Ratio Calculator
Simplify ratios, solve proportions, and compare ratios with step-by-step explanations
Simplify Ratios
Solve Proportions
Divide Quantity by Ratio
What is a Ratio?
A ratio is a mathematical comparison between two or more quantities, showing how many times one value contains another. Ratios are expressed in the form a:b (read as “a to b”), where both numbers represent parts of a whole relationship. For example, if a recipe calls for 2 cups of flour to 1 cup of sugar, the ratio is 2:1.
Where ‘a’ and ‘b’ are the quantities being compared
Ratios appear everywhere in daily life – from cooking recipes and map scales to financial analysis and mixing paint colours. They provide a precise way to describe proportional relationships between different quantities whilst maintaining their relative sizes.
How to Calculate Ratios
Simplifying Ratios
To simplify a ratio to its lowest terms, find the greatest common divisor (GCD) of both numbers and divide each part by this value.
1. Find GCD of 12 and 18 = 6
2. Divide both numbers: 12÷6 = 2, 18÷6 = 3
3. Simplified ratio: 2:3
Solving Proportions
When two ratios are equal, they form a proportion. Use cross multiplication to find missing values in proportions.
1. Cross multiply: 3 × 12 = 4 × x
2. Solve: 36 = 4x
3. Therefore: x = 9
Dividing Quantities by Ratios
To divide a total amount according to a given ratio, find the value of each part and multiply by the ratio components.
1. Total parts = 2 + 3 = 5
2. Value per part = £120 ÷ 5 = £24
3. First share = 2 × £24 = £48
4. Second share = 3 × £24 = £72
Types of Ratios
Part-to-Part Ratios
Compare different parts of a whole. Example: In a class of 30 students with 18 girls and 12 boys, the ratio of girls to boys is 18:12 or 3:2.
Part-to-Whole Ratios
Compare one part to the entire amount. Example: If 18 out of 30 students are girls, the ratio of girls to total students is 18:30 or 3:5.
Unit Ratios
Express ratios with one part equal to 1. Example: 4:2 becomes 2:1, meaning the first quantity is twice the second.
Equivalent Ratios
Different ratios that represent the same relationship. Example: 2:4, 1:2, and 3:6 are all equivalent ratios.
Common Applications
| Application | Example | Ratio |
|---|---|---|
| Cooking & Baking | 3 cups flour to 2 cups sugar | 3:2 |
| Map Scales | 1 cm represents 1000 metres | 1:100,000 |
| Paint Mixing | 2 parts blue to 1 part yellow for green | 2:1 |
| Business Finance | Debt to equity analysis | Variable |
| Screen Resolutions | HD television aspect ratio | 16:9 |
| Investment Portfolios | 60% stocks to 40% bonds | 3:2 |
Step-by-Step Calculation Guide
Method 1: Simplification Process
- Identify the numbers: Write down both quantities in a:b format
- Find common factors: List factors of both numbers
- Determine GCD: Identify the greatest common divisor
- Divide both parts: Divide each number by the GCD
- Verify the result: Check that no further simplification is possible
Method 2: Cross Multiplication for Proportions
- Set up the equation: Write as a:b = c:d or a/b = c/d
- Cross multiply: Calculate a×d = b×c
- Solve for unknown: Isolate the variable
- Check your answer: Substitute back to verify
Frequently Asked Questions
A ratio compares two separate quantities (like 3 apples to 2 oranges), whilst a fraction represents part of a whole (like 3/5 of a pizza). Ratios show relationships between different things, whereas fractions show portions of one thing.
Yes, ratios can include decimals, though they’re often converted to whole numbers for simplicity. For example, 1.5:2.5 can be simplified to 3:5 by multiplying both parts by 2.
For ratios like a:b:c, the same principles apply. Find the GCD of all numbers to simplify, or use the total parts method for division problems. For example, 6:9:12 simplifies to 2:3:4.
If one part of a ratio is zero, it means that quantity doesn’t exist in the relationship. For example, 5:0 means there are 5 of the first item and none of the second.
Ratios can be converted to percentages by making the total equal 100. For a ratio a:b, the percentages are (a/(a+b))×100% and (b/(a+b))×100%.
Pro Tips for Ratio Calculations
Check Units
Always ensure both quantities use the same units before calculating ratios. Convert if necessary.
Simplify Early
Reduce ratios to their simplest form as soon as possible to make calculations easier.
Use Cross Products
For proportion problems, cross multiplication is often the quickest solution method.
Verify Results
Always check your answers by substituting back into the original problem.