Ratio calculator guide
A ratio compares two quantities of the same kind. Written a:b (“a to b”), it records relative size rather than a single measurement: 10:15 means the first amount is two-thirds of the second, the same comparison as 2:3. Equivalent ratios differ by a common multiplier; the simplest form is the pair of whole numbers that share no common factor other than 1.
This calculator has two modes. Simplify divides both integer terms by their greatest common factor (GCF), found with the Euclidean algorithm, and ignores signs so that −10:15 reduces to 2:3. Solve for x treats a:b = c:x as a proportion and cross-multiplies: a × x = b × c, so x = (c × b) / a. Decimals are allowed only in solve mode. For multi-step word problems that mix unit rates and more than two quantities, the related ratio-and-proportion calculator is the conceptual companion.
How to use this ratio calculator
- Choose a mode: Select Simplify to reduce a:b, or Solve for x to complete the proportion a:b = c:x. The active button stays pressed.
- Enter the terms: For Simplify, type two integers. For Solve for x, type a, b, and the known third term c; decimals are allowed. Empty or non-numeric fields are rejected.
- Select Calculate: Press Calculate to update the result. Simplify rejects decimals such as 2.5; Solve for x rejects a first term of 0, because division by zero is undefined.
- Read the result: Simplify shows the reduced ratio and the GCF that was divided out. Solve for x shows the missing term and the completed proportion a:b = c:x.
- Reset or share: Reset restores the example for the current mode (10:15 or 2:3 = 10:x). Copy the link to share the mode and terms — they are encoded in the page URL.
Formula and variables
To simplify, compute the GCF of the two integers with the Euclidean algorithm: replace the pair (a, b) by (b, a mod b) until the remainder is 0; the last non-zero remainder is d. Divide the absolute values by d so the reduced terms are positive and coprime. To solve a proportion, two ratios are equal when their cross-products are equal: a/b = c/x rearranges to a × x = b × c, hence x = (c × b) / a, which is defined only when a is not zero.
a:b → |a|/d : |b|/d, d = gcd(a, b); a:b = c:x ⇒ x = (c × b) / a- a, b — Ratio terms
- The two parts of the given ratio a:b. Simplify requires integers; solve allows real numbers. (unitless)
- d — Greatest common factor
- gcd(a, b) from the Euclidean algorithm; signs are ignored, so d is never negative. (unitless integer)
- c — Known third term
- The first term of the second ratio in a:b = c:x. (unitless)
- x — Unknown term
- The missing fourth term, equal to (c × b) / a. (unitless)
Worked example: simplify 10:15 and solve 2:3 = 10:x
The calculator opens on 10:15 in Simplify mode. Reducing that pair produces 2:3, which is then used as the default proportion 2:3 = 10:x in Solve for x.
- Simplify
- 10 : 15
- Solve for x
- 2 : 3 = 10 : x
- GCF of 10 and 15 by Euclidean steps: gcd(15, 10) = gcd(10, 5) = gcd(5, 0) = 5.
- Divide both terms by 5: 10 ÷ 5 = 2 and 15 ÷ 5 = 3, so 10:15 = 2:3.
- The same GCF method on 8:12 gives gcd(12, 8) = 4, and 8 ÷ 4 : 12 ÷ 4 = 2:3, an equivalent ratio.
- To solve 2:3 = 10:x, cross-multiply: 2 × x = 3 × 10.
- x = 30 ÷ 2 = 15, so 2:3 = 10:15.
Result: 10:15 simplifies to 2:3; 2:3 = 10:x has x = 15
2 and 3 are coprime, so 2:3 is in lowest terms. The solved proportion 10:15 is just 2:3 scaled by 5 — the same factor that was divided out when 10:15 was simplified.
Understanding your results
Simplified ratio
The large pair is a:b in lowest terms. The GCF shown underneath is the positive integer that was divided into both original terms. A GCF of 1 means the ratio was already simplified, as with 7:11.
Solved value of x
x is the fourth term that makes a:b = c:x true. The completed proportion is repeated below the result so you can see a:b = c:x with the number filled in. Because x = (c × b) / a, scaling c scales x by the same factor.
Signs and absolute value
Common factors do not depend on sign, so Simplify reports a positive reduced pair: −10:15 and 10:−15 both become 2:3, with GCF 5. Solve for x keeps the signed inputs and can return a negative x when an odd number of terms is negative.
Assumptions
- Simplify accepts integers only; the GCF is computed from absolute values with the Euclidean algorithm.
- A ratio of 0:0 is undefined and is rejected; 0:n reduces to 0:1.
- Solve for x allows real numbers, including decimals, and requires a ≠ 0.
- The unknown is always the fourth term x in a:b = c:x, not a missing a, b, or c.
Limitations
- Only two-term ratios are simplified; three-part mixes such as 1:2:3 are outside the scope of this page.
- Simplify does not accept decimals or fractions; convert those to integers first, or use Solve for x when the terms are already real.
- Units are not converted: 2 ft : 6 in is not simplified until both quantities use the same unit.
- The related ratio-and-proportion calculator covers unit rates and multi-step scaling word problems that this page does not.
Common mistakes
- Dividing only one term by the GCF, which changes the comparison instead of reducing it.
- Using the LCM rather than the GCF to simplify; the LCM is for common multiples, not lowest terms.
- Entering 2.5:5 in Simplify; the GCF is defined for integers, so decimals belong in Solve for x.
- Setting a = 0 in a proportion, which would require dividing by zero.
- Reading 3:4 as “three-fourths of the whole.” In a group of 3 boys and 4 girls the ratio of boys to girls is 3:4, but boys are 3/7 of the group.
- Cross-multiplying a/b = c/x as a × c = b × x instead of a × x = b × c.
Practical use cases
Scale a recipe or mixture
If flour to sugar is 2:3 and you have 10 cups of flour, Solve for x on 2:3 = 10:x gives 15 cups of sugar, keeping the same proportion at a larger batch.
Reduce map and model scales
A drawing marked 8:12 is the same scale as 2:3 once the GCF of 4 is divided out, which is easier to read and compare with other drawings.
Split a total into parts
A 2:3 split of 25 items is 10 and 15, because 2 + 3 = 5 parts and 25 ÷ 5 = 5 per part. Simplifying 10:15 first confirms the part-counts are already in lowest terms.
Planning and decision guide
Simplify first, then scale
Reducing 10:15 to 2:3 before solving a proportion makes the cross-multiplication smaller and easier to check by hand. The GCF you divided out is the same factor you would multiply back by to restore the original size.
Use the GCF calculator when more than two integers share a factor
This page finds gcd(a, b) for a two-term ratio. To take the greatest common factor of three or more integers, or to need the least common multiple as well, use the GCF and LCM calculator — it is the same Euclidean idea folded across a list.
Choose ratio-and-proportion for unit rates and word problems
Simplify and a:b = c:x cover the two core ratio operations. When a problem asks for a unit rate, a missing term in a different position, or several equivalent ratios in a table, the ratio-and-proportion calculator is the related tool that stays conceptually in the same family.
Frequently asked questions
What is a ratio?
A ratio compares two quantities of the same kind, written a:b, a/b, or “a to b.” 10:15 means the first quantity is to the second as 10 is to 15, which is the same comparison as 2:3.
How do you simplify a ratio?
Divide both terms by their greatest common factor. For 10:15 the GCF is 5, so the simplified ratio is 2:3. This calculator finds that GCF with the Euclidean algorithm and ignores signs.
How do you solve a proportion for x?
If a:b = c:x, the cross-products are equal: a × x = b × c, so x = (c × b) / a. For 2:3 = 10:x, x = (10 × 3) / 2 = 15. The first term a cannot be zero.
Why does Simplify reject decimals such as 2.5?
The GCF is defined for integers: a common factor must divide both terms evenly. Convert 2.5:5 to 5:10 (multiply by 2) and then simplify, or use Solve for x when the terms are already real numbers.
Can a ratio have a negative term?
Yes. Simplify ignores signs, so −10:15 reduces to 2:3 with GCF 5. Solve for x keeps the signed values and returns a signed x from x = (c × b) / a.
What is the difference between a ratio and a fraction?
A ratio compares part to part (or two related amounts). A fraction is one quantity divided by another, often part of a whole. 3 boys and 4 girls give the ratio 3:4 of boys to girls, but boys are the fraction 3/7 of the class.
How is this related to the ratio-and-proportion calculator?
Both pages work with equivalent ratios. This calculator focuses on reducing a:b and solving a:b = c:x. The ratio-and-proportion calculator is the related conceptual tool for unit rates, tables of equivalent ratios, and multi-step word problems.
What if 7:11 does not simplify?
That is a valid result. gcd(7, 11) = 1, so the terms are already coprime and the simplified ratio stays 7:11.
Sources and review
- Ratio — MathWorld — A Wolfram Resource. Accessed 2026-08-23.
- Ratios — Math Is Fun. Accessed 2026-08-23.
- Intro to ratios — Khan Academy. Accessed 2026-08-23.
Reviewed 2026-08-23.