Ratio Calculator
Simplify ratios and solve proportions.
Understanding Ratios
A ratio is a mathematical relationship between two or more quantities, showing how many times one value contains or is contained within another. Ratios are written as "a:b" (read as "a to b") or as a fraction a/b. They're fundamental in mathematics, science, cooking, business, and everyday comparisons.
What is a Ratio?
A ratio compares two or more quantities of the same kind. For example:
- Recipe: 2 cups flour to 1 cup sugar = 2:1 ratio
- Map scale: 1 inch represents 100 miles = 1:100
- Student-teacher ratio: 30 students to 1 teacher = 30:1
- Screen aspect ratio: 16:9 (width to height)
Simplifying Ratios
Simplifying a ratio means reducing it to its smallest whole numbers while maintaining the same relationship. This is done by dividing both terms by their Greatest Common Divisor (GCD).
Example: Simplify the ratio 24:36
- Find the GCD of 24 and 36, which is 12
- Divide both numbers by 12: 24 ÷ 12 = 2, 36 ÷ 12 = 3
- Simplified ratio: 2:3
Proportions and Solving for Unknown Values
A proportion states that two ratios are equal: a:b = c:d. When one value is unknown, you can solve for it using cross-multiplication.
Example: If 3:4 = 9:x, find x
- Cross multiply: 3 × x = 4 × 9
- Simplify: 3x = 36
- Solve: x = 36 ÷ 3 = 12
Real-World Applications of Ratios
Cooking and Recipes
Ratios are essential for scaling recipes up or down:
Finance and Business
| Ratio Type | Formula | Use |
|---|---|---|
| Price-to-Earnings (P/E) | Stock Price : Earnings per Share | Stock valuation |
| Debt-to-Equity | Total Debt : Total Equity | Financial leverage |
| Current Ratio | Current Assets : Current Liabilities | Liquidity measurement |
| Profit Margin | Net Profit : Revenue | Profitability |
Construction and Architecture
Scale drawings use ratios to represent large objects:
- 1:50 scale: 1 cm on drawing = 50 cm in reality
- 1:100 scale: 1 cm on drawing = 100 cm (1 meter) in reality
- 1:200 scale: 1 cm on drawing = 200 cm (2 meters) in reality
Photography and Screens
Aspect ratios define the proportional relationship between width and height:
- 16:9 - Modern widescreen (HDTV, monitors)
- 4:3 - Traditional TV and computer monitors
- 21:9 - Ultra-widescreen monitors
- 3:2 - Classic 35mm film, some DSLRs
- 1:1 - Square (Instagram posts)
Types of Ratios
Part-to-Part Ratios
Compare different parts of a whole. Example: A class has 12 boys and 18 girls. The boy-to-girl ratio is 12:18, which simplifies to 2:3.
Part-to-Whole Ratios
Compare one part to the total. Using the same class: boys to total students = 12:30 = 2:5.
Rate Ratios
Compare quantities with different units:
- Speed: miles per hour (distance:time)
- Price: dollars per pound (cost:weight)
- Fuel efficiency: miles per gallon (distance:fuel)
Working with Three-Part Ratios
Ratios can involve more than two quantities. Example: A concrete mix uses cement, sand, and gravel in a 1:2:3 ratio.
Example: For 60 kg of concrete mix in 1:2:3 ratio
- Total parts: 1 + 2 + 3 = 6 parts
- Cement: (1/6) × 60 = 10 kg
- Sand: (2/6) × 60 = 20 kg
- Gravel: (3/6) × 60 = 30 kg
Common Ratio Mistakes
- Comparing unlike quantities: Don't create ratios with different units unless it's a rate
- Order matters: 2:3 is not the same as 3:2
- Zero in ratios: A ratio cannot have zero as a term
- Adding ratios incorrectly: You can't simply add ratios like fractions
Converting Ratios to Percentages
To express a ratio as a percentage of the total:
Example: In a 3:2 ratio
- Total parts: 3 + 2 = 5
- First part: (3/5) × 100 = 60%
- Second part: (2/5) × 100 = 40%
Quick Reference
Key Formulas:
- Simplified ratio: Divide by GCD
- Proportion: a:b = c:d
- Cross multiply: a×d = b×c
- Solve for x: x = (b×c)/a
Common Ratios
- Golden Ratio: 1:1.618
- Silver Ratio: 1:1.414
- 16:9 - Widescreen
- 4:3 - Standard screen
- 3:2 - Photography
- 1:1 - Square