Ratios and Proportions: Complete Guide with Examples
Ratios and proportions are everywhere—recipes, maps, finance, construction, and more. Understanding how to work with them helps you scale recipes, read maps, mix paint colors, and solve countless real-world problems.
What is a Ratio?
A ratio compares two or more quantities. It shows how much of one thing there is compared to another.
Written as: a:b or a/b or a to b
Mix coffee and milk in ratio 2:3
Meaning: For every 2 cups coffee, use 3 cups milk
Or: Coffee is 2/5 of the mixture, milk is 3/5
How to Simplify Ratios
Simplify ratios the same way you simplify fractions—divide both sides by their GCD.
GCD(12, 18) = 6
12 ÷ 6 = 2
18 ÷ 6 = 3
Simplified: 2:3
GCD(24, 36, 48) = 12
24 ÷ 12 = 2
36 ÷ 12 = 3
48 ÷ 12 = 4
Simplified: 2:3:4
What is a Proportion?
A proportion states that two ratios are equal.
Written as: a:b = c:d or a/b = c/d
2:3 = 4:6 is a proportion
Because 2/3 = 4/6 (both equal 0.667)
Solving Proportions (Finding the Missing Value)
Use cross multiplication: If a/b = c/d, then a × d = b × c
Write as fraction: 3/5 = x/15
Cross multiply: 3 × 15 = 5 × x
45 = 5x
x = 45 ÷ 5 = 9
Answer: 3:5 = 9:15
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Free Ratio Calculator →Real-World Applications
1. Recipe Scaling
Set up proportion: 4:2 = 10:x
Cross multiply: 4x = 20
x = 5 cups
Answer: 5 cups of flour
2. Map Reading
1 cm on map = 50,000 cm in reality
3 cm on map = 3 × 50,000 = 150,000 cm = 1.5 km
3. Mixing Paint Colors
3:2 = 12:x
3x = 24
x = 8 liters
Answer: 8 liters yellow paint
4. Currency Exchange
₹1 : $0.012 = ₹5,000 : x
x = 5,000 × 0.012 = $60
5. Business Partnership
Total parts = 2 + 3 + 5 = 10
Partner 1: (2/10) × 50,000 = ₹10,000
Partner 2: (3/10) × 50,000 = ₹15,000
Partner 3: (5/10) × 50,000 = ₹25,000
Ratios vs Proportions: What's the Difference?
| Ratio | Proportion |
|---|---|
| Compares two quantities | States two ratios are equal |
| Example: 3:4 | Example: 3:4 = 6:8 |
| One comparison | Equation of two ratios |
| Can be simplified | Can be solved for unknown |
Types of Ratios
1. Part-to-Part Ratio
Compares one part to another part.
Example: In a class of 20 boys and 15 girls, boy:girl ratio is 20:15 = 4:3
2. Part-to-Whole Ratio
Compares one part to the total.
Example: Boys to total students = 20:35 = 4:7
3. Rate
Ratio of two different units.
Example: 60 km/hour, ₹50/kg
Common Ratio Problems
Problem Type 1: Dividing a Quantity
Total parts = 3 + 5 = 8
A gets: (3/8) × 1,200 = ₹450
B gets: (5/8) × 1,200 = ₹750
Problem Type 2: Finding Total from Parts
If 5 parts = 20 cups, then 1 part = 4 cups
Sugar = 2 parts = 8 cups
Total = 20 + 8 = 28 cups
Problem Type 3: Comparing Three or More
If 4 parts = 20 years, then 1 part = 5 years
A = 2 parts = 2 × 5 = 10 years
Common Mistakes to Avoid
Mistake 1: Wrong Order
Wrong: "Ratio of boys to girls is 15:20" when there are 20 boys and 15 girls
Right: Always match order! Boys:girls = 20:15
Mistake 2: Not Simplifying
Always simplify ratios: 20:15 should be written as 4:3
Mistake 3: Adding Ratios Directly
Wrong: If A:B = 2:3 and B:C = 3:4, then A:B:C = 2:3:4
Right: Make B the same in both: A:B = 2:3 and B:C = 3:4 → A:B:C = 2:3:4 ✓ (this one works!)
Practice Problems
- Simplify the ratio 45:60
- Solve: 7:x = 14:20
- Divide ₹900 in ratio 2:3:4
- Recipe for 6 uses 4 eggs. How many eggs for 15 servings?
Answers: 1) 3:4, 2) x=10, 3) ₹200, ₹300, ₹400, 4) 10 eggs
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