Ratio Simplifier

What this calculator does

So, you have a ratio like 10 to 15 and you want it in its simplest form. This tool reduces it. Enter the two parts of the ratio and it returns the ratio written in its lowest whole-number terms, still as a ratio.

You type the two numbers with the colon between them in mind, one in each box, and the tool gives back the smallest equivalent ratio.

How to use it

1. Enter the first part of the ratio.
2. Enter the second part of the ratio.
3. Press Calculate.

Both numbers should be positive.

What a ratio is

A ratio compares two quantities, telling you how much of one there is for a given amount of another. Written with a colon, 10 to 15 is 10:15. Ratios show up whenever you mix or share things in a fixed relationship: two parts of this to three parts of that, the ratio of boys to girls in a class, the mix of cement to sand. The key idea is that a ratio is about the relationship between the parts, not their exact sizes, so many different pairs of numbers can express the same ratio.

What simplifying a ratio means

Because a ratio is about the relationship rather than the raw numbers, you can scale both parts by the same amount without changing what the ratio means. The ratio 10:15 says exactly the same thing as 2:3, since both describe two of one thing for every three of another. Simplifying a ratio means finding the smallest whole numbers that still express that relationship. It is the ratio equivalent of reducing a fraction to lowest terms, and it makes the underlying relationship easy to read at a glance.

How the tool does it

The tool reduces the ratio by finding the greatest common divisor of the two parts, the largest number that divides both evenly, and dividing each part by it. For 10 and 15, the greatest common divisor is 5, so dividing both gives 2 and 3, the simplest form. Using the greatest common divisor does the whole reduction in one step, since it strips out every shared factor at once. The result comes back in the same colon form you started with, just with the smallest possible numbers.

A worked example

Enter 10 and 15. The greatest common divisor of 10 and 15 is 5. Dividing each part by 5 gives 2 and 3, so the tool returns 2 : 3. Try 8 and 12, whose greatest common divisor is 4, and it returns 2 : 3 as well, since 8:12 and 10:15 both describe the same relationship.

Ratio form, not fraction form

This tool keeps the answer as a ratio, with the colon, because a ratio and a fraction are subtly different things. A ratio compares two separate parts to each other, while a fraction usually describes a part of a single whole. They are closely related, and 2:3 carries the same numbers as the fraction 2 over 3, but the meaning and the notation differ. If you would rather see the ratio rewritten as a fraction, including as a mixed number when the first part is larger, the ratio to fraction calculator does that conversion.

Questions people ask

What does it mean to simplify a ratio?

To rewrite it with the smallest whole numbers that keep the same relationship. So 10:15 simplifies to 2:3.

How does the tool reduce it?

It divides both parts by their greatest common divisor, the largest number that divides both, which removes all shared factors at once.

Why do different ratios simplify to the same thing?

Because scaling both parts by the same amount does not change the relationship. 8:12 and 10:15 both reduce to 2:3.

Why is the answer a ratio and not a fraction?

Because a ratio compares two parts to each other. For the fraction form, including mixed numbers, use the ratio to fraction calculator.

What if I want to solve a proportion?

The ratio calculator solves proportions, finding a missing value across two equal ratios.

References

On ratios and their reduction. A ratio compares two quantities and is simplified by dividing both parts by their greatest common divisor.

1. Eric W. Weisstein, "Ratio," from MathWorld, a Wolfram resource, on the ratio of two numbers, written with a colon or as a quotient.
2. Euclid, Elements, Book VII, Proposition 2, D. E. Joyce's online edition, Clark University, on the greatest common measure used to reduce to lowest terms.