AFCAT Numerical Ability · Ratio and Proportion

What a Ratio Means, and Its Named Forms

A ratio splits quantities into equal parts. Simplify it like a fraction, and know the inverse, duplicate, triplicate and sub-duplicate forms by name.

All Ratio and Proportion notes

Why this matters

This is the most common page in the chapter's practice bank and the easiest: most of it is one step. Every later page leans on the same idea, that a : b = p : q means a = pk and b = qk, so a mistake here repeats on every page after it.

Concept 1 of 4: A ratio as equal parts

A ratio

3 : 5

does not say there are 3 and 5 of anything. It says the two quantities are built from equal parts, 3 parts against 5 parts. Find the size of one part and every quantity follows.

Definition

In $a : b$ , $a$ is the antecedent and $b$ the consequent. The ratio is the fraction $\dfrac ab$ , so $a : b = 4 : 7$ means $\dfrac ab = \dfrac 47$ .
$a : b = p : q$ means $a = pk$ and $b = qk$ for one number $k$ , the size of one part.
Knowing one term fixes $k$ : if $a : b = 2 : 7$ and $a = 12$ , one part is $6$ and $b = 42$ .
A ratio is also a percentage: $a : b = 2 : 5$ means $a$ is $\dfrac 25 = 40\%$ of $b$ .

A ratio as parts

a : b = p : q \;\text{means}\; a = pk,\ b = qk

Worked example

The ratio of oranges to apples in a basket is

5 : 9

. There are 45 apples. How many oranges are there, and what percentage of the apples is that?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Ratio and Proportion · Ratio Basics and Types of Ratio

If the ratio

a:b = 3:8

and

a = 15

, find the value of

b

The ratio is not the amount

Two numbers in the ratio

3 : 5

adding to 40 are 15 and 25, not 3 and 5. Find one part before you answer.

Order matters

a : b

and

b : a

are different ratios. 'The ratio of X to Y' puts X first, so read which quantity the question names first.

Drill 8 more on a ratio as equal parts

Concept 2 of 4: Simplest form of a ratio

Multiplying or dividing both terms by the same number leaves a ratio unchanged, as with a fraction. So

12 : 16

3 : 4

and

0.75 : 1

are one ratio. Simplest form means whole numbers with no common factor.

Definition

Whole numbers: divide both terms by their HCF ( $18 : 30 = 3 : 5$ , HCF 6).
Decimals: multiply both terms by 10, 100, ... until both are whole ( $0.6 : 1.5 = 6 : 15 = 2 : 5$ ).
Fractions: multiply both by the LCM of the denominators ( $\tfrac 23 : \tfrac 45$ , times 15, is $10 : 12 = 5 : 6$ ).
Mixed numbers: write them as improper fractions first.
Quantities: put both in the same unit first (40 minutes : 1.5 hours is $40 : 90 = 4 : 9$ ).

Scaling both terms

a : b = ka : kb \quad (k \neq 0)

Worked example

Write

1\tfrac 14 : 2\tfrac 16

in simplest form.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Ratio and Proportion · Ratio Basics and Types of Ratio

Express

2\frac{1}{3}:1\frac{2}{5}

in simplest form.

Units first

45 minutes : 2 hours is not

45 : 2

. Convert to one unit,

45 : 120 = 3 : 8

, before simplifying.

A ratio answer keeps both terms

0.75 : 1.25

simplifies to

3 : 5

. Its value 0.6 is a fraction, and an option written as a ratio will not show it.

Drill 11 more on simplest form of a ratio

Concept 3 of 4: Inverse, duplicate, triplicate and sub-duplicate ratios

AFCAT names the ratios you get from

a : b

by flipping it, squaring it, cubing it or taking roots. Each name is one operation applied to both terms.

Definition

Apply the operation to both terms, then simplify. A ratio 'compounded with itself' or 'taken twice' is multiplied by itself, which is the duplicate ratio. (Compounding two different ratios comes on the next page.)

Name	Of a : b	Example
Inverse (reciprocal)	$b : a$	$3 : 10 \to 10 : 3$
Duplicate	$a^2 : b^2$	$2 : 7 \to 4 : 49$ 'The compound ratio of a : b taken twice' is this same ratio.
Triplicate	$a^3 : b^3$	$1 : 3 \to 1 : 27$
Sub-duplicate	$\sqrt a : \sqrt b$	$49 : 64 \to 7 : 8$
Sub-triplicate	$\sqrt[3]{a} : \sqrt[3]{b}$	$8 : 125 \to 2 : 5$

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Ratio and Proportion · Ratio Basics and Types of Ratio

The duplicate ratio of (2a - 3b):(2a + 3b) is:

Duplicate is not double

The duplicate ratio of

3 : 4

9 : 16

, not

6 : 8

. Doubling both terms gives back the same ratio,

3 : 4

Sub-duplicate is not half

The sub-duplicate ratio of

16 : 25

4 : 5

, the square roots, not

8 : 12.5

Drill 18 more on inverse, duplicate, triplicate and sub-duplicate ratios

Concept 4 of 4: Working back from a duplicate or triplicate ratio

If you are given the duplicate ratio, the ratio itself is its sub-duplicate: undo the square with a square root. Figures work the same way. Areas go with the square of lengths and volumes with the cube.

Definition

If $a^2 : b^2 = p : q$ , then $a : b = \sqrt p : \sqrt q$ .
If the sub-duplicate ratio of $a : b$ is $m : n$ , then $a : b = m^2 : n^2$ .
Quantities in the ratio $p : q : r$ have squares in the ratio $p^2 : q^2 : r^2$ .
Similar figures: areas are in the duplicate ratio of lengths, volumes in the triplicate ratio. So lengths are the sub-duplicate of the areas.
A ratio with an unknown that equals a named ratio: work out the named ratio, then cross-multiply.

Undoing a duplicate ratio

a^2 : b^2 = p : q \;\Rightarrow\; a : b = \sqrt p : \sqrt q

Worked example

(x + 2) : (x + 12)

is the duplicate ratio of

2 : 3

. Find

x

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 4 · Ratio and Proportion · Ratio Basics and Types of Ratio

If the duplicate ratio of

a:b

equals the triplicate ratio of

2:3

, find

a:b

One operation per name

If the sub-duplicate ratio of

a : b

2 : 3

, then

a : b = 4 : 9

and its duplicate is

16 : 81

. That is two squarings, so write each step down.

Roots do not always come out whole

a^2 : b^2 = 2 : 5

gives

a : b = \sqrt 2 : \sqrt 5

. Leave the surds in; an option that has 'cleaned them up' to whole numbers is wrong.

Drill 5 more on working back from a duplicate or triplicate 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 (3)

A ratio as equal parts
A ratio as parts
$a : b = p : q \;\text{means}\; a = pk,\ b = qk$
Simplest form of a ratio
Scaling both terms
$a : b = ka : kb \quad (k \neq 0)$
Working back from a duplicate or triplicate ratio
Undoing a duplicate ratio
$a^2 : b^2 = p : q \;\Rightarrow\; a : b = \sqrt p : \sqrt q$

Reference tables (1)

Inverse, duplicate, triplicate and sub-duplicate ratios5 rows

Name	Of a : b	Example
Inverse (reciprocal)	$b : a$	$3 : 10 \to 10 : 3$
Duplicate	$a^2 : b^2$	$2 : 7 \to 4 : 49$ 'The compound ratio of a : b taken twice' is this same ratio.
Triplicate	$a^3 : b^3$	$1 : 3 \to 1 : 27$
Sub-duplicate	$\sqrt a : \sqrt b$	$49 : 64 \to 7 : 8$
Sub-triplicate	$\sqrt[3]{a} : \sqrt[3]{b}$	$8 : 125 \to 2 : 5$

Watch out for (8)

The ratio is not the amount
→ A ratio as equal parts
Order matters
→ A ratio as equal parts
Units first
→ Simplest form of a ratio
A ratio answer keeps both terms
→ Simplest form of a ratio
Duplicate is not double
→ Inverse, duplicate, triplicate and sub-duplicate ratios
Sub-duplicate is not half
→ Inverse, duplicate, triplicate and sub-duplicate ratios
One operation per name
→ Working back from a duplicate or triplicate ratio
Roots do not always come out whole
→ Working back from a duplicate or triplicate ratio

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