Ratio and Proportion

When making tea, a milk : water ratio of 2 : 1 means 1 cup of water for every 2 cups of milk — whether you make 6 cups or 600. A ratio compares two like quantities: how many times one is of the other. The ratio $a : b$ is also written as the fraction $\frac{a}{b}$ ; $a$ is the antecedent and $b$ the consequent. Both quantities must be in the same unit — ₹100 and 500 apples have no ratio.

Proportion means two ratios are equal: $a : b :: c : d$ i.e. $\frac{a}{b} = \frac{c}{d}$ . Every question here needs two tools: the one-part value and cross-multiplication ( $a d = b c$ ).

1

Ratio

Kinds · comparing · sharing

The kinds of ratio, comparing two ratios, and dividing a sum in a ratio.

Kinds of ratio and comparison

Duplicate · triplicate · compound

The rule for each kind, and which of two ratios is bigger.

Kind	Of $a : b$	Example $3 : 4$
Duplicate	$a^{2} : b^{2}$	9 : 16
Sub-duplicate	$a : b$	$4 : 9 \to 2 : 3$
Triplicate	$a^{3} : b^{3}$	27 : 64
Sub-triplicate	$3 a : 3 b$	$8 : 27 \to 2 : 3$
Inverse	$b : a$	4 : 3
Compound	$a ce : b df$	$(2 : 3), (4 : 5) \to 8 : 15$

Comparing: $a : b > c : d$ when $a d > b c$ (cross-multiply). Or write both as fractions over a common denominator. For $2 : 3$ and $4 : 5$ : $2 \times 5 = 10 < 3 \times 4 = 12$ ⟹ $2 : 3 < 4 : 5$ .

Example 1 · Basic

Which is greater,

5 : 7

or

8 : 11

?

Solution, step by step

1Cross-multiply: $5 \times 11 = 55$ and $7 \times 8 = 56$ .
2 $55 < 56$ ⟹ $5 : 7 < 8 : 11$

Answer 8 : 11

Example 2 · Standard

Find the compound ratio of

2 : 3

,

5 : 4

and

3 : 7

, and the sub-duplicate ratio of

4 : 9

.

Solution, step by step

1Compound: $\frac{2 \times 5 \times 3}{3 \times 4 \times 7} = \frac{30}{84} = 5 : 14$ .
2Sub-duplicate: $4 : 9 = 2 : 3$

Answer 5 : 14 and 2 : 3

BasicTry it yourself first

Write the ratio 36 : 60 in its simplest form.

BasicTry it yourself first

Write the ratio 56 : 72 in its simplest form.

BasicTry it yourself first

Which is the greater ratio: 3 : 5 or 5 : 8?

StandardTry it yourself first

Which is the greater ratio: 7 : 9 or 11 : 14?

AdvancedTry it yourself first

Which is the greater ratio: 13 : 17 or 19 : 25?

BasicTry it yourself first

What is the duplicate ratio of 3 : 5?

BasicTry it yourself first

What is the triplicate ratio of 2 : 7?

BasicTry it yourself first

What is the sub-duplicate ratio of 4 : 9?

StandardTry it yourself first

What is the sub-triplicate ratio of 8 : 27?

StandardTry it yourself first

What is the inverse ratio of 5 : 8?

StandardTry it yourself first

What is the compound ratio of 2 : 3, 4 : 5 and 3 : 7?

StandardTry it yourself first

What is the compound ratio of 3 : 5, 10 : 11 and 7 : 9?

AdvancedTry it yourself first

What is the compound ratio of 5 : 6, 8 : 15 and 9 : 14?

Sharing a sum and three-term ratios

Total parts · one part

Sharing a sum in a ratio, using the difference of parts, and getting $A : B : C$ from $A : B$ and $B : C$ .

Add up the total parts. For $a : b : c$ the total is $a + b + c$ parts.
One part $= \frac{total amount}{total parts}$ .
The required share $=$ its parts $\times$ one part.

Trick**

A : B : C

from

A : B

and

B : C

:** equalise the middle term

B

— take the LCM of its two values and scale

A

and

C

to match. For

A : B = 2 : 3

and

B : C = 4 : 5

: LCM of

B

is 12 ⟹

A : B = 8 : 12

,

B : C = 12 : 15

⟹

A : B : C = 8 : 12 : 15

.

Example 3 · Basic

Divide ₹9,000 in the ratio 2 : 3 : 4.

Solution, step by step

1Total parts $= 9$ ; one part $= \frac{9000}{9} = 1000$ .
2Shares: ₹2,000, ₹3,000, ₹4,000

Answer ₹2,000, ₹3,000, ₹4,000

Example 4 · Standard

The sum of three numbers is 195. The ratio of the first to the second is 3 : 4 and of the second to the third 2 : 3. What is the third?

Solution, step by step

1LCM of $B$ is 4 ⟹ $A : B = 3 : 4$ and $B : C = 4 : 6$ ⟹ $A : B : C = 3 : 4 : 6$ .
2Total parts 13; one part $= 15$ ⟹ the third $= 6 \times 15 = 90$ .

Answer 90

BasicTry it yourself first

₹7,200 is divided among A, B in the ratio 3 : 5. What is B's share?

BasicTry it yourself first

₹4,500 is divided among A, B, C in the ratio 1 : 2 : 3. What is C's share?

StandardTry it yourself first

₹12,600 is divided among A, B, C in the ratio 2 : 3 : 4. What is B's share?

StandardTry it yourself first

₹8,400 is divided among A, B in the ratio 5 : 7. What is A's share?

AdvancedTry it yourself first

₹20,100 is divided among A, B, C, D in the ratio 3 : 5 : 7 : 5. What is D's share?

BasicTry it yourself first

A sum is shared in the ratio 5 : 3 and the shares differ by ₹1,200. What is the bigger share?

StandardTry it yourself first

A sum is shared in the ratio 7 : 4 and the shares differ by ₹2,700. What is the bigger share?

AdvancedTry it yourself first

A sum is shared in the ratio 11 : 7 and the shares differ by ₹3,600. What is the bigger share?

BasicTry it yourself first

If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C.

StandardTry it yourself first

If A : B = 3 : 4 and B : C = 6 : 7, find A : B : C.

AdvancedTry it yourself first

If A : B = 5 : 6 and B : C = 8 : 9, find A : B : C.

StandardTry it yourself first

The sum of three numbers is 140. The ratio of the first to the second is 2 : 3 and of the second to the third 4 : 5. What is the second number?

StandardTry it yourself first

The sum of three numbers is 345. The ratio of the first to the second is 3 : 5 and of the second to the third 10 : 7. What is the third number?

AdvancedTry it yourself first

The sum of three numbers is 330. The ratio of the first to the second is 4 : 7 and of the second to the third 14 : 11. What is the first number?

2

Proportion

Means · extremes · three proportionals

The rules of $a : b :: c : d$ , third, fourth and mean proportionals, and componendo-dividendo.

Third, fourth and mean proportionals

Product of means = product of extremes

In $a : b :: c : d$ , $a, d$ are the extremes and $b, c$ the means; $a d = b c$ .

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Componendo, dividendo and friends

Four ways to rewrite a proportion

If $\frac{a}{b} = \frac{c}{d}$ , four related rules.

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3

Practical ratio questions

Changes, savings and coins

Ratio after adding or subtracting, income–expenditure–saving and coin questions.

When the numbers change

Assume

k

, then cross-multiply

The new ratio after adding or subtracting something, or after a percent change.

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Income–expenditure–savings and coins

Different units, one

k

Two ratios with equal savings, and coins of different worth.

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4

Traps and fast tricks

Slips · tricks

Where marks go in ratio questions.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

Assume

k

· LCM · totals

Habits that shorten the work.

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5

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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6

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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Ratio

Kinds of ratio and comparison

Sharing a sum and three-term ratios

Proportion

Third, fourth and mean proportionals

Componendo, dividendo and friends

Practical ratio questions

When the numbers change

Income–expenditure–savings and coins

Traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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