Ratio and Proportion Tricks for Competitive Exams — Solved Examples

Ratio and proportion is one of the most fundamental topics in competitive mathematics — not because it appears frequently as a standalone topic, but because it forms the backbone of nearly every other arithmetic topic. Percentage, profit and loss, time and work, mixture problems, partnership, and speed-distance all reduce to ratio relationships at their core.

A student who genuinely masters ratio and proportion does not just solve ratio problems faster — they solve an entire category of seemingly unrelated problems with greater speed and confidence.

This guide covers every major ratio and proportion concept tested in competitive exams, with the fastest solution method for each type and fully worked examples throughout.

Part 1: Ratio Fundamentals and Fast Simplification

What Is a Ratio?

A ratio a:b compares two quantities of the same kind. It is equivalent to the fraction a/b. Key properties:

• Ratios can be multiplied or divided by the same number without changing their value
• a:b = ka:kb for any non-zero k
• a:b ≠ b:a (order matters)

Simplifying Ratios — Speed Method

Always reduce ratios to lowest terms by dividing by HCF.

Examples:

• 24:36 → HCF=12 → 2:3
• 45:75 → HCF=15 → 3:5
• 84:126 → HCF=42 → 2:3
• 144:180 → HCF=36 → 4:5

Speed tip: If both numbers are divisible by 2, halve both first. Then check divisibility by 3, 5, 7 in order — exactly like fraction simplification.

Ratio Comparison — Without Cross Multiplication Errors

To compare a:b and c:d, compare a×d with b×c.

Which is greater — 7:9 or 11:14?

• 7×14 = 98 vs 9×11 = 99
• 99 > 98 → 11:14 > 7:9

Which is greater — 5:8 or 7:11?

• 5×11 = 55 vs 8×7 = 56
• 56 > 55 → 7:11 > 5:8

Part 2: Types of Ratios

Compound Ratio

The compound ratio of a:b and c:d is ac:bd — multiply corresponding terms.

Example:
Compound ratio of 3:4 and 5:7 = 3×5 : 4×7 = 15:28

Compound ratio of 2:3, 4:5, and 6:7 = 2×4×6 : 3×5×7 = 48:105 = 16:35

Duplicate Ratio

Duplicate ratio of a:b = a²:b²

Examples:

• Duplicate ratio of 3:4 = 9:16
• Duplicate ratio of 5:7 = 25:49

Sub-Duplicate Ratio

Sub-duplicate ratio of a:b = √a:√b

Examples:

• Sub-duplicate ratio of 9:25 = 3:5
• Sub-duplicate ratio of 16:81 = 4:9

Triplicate Ratio

Triplicate ratio of a:b = a³:b³

Example:

• Triplicate ratio of 2:3 = 8:27

Inverse Ratio

Inverse ratio of a:b = b:a

Example:

• Inverse ratio of 5:8 = 8:5

Part 3: Finding Values from Ratios — The One-Part Method

The fastest way to find actual values from a ratio is the one-part method.

The Rule:
If a:b = m:n, and total = T, then:

• One part = T/(m+n)
• Value of a = m × one part
• Value of b = n × one part

Worked Example 1:
Divide ₹5,400 in ratio 5:4.

• One part = 5400/9 = 600
• First share = 5×600 = ₹3,000
• Second share = 4×600 = ₹2,400

Worked Example 2:
Three people share profit in ratio 3:5:7. Total profit = ₹45,000.

• One part = 45000/15 = 3,000
• Shares: 3×3000=9,000 | 5×3000=15,000 | 7×3000=₹21,000

Worked Example 3:
Ratio of boys to girls in a class is 7:5. Total students = 48. How many boys?

• One part = 48/12 = 4
• Boys = 7×4 = 28

Part 4: The Three-Ratio Chain (A:B:C)

When two separate ratios share a common element, they must be combined into a single three-way ratio. This is a frequently tested skill.

The Method:
Make the common element equal in both ratios by finding LCM.

Worked Example 1:
A:B = 2:3 and B:C = 4:5. Find A:B:C.

• B values: 3 and 4 → LCM = 12
• Scale A:B → ×4 → 8:12
• Scale B:C → ×3 → 12:15
• A:B:C = 8:12:15

Worked Example 2:
A:B = 3:5 and B:C = 2:7. Find A:B:C.

• B values: 5 and 2 → LCM = 10
• Scale A:B → ×2 → 6:10
• Scale B:C → ×5 → 10:35
• A:B:C = 6:10:35

Worked Example 3:
P:Q = 5:6, Q:R = 4:9, R:S = 3:2. Find P:Q:R:S.

• Equalize Q: 6 and 4 → LCM=12 → P:Q = 10:12, Q:R = 12:27
• Equalize R: 27 and 3 → LCM=27 → R:S = 27:18
• P:Q:R:S = 10:12:27:18

Part 5: Proportion Types and Shortcuts

Direct Proportion

When two quantities increase or decrease together in the same ratio.

• If a/b = c/d → ad = bc (cross multiplication property)
• "a is to b as c is to d" → a:b::c:d

Speed rule: In direct proportion, set up a/b = c/d and solve for the unknown by cross multiplication.

Worked Example:
If 15 workers complete a job in 12 days, how many days will 20 workers take?

• More workers → fewer days → inverse proportion (not direct)
• 15×12 = 20×d → d = 180/20 = 9 days

Inverse Proportion

When one quantity increases as the other decreases.

• a × b = c × d (product is constant)

Speed rule: In inverse proportion, multiply the known pair and divide by the given value.

Worked Example:
8 pipes fill a tank in 27 hours. How long will 12 pipes take?

• 8×27 = 12×t → t = 216/12 = 18 hours

Continued Proportion

Three quantities a, b, c are in continued proportion when a:b = b:c → b² = ac

Application: Finding the mean proportional between two numbers.

• Mean proportional between 4 and 25 = √(4×25) = √100 = 10
• Mean proportional between 9 and 16 = √(9×16) = √144 = 12

Third Proportional

If a:b = b:x, then x = b²/a

Example: Third proportional to 4 and 6 = 6²/4 = 36/4 = 9

Fourth Proportional

If a:b = c:x, then x = bc/a

Example: Fourth proportional to 3, 7, 9 = 7×9/3 = 21

Part 6: Partnership Problems

Partnership problems are ratio problems in disguise. Profit is divided in the ratio of (capital × time).

Simple Partnership (Same Time Period)

Profit ratio = Capital ratio directly.

Worked Example:
A invests ₹12,000 and B invests ₹18,000. Total profit = ₹25,000. Find each share.

• Ratio = 12:18 = 2:3
• One part = 25000/5 = 5,000
• A's share = 2×5000 = ₹10,000
• B's share = 3×5000 = ₹15,000

Compound Partnership (Different Time Periods)

Profit ratio = Capital × Time for each partner.

Worked Example 1:
A invests ₹15,000 for 8 months, B invests ₹20,000 for 6 months. Divide profit of ₹34,000.

• A's ratio value = 15,000×8 = 1,20,000
• B's ratio value = 20,000×6 = 1,20,000
• Ratio = 1:1
• Each gets ₹34,000/2 = ₹17,000

Worked Example 2:
A invests ₹24,000 for 12 months, B invests ₹16,000 for 9 months, C invests ₹20,000 for 6 months. Total profit = ₹1,14,000.

• A = 24,000×12 = 2,88,000
• B = 16,000×9 = 1,44,000
• C = 20,000×6 = 1,20,000
• Ratio = 288:144:120 = 12:6:5
• One part = 1,14,000/23 = 4,956 (approx)
• Wait — let's simplify: 288:144:120 → ÷24 → 12:6:5
• One part = 1,14,000/23 = ₹4,956.52
• A = 12×4956.52 ≈ ₹59,478, B ≈ ₹29,739, C ≈ ₹24,783

Worked Example 3:
A and B start a business. A invests ₹10,000 for the full year. B joins after 4 months with ₹15,000. Annual profit = ₹22,000. Find B's share.

• A = 10,000×12 = 1,20,000
• B = 15,000×8 = 1,20,000
• Ratio = 1:1 → B's share = ₹11,000

Part 7: Ratio in Mixture Problems

Mixture problems are among the most ratio-intensive question types in exams.

Type 1: Simple Mixing

Two solutions of different concentrations are mixed. Find the resulting concentration.

Worked Example:
20 litres of 30% acid solution mixed with 30 litres of 50% acid solution. Final concentration?

• Acid from first = 20×0.30 = 6 litres
• Acid from second = 30×0.50 = 15 litres
• Total acid = 21 litres, Total solution = 50 litres
• Concentration = 21/50 × 100 = 42%

Type 2: Replacement Problems

A container has a mixture. Some is removed and replaced with pure liquid. Find the concentration after k replacements.

Formula: Final concentration = Initial concentration × (1 − r/V)^k

Where r = amount replaced each time, V = total volume, k = number of replacements.

Worked Example:
A 40-litre container has pure milk. 8 litres replaced with water 3 times. Final milk concentration?

• = 1 × (1 − 8/40)³ = (32/40)³ = (4/5)³ = 64/125
• Milk = 64/125 × 40 = 20.48 litres
• Concentration = 51.2%

Type 3: Alligation for Mixing Ratios

Find the ratio in which two items must be mixed to achieve a target average price or concentration.

Worked Example:
In what ratio must coffee at ₹80/kg be mixed with coffee at ₹120/kg to get a mixture at ₹96/kg?

80                 120
            96
 120-96       96-80
  = 24            = 16

Ratio = 24:16 = 3:2

Part 8: Ratio Word Problems — Quick Templates

Template 1: Ratio Changes When a Quantity Is Added/Removed

Problem: Ratio of A:B = 3:5. If 10 is added to each, ratio becomes 5:7. Find A and B.

• Let A = 3x, B = 5x
• (3x+10)/(5x+10) = 5/7
• 7(3x+10) = 5(5x+10)
• 21x+70 = 25x+50
• 4x = 20 → x = 5
• A = 15, B = 25

Template 2: Ratio of Ages

Problem: Present ratio of ages of A and B is 4:5. After 10 years ratio will be 6:7. Find present ages.

• Let A = 4x, B = 5x
• (4x+10)/(5x+10) = 6/7
• 28x+70 = 30x+60
• 2x = 10 → x = 5
• A = 20 years, B = 25 years

Template 3: Division with Conditions

Problem: ₹3,600 divided among A, B, C such that A gets twice B and B gets twice C. Find each share.

• Let C = x, B = 2x, A = 4x
• 4x+2x+x = 3600 → 7x = 3600 → x = 514.28
• C = ₹514.28, B = ₹1,028.57, A = ₹2,057.14

Common Ratio Mistakes and How to Avoid Them