❌ Ratio and Proportion - Common Mistakes to Avoid

🎯 Overview

Ratio and Proportion is fundamental to many quantitative topics, but students often make systematic errors in understanding and applying concepts. This guide covers common mistakes and their solutions.

🔥 Critical Mistake Categories

Mistake 1: Incorrect Ratio Simplification

Common Error:

Not reducing ratios to simplest form or wrong reduction

Example 1: Wrong Simplification

Question: Simplify the ratio 24:36

Wrong: 24:36 = 12:18 = 6:9 ❌ (not fully simplified)
Correct: 24:36 = 2:3 ✅ (divided by 12)

Example 2: Decimal Ratios

Question: Simplify 1.5:2.5

Wrong: 1.5:2.5 = 15:25 ❌
Correct: 1.5:2.5 = 3:5 ✅ (multiplied by 2 to remove decimals)

Simplification Rules:

• Divide by GCD (Greatest Common Divisor)
• Remove decimals by multiplying by appropriate power of 10
• Keep integers in the ratio

Mistake 2: Wrong Cross-Multiplication

Common Error:

Incorrect cross-multiplication in proportion problems

Example:

Question: If 2:3 = 8:x, find x

Wrong: 2 × x = 3 × 8, so x = 12 ❌
Correct: 2/3 = 8/x, so 2x = 24, x = 12 ✅

(Actually both give same answer, but method is important)

Example 2: Complex Proportion

Question: If (x+2):(x-1) = 5:2, find x

Wrong: (x+2) × 2 = (x-1) × 5
2x + 4 = 5x - 5
9 = 3x, x = 3 ✅ (This is correct)

Common mistake would be: x+2 = 5, x-1 = 2 ❌

Proportion Rules:

• If a:b = c:d, then ad = bc
• Cross-multiply only when both sides are ratios
• Solve equations systematically

Mistake 3: Ratio Division/Combination Errors

Common Error:

Wrong method when dividing quantities in given ratios

Example 1: Simple Division

Question: Divide ₹720 in ratio 2:3

Wrong: First part = 720/2 = 360, Second part = 720/3 = 240 ❌
Correct: Total parts = 2 + 3 = 5
First part = (2/5) × 720 = 288
Second part = (3/5) × 720 = 432 ✅

Example 2: Three-way Division

Question: Divide 1080 in ratio 2:3:4

Wrong: Part 1 = 1080/2 = 540, Part 2 = 1080/3 = 360 ❌
Correct: Total parts = 2 + 3 + 4 = 9
Part 1 = (2/9) × 1080 = 240
Part 2 = (3/9) × 1080 = 360
Part 3 = (4/9) × 1080 = 480 ✅

Division Formula:

• Find total parts = sum of ratio numbers
• Each part = Total quantity × (ratio number/total parts)

Mistake 4: Proportion Word Problems

Common Error:

Setting up wrong proportions from word problems

Example 1: Worker Problem

Question: 5 workers can complete a job in 12 days. How many days for 8 workers?

Wrong: 5:12 = 8:x, so 5x = 96, x = 19.2 days ❌
Correct: Workers × Days = constant (inverse proportion)
5 × 12 = 8 × x
60 = 8x, x = 7.5 days ✅

Example 2: Mixture Problem

Question: Ratio of milk to water is 3:2. If total mixture is 25 liters, find milk.

Wrong: Milk = 25 × 3 = 75 liters ❌
Correct: Total parts = 3 + 2 = 5
Milk = (3/5) × 25 = 15 liters ✅

📊 Advanced Concept Mistakes

Mistake 5: Continued Proportion Errors

Common Error:

Wrong handling of three-term continued proportions

Example:

Question: If 2:3 = 6:x = x:12, find x

Wrong: 2:3 = 6:x, so 2x = 18, x = 9 ❌
Correct: From 2:3 = 6:x, x = 9
Check if 6:9 = 9:12
6:9 = 2:3 and 9:12 = 3:4 ❌
So no such x exists ✅

Continued Proportion:

• a:b = b:c = c:d
• Middle term repeated: a:b = b:c
• All ratios must be equal

Mistake 6: Proportion Properties Errors

Common Error:

Misapplying componendo and dividendo rules

Example:

Question: If a/b = 3/4, find (a+b)/(a-b)

Wrong: (a+b)/(a-b) = (3+4)/(3-4) = 7/(-1) = -7 ❌
Correct: Using componendo and dividendo:
(a+b)/(a-b) = (3+4)/(3-4) = 7/(-1) = -7 ✅
(This actually gives correct result, but students often confuse the rule)

Important Properties:

• Alternendo: a/b = c/d → a/c = b/d
• Invertendo: a/b = c/d → b/a = d/c
• Componendo: a/b = c/d → (a+b)/b = (c+d)/d
• Dividendo: a/b = c/d → (a-b)/b = (c-d)/d

Mistake 7: Age Ratio Problems

Common Error:

Not understanding that ratios change over time

Example:

Question: Current age ratio of A:B is 3:4. After 5 years, ratio is 4:5. Find current ages.

Wrong: Current ages = 3x and 4x
After 5 years: 3x+5 : 4x+5 = 4:5
3x+5 = 4, 4x+5 = 5 ❌

Correct: 3x+5 : 4x+5 = 4:5
(3x+5)/(4x+5) = 4/5
5(3x+5) = 4(4x+5)
15x + 25 = 16x + 20
x = 5
Current ages: A = 15, B = 20 years ✅

Mistake 8: Partnership Ratio Errors

Common Error:

Not considering time factor in partnership

Example:

Question: A invests ₹2000 for 6 months, B invests ₹3000 for 4 months. Find profit ratio.

Wrong: Ratio = 2000:3000 = 2:3 ❌
Correct: A's capital-months = 2000 × 6 = 12000
B's capital-months = 3000 × 4 = 12000
Ratio = 12000:12000 = 1:1 ✅

🔢 Application-Based Mistakes

Mistake 9: Mixture Ratio Problems

Common Error:

Wrong application of alligation method

Example:

Question: Mix rice at ₹30/kg with rice at ₹50/kg to get mixture at ₹40/kg. Find ratio.

Wrong: Ratio = 50:30 = 5:3 ❌
Correct: Using alligation:
        30
          \
           40
          /
        50

Ratio = (50-40) : (40-30) = 10:10 = 1:1 ✅

Alligation Method:

• Mean value in center
• Differences on sides
• Ratio of differences = ratio of quantities

Mistake 10: Scale and Map Problems

Common Error:

Wrong interpretation of scale ratios

Example:

Question: Map scale is 1:50000. Distance on map is 5cm. Find actual distance.

Wrong: Actual = 5 × 50000 = 250000 cm ❌
Correct: Actual = 5 cm × 50000 = 250000 cm = 2.5 km ✅

⚡ Quick Verification Methods

Method 1: Logic Check

If a:b = 2:3, then a should be smaller than b ✓
If quantities are divided, sum should equal total ✓
If ratio is simplified, no common factors ✓

Method 2: Back Calculation

If ₹720 divided in 2:3 gives 288 and 432
Check: 288 + 432 = 720 ✓
Check: 288:432 = 2:3 ✓

Method 3: Extreme Values Check

If ratio increases, value should increase ✓
If ratio decreases, value should decrease ✓

📝 Exam Strategy Tips

Question Approach

1. Identify the ratio type: Simple, compound, or continued
2. Find total parts: Sum of ratio numbers
3. Apply correct formula: Based on what’s given and asked
4. Check units: Ensure consistency
5. Verify answer: Logic check

Common Question Patterns

1. Simple ratio simplification
2. Division of quantities
3. Proportion problems
4. Age-related ratios
5. Partnership problems
6. Mixture problems
7. Scale and map problems

Time Management

• Simple problems: 30-45 seconds
• Medium problems: 60-90 seconds
• Complex problems: 2 minutes maximum

🔗 Related Topics

• Percentage - Percentage calculations
• - Partnership business
• - Mixture problems
• - Average calculations

📚 Quick Reference Sheet

Essential Formulas

1. a:b = c/d → ad = bc
2. If a:b = k, then a = bk, b = a/k
3. Division: Part = (ratio number/total parts) × total quantity
4. Partnership ratio = Investment₁ × Time₁ : Investment₂ × Time₂

Important Properties

1. a:b = b:a → a = b
2. a:b = ka:kb (multiplication property)
3. a:b = (a/k):(b/k) (division property)

Red Flags

• Answer not in simplest form
• Sum of parts not equaling total
• Wrong cross-multiplication
• Missing time factor in partnership
• Incorrect proportion setup

🎯 Next Steps

Master ratio and proportion:

1. Practice basic ratio problems
2. Focus on word problems
3. Learn alligation method
4. Master partnership concepts