
Income, expenditure and savings
Savings
$\text{savings} = \text{income} - \text{expenditure}$

Watch out for (5)

Add to the quantities, not to the ratio
→ Adding or subtracting the same number from both terms
Both people age
→ Age problems with ratios
'Ago' subtracts
→ Age problems with ratios
A percentage multiplies the term
→ When each term changes by a percentage
Two ratios need two letters
→ Income, expenditure and savings

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