Ratio and Proportion

Prerequisites

• Percentages — a percentage is a ratio with second term 100; the “base” idea carries over.
• HCF and LCM of small numbers — you will use them to simplify and combine ratios.

The Hook: Sharing the Self-Help Group’s Profit

Why it matters for CGPSC: ratio (अनुपात) and proportion (समानुपात) questions in Prelims Paper II are short and score well once the method is automatic. The same thinking runs through partnership, mixtures, ages, speed and Data Interpretation (“ratio of production in two years”), and through the reasoning-and-mathematics part of Mains Paper IV (GS-II).

Ratio: The Basics

A ratio compares two quantities of the same kind, in the same units:

$$\boxed{a : b = \frac{a}{b}} \qquad (b \neq 0)$$

• \( a \) is the antecedent, \( b \) the consequent.
• Multiplying or dividing both terms by the same non-zero number does not change a ratio: \( 45 : 60 = 3 : 4 \) (divide by the HCF, 15).
• Convert units first: 50 paise : ₹2 = 50 : 200 = 1 : 4, not 50 : 2.

Kinds of ratio

Name	Of \( a : b \)	Example (2 : 3)
Duplicate ratio	\( a^2 : b^2 \)	4 : 9
Sub-duplicate ratio	\( \sqrt{a} : \sqrt{b} \)	\( \sqrt{2} : \sqrt{3} \)
Triplicate ratio	\( a^3 : b^3 \)	8 : 27
Inverse (reciprocal) ratio	\( \frac{1}{a} : \frac{1}{b} = b : a \)	3 : 2
Compound ratio of \( a:b \) and \( c:d \)	\( ac : bd \)	with 4 : 5 → 8 : 15

Comparing ratios

To ask “which is bigger, 3 : 5 or 4 : 7?”, cross-multiply: \( 3 \times 7 = 21 \) and \( 5 \times 4 = 20 \). Since \( 21 > 20 \), \( 3 : 5 > 4 : 7 \).

Combining Ratios (A : B : C)

Given \( A : B \) and \( B : C \), make the common term \( B \) equal in both, then read off \( A : B : C \).

Dividing a Quantity in a Given Ratio

To divide \( N \) in the ratio \( a : b : c \):

$$\boxed{\text{Share of first} = \frac{a}{a+b+c}\times N}$$

and similarly for the others. If the ratio is given as fractions, clear them with the LCM first.

Proportion

Four quantities are in proportion when two ratios are equal:

$$\boxed{a : b :: c : d \iff \frac{a}{b} = \frac{c}{d} \iff ad = bc}$$

“Product of extremes (\( a, d \)) = product of means (\( b, c \)).”

Term	Definition	Formula	Example
Fourth proportional to \( a, b, c \)	\( a : b :: c : d \)	\( d = \frac{bc}{a} \)	of 2, 3, 4 → 6
Third proportional to \( a, b \)	\( a : b :: b : x \)	\( x = \frac{b^2}{a} \)	of 4, 6 → 9
Mean proportional of \( a, b \)	\( a : x :: x : b \)	\( x = \sqrt{ab} \)	of 9, 16 → 12

Direct and inverse proportion

Type	Meaning	Rule	Example
Direct	Both rise together	\( \frac{x_1}{y_1} = \frac{x_2}{y_2} \)	More quintals of paddy, more money at a fixed price
Inverse	One rises, other falls	\( x_1 y_1 = x_2 y_2 \)	More workers, fewer days for the same job

Applications CSAT Loves

Mixtures: changing a ratio by adding one component

Fix the quantity that does not change, then find the new total for the other.

Coins: ratio of number vs ratio of value

Partnership: profit in the ratio of capital × time

$$\boxed{\text{Profit share} \propto \text{Capital} \times \text{Time}}$$

If all partners invest for the same time, profit is shared simply in the ratio of capital (as in the hook).

Ages

Write present ages as \( ax \) and \( bx \), shift both by the same number of years, and set up one equation in \( x \).

Memory Tricks

Don’t Confuse

This	vs	That
Ratio of number of coins (2 : 3 : 4)		Ratio of value of coins (2 : 1.5 : 1)
Duplicate ratio \( a^2 : b^2 \)		Double of a ratio — still \( a : b \) (\( 2a : 2b \))
Third proportional of 4, 6 = \( \frac{36}{4} = 9 \)		Fourth proportional of 2, 3, 4 = \( \frac{12}{2} = 6 \)
Direct proportion: \( \frac{x}{y} \) constant		Inverse proportion: \( xy \) constant

Common Mistakes

Practice

Level 1: Foundation

1. Simplify 45 : 60.
2. Write the duplicate ratio of 2 : 3.
3. Write the inverse ratio of 5 : 7.
4. Divide ₹720 in the ratio 4 : 5.
5. If \( 3 : x :: 9 : 15 \), find \( x \).

Answers

1. 3 : 4 (HCF 15).
2. 4 : 9.
3. 7 : 5.
4. 9 parts of ₹80: ₹320 and ₹400.
5. \( 9x = 45 \Rightarrow x = 5 \).

Level 2: Prelims MCQs

All money figures below are hypothetical practice values.

Q1. If A : B = 2 : 3 and B : C = 4 : 5, then A : C is:

(a) 2 : 5 (b) 8 : 15 (c) 6 : 5 (d) 15 : 8

Answer: (b) 8 : 15

Make B equal: A : B = 8 : 12, B : C = 12 : 15, so A : C = 8 : 15. Directly: \( (2\times 4) : (3\times 5) \). Option (d) is the reversed-ratio trap.

Q2. A and B start a tent-house business in Raigarh. A invests ₹40,000 for 12 months; B invests ₹60,000 but joins for only the last 6 months. Out of an annual profit of ₹21,000, A’s share is:

(a) ₹8,400 (b) ₹9,000 (c) ₹10,500 (d) ₹12,000

Answer: (d) ₹12,000

Profit ratio = capital × time: \( 40000 \times 12 : 60000 \times 6 = 480000 : 360000 = 4 : 3 \).

A’s share \( = \frac{4}{7}\times 21000 = 12000 \). Option (a) is what you get by using capital alone (2 : 3) and ignoring time.

Q3. Consider the following statements:

1. The mean proportional between 9 and 16 is 12.
2. The third proportional to 4 and 6 is 9.
3. The fourth proportional to 2, 3 and 4 is 8.

Which of the statements given above is/are correct?

(a) 1 only (b) 1 and 2 only (c) 2 and 3 only (d) 1, 2 and 3

Answer: (b) 1 and 2 only

1. Mean proportional \( = \sqrt{9 \times 16} = \sqrt{144} = 12 \) — correct.
2. Third proportional \( = \frac{6^2}{4} = 9 \) — correct.
3. Fourth proportional \( = \frac{3 \times 4}{2} = 6 \), not 8 — incorrect.

Q4. A 60-litre mixture contains milk and water in the ratio 2 : 1. How much water must be added to make the ratio of milk to water 1 : 2?

(a) 20 L (b) 40 L (c) 60 L (d) 80 L

Answer: (c) 60 L

Milk \( = \frac{2}{3}\times 60 = 40 \) L, water = 20 L. Milk stays 40 L; for 1 : 2 water must be 80 L. Water to add \( = 80 - 20 = 60 \) L. Option (d) is the final water quantity, not the amount added.

Q5. A bag contains ₹1, 50-paise and 25-paise coins in the ratio 2 : 3 : 4. If the total value is ₹180, the number of 25-paise coins is:

(a) 40 (b) 80 (c) 120 (d) 160

Answer: (d) 160

Value ratio \( = 2 : 1.5 : 1 \), total \( 4.5x = 180 \Rightarrow x = 40 \). Number of 25-paise coins \( = 4x = 160 \). Option (a) is \( x \) itself — a common slip.

Level 3: Speed Drill — Solve in 60 Seconds

The reasoning-and-mathematics part (Part 2) of Mains Paper IV (GS-II) asks for worked steps; Prelims Paper II asks for speed. Train both: solve each item below in under 60 seconds, then write the two-line method you used.

1. Divide ₹1,300 among A, B, C in the ratio \( \frac{1}{2} : \frac{1}{3} : \frac{1}{4} \).
2. Present ages are in the ratio 3 : 4; after 5 years they will be 4 : 5. Find the present ages.
3. 12 workers build a road stretch in 10 days. How long for 15 workers?
4. Find the compound ratio of 2 : 3, 4 : 5 and 5 : 8.
5. Two numbers are in the ratio 3 : 5. If 10 is added to each, the ratio becomes 5 : 7. Find the numbers.

Answers with the shortcut used

1. ₹600, ₹400, ₹300 — LCM 12 turns it into 6 : 4 : 3; ₹100 per part.
2. 15 and 20 years — each term rose by 1 part over 5 years, so 1 part = 5.
3. 8 days — inverse proportion: \( 12 \times 10 = 15 \times 8 \).
4. 1 : 3 — \( (2\cdot 4\cdot 5) : (3\cdot 5\cdot 8) = 40 : 120 \).
5. 15 and 25 — \( 7(3x+10) = 5(5x+10) \Rightarrow 4x = 20 \Rightarrow x = 5 \).

Quick Revision Table

Idea	Formula / Fact
Ratio	\( a : b = \frac{a}{b} \), same units
Combine A : B and B : C	Equalise B (LCM); A : C \( = \) product of antecedents : product of consequents
Share of \( a \) in \( a : b : c \)	\( \frac{a}{a+b+c}\times N \)
Proportion	\( a : b :: c : d \iff ad = bc \)
Fourth / third / mean proportional	\( \frac{bc}{a} \) / \( \frac{b^2}{a} \) / \( \sqrt{ab} \)
Compound ratio	\( ac : bd \)
Partnership	Profit \( \propto \) capital \( \times \) time
Inverse proportion	\( x_1 y_1 = x_2 y_2 \)

Key Takeaways

What’s Next

• Averages — weighted averages are ratios in action.
• Percentages — revisit successive change with ratio thinking.
• Data Interpretation — “ratio of A to B across years” is a standard DI question.
• General Mental Ability and Logical Reasoning — the non-arithmetic half of CSAT.
• Basic Numeracy — Quick Revision — the whole chapter on one page.