Ratio Calculator
Simplify a ratio to lowest terms and split a total by it — two terms or three.
Updated
Leave at zero for a two-term ratio.
In lowest terms
16 : 9
16 : 9 reduced by their greatest common factor.
- As a decimal
- 1.7778
- Total of the terms
- 25.00
Share of the total
A ratio compares parts to each other; a fraction compares one part to the whole. For 16:9 the first term is 64.0% of the total, not 16/9 of it.
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In short
How do you simplify a ratio?
Divide every term by their greatest common factor. For 12:18 that factor is 6, giving 2:3. Decimal terms have to be scaled up to whole numbers first, so 0.5:1.5 becomes 5:15 and then reduces to 1:3.
A ratio compares parts to each other. A fraction compares one part to the whole, which is a different question.
How to use the ratio calculator
A ratio says how much of one thing there is compared with another. Simplifying it means finding the smallest whole numbers that express the same comparison, and that is done by dividing every term by the largest number that divides all of them. For 12:18 that number is 6, so the ratio is 2:3 — the same relationship, easier to hold in mind.
The step that trips people up is decimals. Running the greatest-common-factor algorithm directly on 0.5 and 1.5 produces nonsense, because the algorithm works on whole numbers by construction. The terms have to be multiplied up first: 0.5:1.5 becomes 5:15, which reduces cleanly to 1:3. This page does that scaling automatically and reports the multiplier it used.
2 : 3
Twelve to eighteen
both divided by 6
1 : 3
Nought point five to one point five
scaled by ten first
25%
The first part of 1:3
one of four, not one of three
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That third figure is the distinction worth learning. In the ratio 1:3 the first part is not a third of the total — it is a quarter, because the total is four parts. A ratio compares the parts to each other, while a fraction compares one part to the whole. Reading a part-to-part ratio as a fraction is the commonest error in this area and it always understates the larger share.
Ratios also scale. A recipe written for 2:3 flour to water works identically at 200 g to 300 g or at 2 kg to 3 kg, because only the relationship is fixed. That is what makes ratios useful for scaling anything up or down: the numbers change and the result does not.
- As a ratio
- 1 : 3
- As parts of a whole
- 1 part of 4, and 3 parts of 4
- As fractions
- 1/4 and 3/4
- As percentages
- 25% and 75%
- As a decimal quotient
- 0.333 — the first divided by the second
The last row is the one that looks like a fraction and is not the same thing. It is the quotient of the two parts, not either part as a share of the total.
Scale a ratio to a new size
When you know three terms of a proportion and need the fourth, cross multiplication solves it in one step. That page does it and shows both cross products.
Open the proportion calculator →Common ratios reduced to lowest terms, with each part shown as its share of the total so the part-to-part and part-to-whole readings sit side by side.
| Ratio | Simplified | Shares of the total |
|---|---|---|
| 12 : 18 | 2 : 3 | 40% and 60% |
| 1 : 3 | 1 : 3 | 25% and 75% |
| 1 : 1 | 1 : 1 | 50% and 50% |
| 2 : 1 | 2 : 1 | 66.7% and 33.3% |
| 0.5 : 1.5 | 1 : 3 | 25% and 75% |
| 16 : 9 | 16 : 9 | 64% and 36% |
| 4 : 3 | 4 : 3 | 57.1% and 42.9% |
| 100 : 75 | 4 : 3 | 57.1% and 42.9% |
| 10 : 20 : 30 | 1 : 2 : 3 | 16.7%, 33.3%, 50% |
| 2 : 3 : 5 | 2 : 3 : 5 | 20%, 30%, 50% |
Part to part, and part to whole
Two ratios can describe the same mixture and look completely different, because they are measuring against different things. A part-to-part ratio compares the ingredients with each other. A part-to-whole ratio compares one ingredient with the finished total. Neither is wrong, and reading one as the other is how dilution goes astray.
- Part to part
- 1 : 3, concentrate to water
- Total parts
- 4
- Part to whole
- 1 : 4, concentrate to mixture
- As a fraction of the total
- 1/4
- Concentrate
- 25% of the finished mixture
Product labels use both conventions and rarely say which. "Dilute 1:3" almost always means one part concentrate to three parts water; "a 1 in 4 dilution" means one part concentrate making four parts in total. They describe the same bottle.
The safe habit is to add the terms. Whatever the notation, if the parts sum to four then each part is a quarter of the whole, and no amount of ambiguity in the label changes that arithmetic.
Where the numbers come from
Aspect ratios describe the shape of a screen or an image: 16:9 is the widescreen standard, 4:3 the older television shape, 3:2 the proportion of a 35 mm frame. They are ratios rather than sizes, which is exactly why a 16:9 image fits a phone and a cinema screen without changing.
Do
- Add the terms to find the total before working out shares
- Scale decimal terms to whole numbers before reducing
- Keep the order of the terms — 1:3 and 3:1 are different mixtures
- State whether a dilution is part-to-part or part-to-whole
Don't
- Read the first term of a:b as a/b of the total
- Reduce a ratio by dividing only some of its terms
- Assume 1:3 and 1 in 3 mean the same thing
- Compare two ratios without putting them on the same basis first
Financial ratios work the same way and carry the same trap. A debt-to-equity ratio of 2:1 means debt is two thirds of the total capital, not two times the total. The ratio is against the other term, and the share is against the sum, which is the same distinction in a suit.
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Full guide
Scale Factor: Why Doubling the Size Gives Four Times the Area
Twice as big means four times the paint and eight times the water. Lengths take k, areas take k², volumes take k³, and every scaling error is the wrong exponent.
Read the full guide →The formula, worked line by line
One division for the reduction and one for each share. The only subtlety is which denominator each uses, and that is the whole difference between a ratio and a fraction.
“The ratio divides by the other part. The share divides by the total. Two different denominators, and only one of them is what somebody usually wants.”
Once that is fixed, ratios become the most portable way to express a relationship: they survive any change of scale or unit, which is why recipes, screens, maps and balance sheets all reach for them.
simplified = each term ÷ gcf(all terms)
total = sum of all terms
share of term a = a ÷ total
a as a percentage of the total = a ÷ total × 100
quotient of a : b = a ÷ b- Total parts
- 2 + 3 + 5 = 10
- On a bill of £240, one part is
- £24
- First share
- 2 × 24 = £48
- Second share
- 3 × 24 = £72
- Third share
- 5 × 24 = £120
- Check
- 48 + 72 + 120 = £240
Finding the value of one part first is the reliable method for any number of terms. The check at the end is not optional: if the shares do not sum back to the total, a part was miscounted somewhere and the error is usually in the sum of the terms.
Why the order matters
A ratio is an ordered comparison, so 1:3 and 3:1 describe different things entirely — one part in four against three parts in four. Fractions carry the same information in their numerator and denominator, and reversing them inverts the value. Whenever a ratio is written down, the labels matter as much as the numbers.
Aspect ratios, which are ratios and nothing else
- 16 : 9
- widescreen, the current television standard
- 4 : 3
- the older television and monitor shape
- 3 : 2
- the 35 mm photographic frame
- 1 : 1
- square
- 21 : 9
- ultrawide, marketing shorthand rather than exact
None of these is a size. A 16:9 image is the same shape at 1,280 by 720 and at 3,840 by 2,160, which is precisely what makes the ratio the useful thing to quote.
The last row is worth a caveat: "21:9" is a marketing label rather than an exact ratio. The common ultrawide resolutions work out closer to 64:27 or 43:18, and neither reduces to 21:9. It is a name, not arithmetic.
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Questions people ask
How do you simplify a ratio?
Divide every term by the greatest common factor of all of them. For 12:18 the greatest common factor is 6, so the simplified ratio is 2:3. If the terms are decimals, multiply them all up to whole numbers first — 0.5:1.5 becomes 5:15, which then reduces to 1:3.
Covered in depth in Scale Factor: Why Doubling the Size Gives Four Times the Area →
Is a ratio the same as a fraction?
They are closely related but not the same. A ratio compares two quantities with each other; a fraction expresses one quantity as a share of a whole. In the ratio 1:3 the first part is one of four parts in total, so it is a quarter of the whole rather than a third, even though the ratio is written with a 1 and a 3.
What does a 1:3 dilution mean?
Conventionally, one part concentrate to three parts water, giving four parts in total with the concentrate at 25 percent. A "1 in 4" dilution describes the identical mixture from the other direction — one part in a total of four. Since labels use both wordings, adding the parts to find the total is the way to be certain.
How do you split an amount in a given ratio?
Add the terms to find the total number of parts, divide the amount by that total to get the value of one part, then multiply by each term. Splitting £240 in the ratio 2:3:5 gives ten parts of £24, so the shares are £48, £72 and £120. They should always sum back to the original amount.
Can a ratio have three or more terms?
Yes, and it reduces the same way: divide every term by the greatest common factor of all of them together. So 10:20:30 becomes 1:2:3. The only rule that changes is that you cannot express a three-term ratio as a single fraction, because a fraction compares exactly two quantities.
Sources
Where the constants and formulas on this page come from. Each line names the figure it backs.
A ratio describes a relationship between two quantities, distinct from a fraction expressing a part of a whole.
CCSS.Math.Content.6.RP — Ratios and Proportional Relationships — Common Core State Standards Initiative
The greatest common factor of a set of whole numbers is the largest number that divides all of them.
Greatest Common Divisor — Wolfram MathWorld
The 16:9 aspect ratio is the standard picture format for high-definition television.
Recommendation ITU-R BT.709 — Parameter values for HDTV standards — International Telecommunication Union
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