Simplify Ratio Calculator

Reduce a ratio of two to six terms to its lowest whole numbers. Decimals and fractions are cleared first, then every term is divided by the greatest common divisor — with the GCD and each step shown.

Formula divide by GCD

Type the ratio with colons, commas or spaces — 45:60, 12, 18, 30 and 1/2 : 2/3 all parse. Or use the numbered fields below. Convert to the same unit first: 30 minutes to 2 hours is 30 : 120, not 30 : 2.

Try an example
Method

Clear, then divide by the GCD

Two stages: get every term to a whole number, then cancel the largest factor they all share.

Put every quantity in the same unit

This has to happen before any arithmetic. 30 minutes to 2 hours becomes 30 : 120, not 30 : 2. Grams against kilograms, centimetres against metres, pence against pounds — all need converting first, and the answer is wrong by exactly the conversion factor if you skip it.

Clear any fractions with the LCM

Multiply every term by the lowest common multiple of all denominators. For 1/2 : 2/3 that is 6, giving 3 : 4. Multiplying every term by the same number leaves the ratio unchanged, which is what makes this legal.

Clear any decimals with a power of ten

Count the most decimal places in any term and multiply everything by that power of ten. 1.5 : 2.5 needs one decimal place, so multiply by 10 to get 15 : 25.

Find the greatest common divisor

For 15 : 25 the GCD is 5. With more than two terms, take the GCD of the first two, then the GCD of that result with the third, and so on — the answer is the same whichever order you go in.

Divide every term by it

15 ÷ 5 : 25 ÷ 5 gives 3 : 5. Verify by confirming the quotient is unchanged: 1.5/2.5 = 0.6 and 3/5 = 0.6.

A : B : C → A/g : B/g : C/g

Simplification. Where g = gcd(A, B, C). Dividing every term by the same number preserves the ratio.

gcd(a, b) = gcd(b, a mod b)

The Euclidean algorithm. Repeat until the remainder is zero. gcd(60, 45) → gcd(45, 15) → gcd(15, 0) = 15. Around 2300 years old and still the fastest way.

Three terms cannot be reduced in pairs

For 12 : 18 : 30, cancelling 12 and 18 by 6 while leaving 30 alone breaks the ratio. Every term must be divided by the same number — the GCD of all three, which here is also 6, giving 2 : 3 : 5.

Worked examples

Whole numbers, decimals, fractions and units

Whole numbers

Simplify 45 : 60

Factors of 45
1, 3, 5, 9, 15, 45
Factors of 60
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Greatest common
15
Divide both
45÷15 : 60÷15
Check
45/60 = 0.75, 3/4 = 0.75 ✓

3 : 4

Decimals

Simplify 1.5 : 2.5

Decimal places
1, so multiply by 10
Cleared
15 : 25
GCD
5
Divide both
15÷5 : 25÷5
Check
1.5/2.5 = 0.6, 3/5 = 0.6 ✓

3 : 5

Fractions

Simplify ½ : ⅔

Denominators
2 and 3
LCM
6
Multiply both
(1/2)×6 : (2/3)×6
Cleared
3 : 4
GCD of 3, 4
1 — already simplest

3 : 4

Mismatched units

Express 30 minutes to 2 hours as a ratio.

Convert
2 hours = 120 minutes
Ratio
30 : 120
GCD
30
Divide both
30÷30 : 120÷30
Wrong answer if unconverted
30 : 2 = 15 : 1 ✗

1 : 4

Reference

Common ratios in simplest form

Ratio GCD Simplest form Decimal Note
2 : 421 : 20.5Halving
45 : 60153 : 40.75Three quarters
1920 : 108012016 : 91.7778Full HD aspect ratio
1024 : 7682564 : 31.3333Classic monitor ratio
100 : 25254 : 14Four times as much
12 : 18 : 3062 : 3 : 5—Three-term ratio
7 : 917 : 90.7778Coprime, cannot reduce
1.5 : 2.55 after ×103 : 50.6Decimals cleared first
1/2 : 2/31 after ×63 : 40.75Fractions cleared with the LCM
8 : 12 : 20 : 3242 : 3 : 5 : 8—Four-term ratio
Background

Why simplest form is worth reaching

Every ratio has infinitely many equivalent forms and exactly one simplest form. That uniqueness is what makes simplification useful: two ratios are equivalent if and only if they reduce to the same pair. So reducing both is a reliable way to compare them, and it is the method behind the equivalent ratio calculator's equality check.

It also makes ratios legible. 1920 : 1080 tells you the pixel dimensions of a screen; 16 : 9 tells you its shape, which is the part you actually reason about. A recipe listed as 250 : 100 : 50 grams is harder to hold in mind than 5 : 2 : 1, even though they specify the same mixture — and the simplified version is the one you can scale in your head.

How the Euclidean algorithm works

Listing all the factors of both numbers works for small values but becomes impractical quickly. Euclid's method instead uses the fact that any common divisor of two numbers also divides their difference. Repeatedly replacing the larger number with the remainder of division shrinks the problem fast: gcd(1920, 1080) becomes gcd(1080, 840), then gcd(840, 240), gcd(240, 120), gcd(120, 0) = 120. Five steps rather than factorising two four-digit numbers.

Simplified is not always what you want

Sometimes the unsimplified form carries information that reducing throws away. A screen resolution of 1920 : 1080 is not interchangeable with 16 : 9 if you need the actual pixel count. A mixing instruction of 250 : 100 grams tells you the batch size as well as the proportions. Simplify when you want the shape of the relationship; keep the original when the absolute quantities matter.

Related tools

To go the other way and generate larger equivalent ratios, use the equivalent ratio calculator. For converting a simplified ratio into percentages, the ratio to percentage calculator handles it. For dividing a total according to a simplified ratio, the 3-number and 4–6 number calculators do the share-out. And for screen and image dimensions specifically, the width and height ratio calculator adds the standard aspect-ratio presets.

Questions

Simplifying ratios — questions

Decimals, fractions, units and multi-term ratios — the four cases that need extra care.

How do you simplify a ratio?
Divide every term by the greatest common divisor of all the terms. For 45 : 60 the GCD is 15, so dividing gives 3 : 4. The ratio is in simplest form when the only number that divides every term is 1. Dividing by any common factor works, but using the GCD gets you there in one step.
What is the greatest common divisor?
The largest number that divides every term exactly. For 45 and 60 the common divisors are 1, 3, 5 and 15, so the GCD is 15. The quickest way to find it by hand is the Euclidean algorithm: repeatedly replace the larger number with the remainder of dividing it by the smaller, until one becomes zero. The other is then the GCD.
How do you simplify a ratio containing decimals?
Multiply every term by a power of ten until they are all whole numbers, then reduce as usual. For 1.5 : 2.5, multiplying by 10 gives 15 : 25, whose GCD is 5, giving 3 : 5. The number of tens you need is set by the term with the most decimal places. This calculator does that scaling automatically.
How do you simplify a ratio with fractions?
Multiply every term by the lowest common multiple of the denominators to clear all the fractions at once, then reduce. For 1/2 : 2/3 the LCM of 2 and 3 is 6, so multiplying gives 3 : 4, which is already simplest. Mixed numbers such as 1 1/2 should be converted to improper fractions first, which the calculator handles for you.
Can a simplified ratio still contain a decimal?
Not once the terms have been scaled to integers, which is the standard convention. There is one exception in normal use: the 1 : n form deliberately keeps a decimal in the second term so that the first is exactly 1, because n then works as a direct multiplier. This calculator reports that form alongside the whole-number one.
Do the units have to match before simplifying?
Yes, and this is a common source of wrong answers. A ratio of 30 minutes to 2 hours is not 30 : 2. Convert to the same unit first — 2 hours is 120 minutes — and the ratio is 30 : 120, which simplifies to 1 : 4. Ratios are only meaningful between quantities measured the same way.
Can I simplify a ratio with more than two terms?
Yes. The method is identical: find the GCD of all the terms and divide each by it. For 12 : 18 : 30 the GCD is 6, giving 2 : 3 : 5. This calculator handles two to six terms. Note that a three-term ratio cannot generally be simplified pairwise — you must use the GCD of the whole set, or you will not reach the simplest form.
What if the terms share no common factor?
Then the ratio is already in simplest form and the calculator says so. The GCD comes out as 1 and no division is needed. 7 : 9 is an example: 7 is prime and does not divide 9, so nothing can be cancelled. Two numbers with a GCD of 1 are called coprime.