AFCAT Numerical Ability · Ratio and Proportion

Joining, Compounding and Comparing Ratios

Join A : B and B : C through the shared term, turn equal multiples into a ratio, multiply ratios to compound them, and compare two ratios as fractions.

All Ratio and Proportion notes

Why this matters

The hardest page in the chapter's practice bank: a third of its questions are HARD, almost all of them compound ratios written with fractions, roots or letters. The method never changes. Make each ratio one simple fraction, then multiply.

Concept 1 of 5: Joining ratios through a common term

A : B

and

B : C

can be joined only when

B

has the same number in both. Scale each ratio so the shared term matches, then read off

A : B : C

Definition

To join $A : B = p : q$ and $B : C = r : s$ , scale to $A : B : C = pr : qr : qs$ . Making $B$ the LCM of $q$ and $r$ keeps the numbers small.
For $A : C$ alone, multiply the fractions: $\dfrac AC = \dfrac AB \times \dfrac BC$ . A longer chain multiplies the same way.
A chain that comes back to its start multiplies to 1: $\dfrac xy \cdot \dfrac yz \cdot \dfrac zx = 1$ .
Once you have $A : B : C$ , a total splits by finding one part: the total divided by the sum of the terms.

Joining two ratios

A : B = p : q,\ B : C = r : s \;\Rightarrow\; A : B : C = pr : qr : qs

Worked example

P : Q = 3 : 4

and

Q : R = 6 : 7

, find

P : Q : R

. If the three numbers add up to 140, find

R

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Ratio and Proportion · Combining Ratios

The sum of three numbers is 136. If the ratio between the first and second is 2 : 3 and that between the second and third is 5 : 3, then find the second number.

Scale, do not stitch

Joining

2 : 3

and

4 : 5

2 : 3 : 5

is wrong: the shared term is 3 in one and 4 in the other. Make it equal first, which gives

8 : 12 : 15

Read which term comes first

Given 'the ratio of the second to the third is

5 : 3

', the second number takes the 5. Writing it the other way round joins the wrong terms.

Drill 12 more on joining ratios through a common term

Concept 2 of 5: Ratios from equal multiples

When

2A = 3B = 4C

, set all three equal to a convenient number, usually the LCM of the multipliers, and read off

A

B

and

C

. The same works for

\dfrac x4 = \dfrac y6 = \dfrac z9

Definition

$pA = qB = rC$ gives $A : B : C = \dfrac 1p : \dfrac 1q : \dfrac 1r$ . Clear the fractions by multiplying by the LCM of $p, q, r$ .
$\dfrac xa = \dfrac yb = \dfrac zc$ gives $x : y : z = a : b : c$ directly.
Two separate statements, such as $2X = 3Y$ and $4Y = 5Z$ , are two ratios with a shared term: join them as in the last concept.

Equal multiples

pA = qB = rC \;\Rightarrow\; A : B : C = \tfrac 1p : \tfrac 1q : \tfrac 1r

Worked example

2A = 3B = 4C

, find

A : B : C

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Ratio and Proportion · Combining Ratios

If 6A = 4B = 9C, then A:B:C is:

Equal multiples flip the ratio

3A = 5B

means

A : B = 5 : 3

, not

3 : 5

. The bigger multiplier goes with the smaller quantity.

Drill 3 more on ratios from equal multiples

Concept 3 of 5: Compound ratio: multiplying ratios

The compound ratio of

a : b

and

c : d

ac : bd

: multiply the first terms together and the second terms together, then simplify. It is what you get when two effects stack, such as the number of passengers and the fare each pays.

Definition

The compound ratio of $a : b$ , $c : d$ and $e : f$ is $ace : bdf$ .
Cancel before you multiply, so the numbers stay small.
With fractions, turn each ratio into one fraction first ( $\tfrac 12 : \tfrac 13 = \tfrac 32$ ), then multiply.
With roots or letters, simplify each ratio first ( $\sqrt 8 : \sqrt{18} = 2 : 3$ ). Letters cancel like numbers.
An amount is a count times a rate, so amounts are in the compound ratio of the counts and the rates.

Compound ratio

(a : b),\ (c : d) \;\Rightarrow\; ac : bd

Worked example

Find the compound ratio of

3 : 4

8 : 9

and

15 : 16

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Ratio and Proportion · Combining Ratios

The ratio between the number of passengers travelling by I and II class between the two railway stations is 1 : 50, whereas the ratio of I and II class fares between the same stations is 3 : 1. If on a particular day, ₹1325 were collected from the passengers travelling between these stations, then what was the amount collected from the II class passengers?

Compounding multiplies, it does not add

The compound ratio of

2 : 3

and

4 : 5

8 : 15

. Adding the terms to get

6 : 8

has no meaning.

A ratio of fractions is one division

\tfrac 12 : \tfrac 13

\tfrac 12 \div \tfrac 13 = 3 : 2

, not

1 : 1

(the numerators) or

2 : 3

(the denominators).

Drill 19 more on compound ratio: multiplying ratios

Concept 4 of 5: Finding a missing ratio from a compound ratio

When the compound ratio is known and every ratio but one is known, divide: the missing ratio is the compound ratio divided by the product of the others.

Definition

If $\dfrac ab \times \dfrac cd = \dfrac mn$ , then $\dfrac cd = \dfrac mn \div \dfrac ab$ .
With an unknown inside a term, write the product as one fraction equal to the given ratio and cross-multiply.
The compound ratio of $x : y$ and $y : z$ is $x : z$ : the middle term cancels.

The missing ratio

\dfrac ab \times \dfrac cd = \dfrac mn \;\Rightarrow\; \dfrac cd = \dfrac mn \div \dfrac ab

Worked example

The compound ratio of

5 : 4

and

c : d

15 : 16

. Find

c : d

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 4 · Ratio and Proportion · Combining Ratios

If the compound ratio of

a:3

4:b

, and

c:5

2:1

, and

a = 6

b = 12

, find

c

Divide, do not subtract

If the compound ratio is

6 : 5

and one ratio is

3 : 2

, the other is

\tfrac 65 \div \tfrac 32 = 4 : 5

. Subtracting the terms to get

3 : 3

is wrong.

Drill 5 more on finding a missing ratio from a compound ratio

Concept 5 of 5: Comparing two ratios

To tell which ratio is larger, compare them as fractions: turn each into a decimal, or cross-multiply.

Definition

For positive terms, $a : b > c : d$ exactly when $ad > bc$ .
For a list, decimals are quickest: $7 : 20 = 0.35$ .
Or give them a common second term: $3 : 4 = 15 : 20$ and $4 : 5 = 16 : 20$ , so $4 : 5$ is larger.

Cross-multiplication test

\dfrac ab > \dfrac cd \;\text{exactly when}\; ad > bc \quad (b, d > 0)

Worked example

Which is greater,

5 : 7

7 : 10

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 5 · Ratio and Proportion · Combining Ratios

Out of the ratios 7 : 20, 13 : 25, 17 : 30 and 11 : 15, the smallest one is

Bigger terms are not a bigger ratio

17 : 30

has bigger terms than

2 : 3

but is the smaller ratio,

0.57

against

0.67

. Compare values, not the size of the terms.

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 (5)

Joining ratios through a common term
Joining two ratios
$A : B = p : q,\ B : C = r : s \;\Rightarrow\; A : B : C = pr : qr : qs$
Ratios from equal multiples
Equal multiples
$pA = qB = rC \;\Rightarrow\; A : B : C = \tfrac 1p : \tfrac 1q : \tfrac 1r$
Compound ratio: multiplying ratios
Compound ratio
$(a : b),\ (c : d) \;\Rightarrow\; ac : bd$
Finding a missing ratio from a compound ratio
The missing ratio
$\dfrac ab \times \dfrac cd = \dfrac mn \;\Rightarrow\; \dfrac cd = \dfrac mn \div \dfrac ab$
Comparing two ratios
Cross-multiplication test
$\dfrac ab > \dfrac cd \;\text{exactly when}\; ad > bc \quad (b, d > 0)$

Watch out for (7)

Scale, do not stitch
→ Joining ratios through a common term
Read which term comes first
→ Joining ratios through a common term
Equal multiples flip the ratio
→ Ratios from equal multiples
Compounding multiplies, it does not add
→ Compound ratio: multiplying ratios
A ratio of fractions is one division
→ Compound ratio: multiplying ratios
Divide, do not subtract
→ Finding a missing ratio from a compound ratio
Bigger terms are not a bigger ratio
→ Comparing two ratios

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