What the simplest form of a ratio is
A ratio in lowest terms is the smallest set of whole numbers that carries the same proportion as the numbers you started with. 150 : 225 and 2 : 3 describe exactly the same mixture; the second is easier to hold in your head, easier to scale, and easier to check against a specification.
The proportion survives because multiplying or dividing every term by the same non-zero number does not change the relationship between them. That is the only rule at work here, and it runs in both directions. It lets you clear decimals by multiplying up — 1.5 : 2.25 becomes 150 : 225 when you multiply both by 100 — and it lets you reduce by dividing down. What you must never do is add or subtract the same number from every term. Adding 1 to each side of 2 : 3 gives 3 : 4, which is a different mixture entirely.
Three forms of the same ratio get used in practice and it is worth knowing which one your context expects. Lowest terms (2 : 3) is what a math answer wants. n : 1 form (0.667 : 1) is what drafting, gearing and optics use, because a single number then describes the whole relationship. Parts of a total (2 parts of 5) is what mixing wants, because you divide the batch by the total and count out parts. This calculator gives you all three.
How the reduction works: scaling, then Euclid
Two steps, in this order.
Step one: clear the decimals. Count the decimal places in the longest term, call it k, and multiply every term by 10k. For 1.5 : 2.25 the longest is two places, so multiply through by 100 to get 150 : 225. This is exact, not an approximation, because you are applying the same factor to every term.
Step two: divide by the greatest common divisor. The gcd of a set of integers is the largest number that divides all of them without remainder, and dividing by it is guaranteed to leave terms with no common factor — which is what makes the result lowest and not merely lower. Euclid's algorithm, from Book VII of the Elements, finds it by repeated remainders:
gcd(225, 150) → gcd(150, 75) → gcd(75, 0) = 75
Each line replaces the pair with the smaller number and the remainder of dividing the larger by it, and stops when the remainder is zero. For three or more terms, take the gcd of the first two and then the gcd of that result with the next term; gcd is associative, so the order does not matter.
The algorithm has one convention that matters here: gcd(x, 0) = x. That is why a ratio like 7 : 0 reduces to 1 : 0 rather than failing. It only breaks down when every term is zero, at which point there is no proportion to describe and this calculator says so instead of returning a number. Our greatest common factor calculator runs the same algorithm on its own if you want the intermediate remainders spelled out.
Why not just divide by the smallest term? Because that produces the n : 1 form, not whole numbers. 150 : 225 divided by 150 gives 1 : 1.5, which is a correct statement of the proportion but is not in lowest whole-number terms. Both are useful; they are answers to different questions.
Worked example: a 1.5 : 2.25 mortar mix scaled to a batch
A specification calls for 1.5 parts of one component to 2.25 parts of another, and you need to mix 40 litres.
- Clear the decimals. The longest term has two decimal places, so multiply both by 10² = 100: 150 : 225.
- Run Euclid. 225 ÷ 150 leaves remainder 75. 150 ÷ 75 leaves remainder 0. The gcd is 75.
- Divide through. 150 ÷ 75 = 2 and 225 ÷ 75 = 3, so the ratio is 2 : 3.
- Count the parts. 2 + 3 = 5 parts in total.
- Size one part. 40 ÷ 5 = 8 litres per part.
- Measure it out. 2 × 8 = 16 litres of the first, 3 × 8 = 24 litres of the second. Check: 16 + 24 = 40 litres. Correct.
The first component's share is 2 ÷ 5 = 0.40, or 40% of the batch, and the n : 1 form is 1.5 ÷ 2.25 = 0.6667 : 1. All three describe the same mix. Notice that 16 : 24 also reduces to 2 : 3 — gcd(16, 24) = 8 — which is the check that the scaling was done correctly.
Now a three-term case. A blend specified as 18 : 24 : 30 has gcd 6 — gcd(18, 24) = 6, then gcd(6, 30) = 6 — so it reduces to 3 : 4 : 5, 12 parts in total, with the first component at 3 ÷ 12 = 25% of the mix. Anyone who has laid out a right angle with a builder's triangle will recognise the result.
Which form to quote, and how to check it
Use lowest terms when you are communicating a specification. Whole numbers are unambiguous, they survive photocopying and dictation, and they make the parts countable. A mix quoted as 1 : 2 : 4 tells a labourer to count buckets; the same mix quoted as 14.3% : 28.6% : 57.1% does not.
Use n : 1 when you are comparing ratios to each other. A single number sorts. Aspect ratios, gear ratios, compression ratios, contrast ratios and lens speeds are all quoted this way, because 1.78 : 1 against 1.33 : 1 is an instant comparison while 16 : 9 against 4 : 3 needs a moment's arithmetic. The cost is that the number is usually not exact: 16 : 9 is 1.7778 : 1 to four places and never terminates.
Use shares of a total when you are dividing something up. The total-parts figure is the divisor, and each term is a multiplier. This is the form to use for splitting a bill, allocating shares, or apportioning a budget, and the ratio share calculator takes it further by doing the division for you.
One check catches nearly every mistake: multiply the simplified ratio back up and see if you recover something proportional to what you typed. If 2 : 3 is right for 1.5 : 2.25, then 2 × 0.75 = 1.5 and 3 × 0.75 = 2.25, with the same factor of 0.75 in both. A different factor for each term means the reduction is wrong.
Watch the units. A ratio is only meaningful when all terms are measured the same way. Two parts sand to three parts cement is a different mix by volume than by weight, and the two disagree by the ratio of the bulk densities. Specifications almost always say which, and the answer changes the batch.
Common ratios in each of the three forms
| As given | Lowest terms | As n : 1 | First term's share |
|---|---|---|---|
| 1920 : 1080 | 16 : 9 | 1.7778 : 1 | 64.00% |
| 4 : 3 | 4 : 3 | 1.3333 : 1 | 57.14% |
| 1.5 : 2.25 | 2 : 3 | 0.6667 : 1 | 40.00% |
| 0.75 : 0.5 | 3 : 2 | 1.5000 : 1 | 60.00% |
| 60 : 40 | 3 : 2 | 1.5000 : 1 | 60.00% |
| 1 : 1.618 | 500 : 809 | 0.6180 : 1 | 38.20% |
| 18 : 24 : 30 | 3 : 4 : 5 | 0.7500 : 1 | 25.00% |
| 100 : 35 : 15 | 20 : 7 : 3 | 2.8571 : 1 | 66.67% |
Rows four and five reduce to the same ratio from very different inputs, which is exactly the point of a simplest form. The golden-ratio row shows what happens to an irrational proportion: 1 : 1.618 is a truncated decimal, so its whole-number reduction is large.
Mistakes that change the mixture
- Adding or subtracting instead of multiplying. Taking 1 off each side of 3 : 5 gives 2 : 4 = 1 : 2, a completely different proportion. Only multiplication and division preserve a ratio.
- Dividing by a common factor that is not the greatest one. 150 : 225 divided by 5 gives 30 : 45, which is simpler but not simplest. Keep going until the terms are coprime.
- Mixing units between terms. 500 g to 1 kg is 1 : 2, not 500 : 1. Convert to a common unit before reducing.
- Confusing a ratio with a fraction of the whole. A ratio of 2 : 3 means the first part is two fifths of the total, not two thirds. The denominator of the share is the sum of the terms, not the other term.
- Rounding an irrational or repeating proportion before reducing. Truncating the golden ratio to 1.618 and reducing gives 500 : 809, which looks precise and is not. Keep such ratios in n : 1 form.
- Reducing a ratio that encodes a count. If 12 : 18 means twelve items and eighteen items, 2 : 3 is the proportion but you still have thirty objects. Reduce for the relationship, not for the inventory.
Ratios, fractions, proportions and rates
These four words are used loosely in conversation and precisely in mathematics, and the distinction decides which tool you reach for.
A ratio compares parts to parts: 2 : 3 sand to cement. It may have more than two terms, and none of them is a whole.
A fraction compares a part to a whole: the sand is 2/5 of the mix. Reducing a fraction uses the same gcd machinery, which is why the simplifying fractions calculator looks so similar — and why converting between the two is a matter of choosing a denominator, either the other term or the total.
A proportion is a statement that two ratios are equal, such as 2 : 3 = 8 : 12, and solving one for a missing term is what the proportion solver does by cross-multiplication.
A rate is a ratio between quantities in different units, such as 60 miles per hour. Rates are almost always quoted in n : 1 form with the “1” implied by the word “per”, and they cannot be reduced to whole numbers in the same way because the units do not cancel.
Two special cases are worth a mention. Ratios that reduce to nothing smaller are called coprime, and coprimality is a property with real consequences: gear pairs with coprime tooth counts distribute wear evenly because every tooth eventually meets every other tooth, which is why gear designers deliberately avoid ratios that reduce. And proportions that are irrational — the golden ratio, the 1 : √2 of ISO paper sizes — have no exact whole-number form at all. Any reduction of them is a reduction of a decimal approximation, as the golden ratio calculator makes explicit. Quote those in n : 1 form and say how many digits you are relying on.