How to Solve a Proportion
A proportion says that two ratios are equal, such as 2 : 3 = x : 12. To solve it, cross-multiply: multiply the top of each fraction by the bottom of the other, set the two products equal, and divide. Here 2 × 12 = 24 and 3 × x = 24, so x = 8. The calculator does exactly this, shows the cross-multiplication, and then checks the answer by comparing the two fractions.
Any one of the four positions can be the unknown. If the ratio 5 : 8 is the same as 20 : y, the missing value is 32. If the unknown is the first term, as in x : 4 = 9 : 6, the answer is 6. Leave exactly one box empty and the tool works out which one to solve. This is the same method used for scaling recipes, converting units, reading map scales and comparing prices, as long as both ratios compare the same kinds of things in the same order.
Here is a word problem. A recipe calls for 3 cups of flour for 5 servings, and you need 12 servings. Write flour : servings as 3 : 5 = x : 12. Cross-multiplying gives 5 × x = 36, so x = 36/5 = 7.2 cups of flour. Check the order: flour is on top in both ratios and servings are on the bottom in both. Then check the size: 7.2 cups for 12 servings is 0.6 cup per serving, the same as 3 cups for 5 servings, so the proportion holds. Map scales, unit prices and exchange rates are solved the same way.
Keep the order consistent. If the first ratio is cups of flour to cups of sugar, the second must also be flour to sugar. Swapping the order in one ratio gives a wrong answer that still looks plausible, which is the easiest proportion mistake to make. Writing a label above each term, or reading the equation aloud, catches it.
There are two edge cases worth knowing. A denominator cannot be zero, so B and D are rejected if they are 0, because a ratio such as 3 : 0 has no meaning as a fraction. And a proportion such as 0 : 3 = 0 : x has no single answer: every nonzero value of x works, because 0 divided by any nonzero number is 0. The calculator reports that instead of picking one number arbitrarily.
How to Simplify a Ratio
A ratio is in simplest form when its terms have no common factor other than 1. To simplify, divide every term by their greatest common factor. For 45 : 60 the greatest common factor is 15, so the ratio is 3 : 4. For three terms the method is the same: 12 : 18 : 30 shares a factor of 6, which gives 2 : 3 : 5.
Decimals and fractions are handled by first clearing the decimals. The ratio 1.5 : 2.25 becomes 150 : 225 after multiplying both terms by 100, and dividing by 75 leaves 2 : 3. The same ratio can be written as the fraction 2/3 is about 0.6666666667. Simplifying never changes what the ratio means; it only removes redundant size, so 2 : 3 and 20 : 30 describe exactly the same relationship. Finding the greatest common factor by hand is covered in the GCF calculator, which uses the same idea.
How to Scale a Ratio
Scaling keeps the shape of a ratio while changing its size. To enlarge, multiply every term by the same factor. Enlarging 2 : 3 by 4 gives 8 : 12, and shrinking 8 : 12 by dividing by 4 returns 2 : 3. The factor can be a decimal: scaling 3 : 2 : 1 by 2.5 gives 15/2 : 5 : 5/2.
Scaling is how you resize a recipe, a paint mix, a concrete mix or a model. If a mix is 3 parts sand to 2 parts cement to 1 part water, doubling the batch means doubling all three terms, not just one. The ratio bars in the tool draw the original and scaled ratios side by side so you can see they have the same proportions even though the numbers differ.
Ratios, Fractions and Percentages
A ratio compares two quantities, a fraction compares a part with a whole, and a percentage is a fraction out of 100. The ratio 2 : 3 compares one amount with another; the fraction 2/3 would describe a part of a total of 3. If the ratio is of parts to the whole, such as 2 parts of a total of 5, the first amount is 2/5, or 40 percent. Converting between the three is straightforward once you decide whether each number is a part or a whole.
Part-to-part and part-to-whole ratios are easy to confuse. A class with 12 boys and 18 girls has a boys-to-girls ratio of 2 : 3, but boys make up 12 out of 30, which is 2 out of 5. Always check which comparison a problem is asking for before choosing the numbers to enter.
Frequently asked questions
How do I solve a proportion with an unknown?
Cross-multiply and divide. For a : b = c : x, multiply b by c, then divide by a to get x. The calculator shows these steps and checks that the two ratios match.
What does it mean to simplify a ratio?
It means dividing all the terms by their greatest common factor so the numbers are as small as possible while the relationship stays the same. For example 45 : 60 simplifies to 3 : 4.
Can a ratio have decimals?
Yes. The calculator clears the decimals by multiplying all terms by a power of 10, then simplifies, so 1.5 : 2.25 becomes 2 : 3.
Why can't a denominator be zero?
A proportion is a statement about two fractions, and a fraction with a zero denominator is undefined. If a term is zero in the numerator position, such as 0 : 3, that is allowed.
How do I scale a ratio up or down?
Multiply every term by the same number to enlarge it, or divide every term by it to shrink it. The proportions stay identical, so 2 : 3 scaled by 4 is 8 : 12.
Related tools: the fraction calculator for exact arithmetic, the GCF calculator for the common factor, and the math directory for every math tool on the site.