Ratio Calculator
This free ratio calculator handles three ratio operations: simplify a ratio to its lowest terms, scale a ratio by any factor, or check if two ratios are equivalent. Choose a mode, enter your values, and get the result instantly with full step-by-step working.
All calculations use standard published formulas. Results are for informational use only.
How to use this ratio calculator
- Choose a mode: Simplify, Scale, or Equivalent?
- Enter the values for the selected mode and the result appears immediately.
- Use the step-by-step section to see exactly how the answer was reached.
Simplify mode reduces a ratio A:B to its simplest whole-number form using the GCD.
Scale mode multiplies both parts of a ratio by the same factor to get an equivalent ratio at a different magnitude.
Equivalent mode checks whether A:B = C:D using cross-multiplication.
What is a ratio?
A ratio is a way of comparing two quantities to show their relative sizes. Written as A:B, it tells you how much of A there is for every unit of B. For example, a ratio of 3:5 means for every 3 units of the first quantity, there are 5 units of the second.
Ratios appear everywhere: mixing paint colors (3 parts blue to 1 part white), scaling recipes, map scales (1:50,000), gear ratios in engineering, financial ratios like debt-to-equity, and odds in probability.
Unlike fractions, a ratio does not need to represent a part of a whole. The ratio 3:5 could compare two separate groups - it does not imply that 3 and 5 are parts of an 8-unit total, though in some contexts (like mixing) that interpretation makes sense.
How to simplify a ratio
To simplify a ratio, divide both parts by their greatest common divisor (GCD). The GCD is the largest whole number that divides both A and B exactly.
Example: Simplify 18:24.
- GCD(18, 24) = 6
- 18 / 6 = 3
- 24 / 6 = 4
- Simplified: 3:4
A ratio is fully simplified when both parts share no common factor greater than 1 - when their GCD equals 1.
How to check if two ratios are equivalent
Two ratios A:B and C:D are equivalent ratios if they express the same proportion. The most reliable test is cross-multiplication: multiply A by D and multiply B by C. If the products are equal, the ratios are equivalent.
Example: Are 2:3 and 8:12 equivalent?
- A x D = 2 x 12 = 24
- B x C = 3 x 8 = 24
- 24 = 24 - Yes, they are equivalent
Alternatively, simplify both ratios to their lowest terms. If they share the same simplified form, they are equivalent: 2:3 and 8:12 both simplify to 2:3.
Where ratios are used in real life
Cooking and recipes
Recipes are ratio-based. A basic bread recipe might call for a 5:3 flour-to-water ratio. Scaling a recipe for a larger group means multiplying both parts of the ratio by the same factor - exactly what the scale mode does. Keeping the ratio constant is what preserves flavour and texture when quantities change.
Maps and scale drawings
A map scale of 1:50,000 means every 1 unit on the map represents 50,000 units in reality. Architects and engineers use scale ratios in blueprints to represent large structures on paper. Understanding scale ratio means you can calculate real distances from drawing measurements without guessing.
Finance and business
Financial analysis relies heavily on ratios. A debt-to-equity ratio of 2:1 means the company has twice as much debt as equity. A current ratio of 3:1 means three times more current assets than current liabilities - a sign of good short-term liquidity. Price-to-earnings (P/E) ratios compare a stock's price to its earnings to gauge market valuation.
Reading and interpreting ratio results
Simplifying vs. changing the ratio
Simplifying a ratio does not change its meaning - 6:10 and 3:5 describe exactly the same proportion. Simplification only removes the common factor to make the numbers smaller and easier to work with. The relationship between the two quantities remains identical.
Ratios with decimal values
When you enter decimal values, the GCD calculation rounds to the nearest integer internally. If precision matters - for example, in a ratio like 1.5:2.5 - multiply both parts by 10 first (15:25) to work in whole numbers, then simplify. The simplified form of 15:25 is 3:5.
Order matters in a ratio
A:B and B:A are different ratios unless A equals B. A win-loss ratio of 4:1 means 4 wins for every 1 loss - very different from 1:4, which means the opposite. Always identify which quantity goes first before writing or interpreting a ratio.
Quick tips for ratio calculations
- Simplify by finding the GCD - do not just guess a common factor; the GCD gives the fully reduced form in one step.
- Scale by multiplying both parts - the factor must be the same for both parts or the proportion changes.
- Use cross-multiplication for equivalence - it works for any numbers, including decimals and large values.
- Order is significant - 3:1 and 1:3 are entirely different ratios.
- Convert decimals to integers before simplifying for cleaner results.
Common ratio mistakes
Dividing by a common factor that is not the GCD
For ratio 12:18, dividing by 2 gives 6:9, which looks simpler but is not fully reduced - GCD(6,9) = 3, so the true simplest form is 2:3. Always divide by the GCD, not just any shared factor.
Swapping the order when comparing
For cross-multiplication, A x D = B x C. If you accidentally compute A x C = B x D, you will get wrong equivalence results. Keep track of which values correspond to which ratio.
Using different units
A ratio only makes sense when both parts share the same unit. You cannot form a meaningful ratio of meters to kilograms. Always convert to compatible units before creating a ratio.
Frequently Asked Questions
How do you simplify a ratio?
To simplify a ratio A:B, find the greatest common divisor (GCD) of A and B, then divide both values by the GCD. For example, 12:8 - GCD is 4, so 12/4 : 8/4 = 3:2. The simplified ratio represents the same proportion but in the smallest whole numbers.
What does it mean for two ratios to be equivalent?
Two ratios A:B and C:D are equivalent if they represent the same proportion, meaning A/B = C/D. You can also cross-multiply: A x D = B x C. For example, 2:3 and 8:12 are equivalent because 2/3 = 8/12 = 0.667. Equivalent ratios are the same proportion expressed at different scales.
How do you scale a ratio by a factor?
To scale ratio A:B by factor N, multiply both parts by N. For example, scaling 3:4 by a factor of 5 gives 15:20. Scaling preserves the proportion while changing the actual quantities. This is useful for recipes, maps, blueprints, and any situation where you need to keep proportions fixed while changing total amounts.
What is a ratio in math?
A ratio is a comparison of two quantities that shows their relative sizes. Written as A:B, it tells you how much of A there is for every B. For example, a ratio of 2:5 means for every 2 units of A, there are 5 units of B. Ratios can be written as fractions (2/5), with a colon (2:5), or in words ('2 to 5').
How is a ratio different from a fraction?
A fraction represents a part of a whole - for example, 2/5 means 2 out of 5. A ratio compares two separate quantities - for example, 2:5 might mean 2 cups of oil to 5 cups of water, where neither 2 nor 5 needs to be parts of the same total. Ratios can be converted to fractions, but they represent different concepts - part-to-whole vs. part-to-part.
Why does the ratio simplifier divide by the GCD?
The GCD (greatest common divisor) is the largest number that divides both parts of the ratio exactly. Dividing both parts by the GCD reduces the ratio to its simplest form while keeping the proportion identical. For 18:24, GCD = 6, so simplified = 3:4. Any other common factor would only partially reduce the ratio.