Ratio Calculator
Result
Simplified ratio
- As a decimal
- 1.5000
- With the first term as 1
- 1 : 0.6667
A ratio calculator simplifies a ratio into its lowest terms and then reads the same pair two more ways. Give it 6 : 4 and both terms divide by two, so the simplified ratio is 3 : 2; the same pair is 1.5 when the first term is divided by the second, and 1 : 0.6667 when the ratio is turned round and written with the first term as 1. Those three readings are not three answers to three questions. A ratio is a pair of numbers rather than a single division, so 6 : 4, 3 : 2 and 1.5 all describe one relationship, and the reason to print more than one of them is that different jobs need different spellings. A recipe, a scale drawing or a gear train can only be written as whole numbers, which is what the simplified form is for. A slope or a rate that is about to be multiplied wants the decimal. And a sentence that explains a ratio to somebody usually wants the per-one form, because for every one of the first term there are 0.6667 of the second is easier to act on than a bare pair. Terms that are not whole numbers are worked with rather than refused. Both terms are scaled by the same power of ten until they are integers, and only then is the common factor cancelled: 2.5 : 5 becomes 25 : 50 and then 1 : 2. Equivalent ratios are therefore recognised as such, and a decimal-heavy pair like 0.5 : 2 comes back as the clean 1 : 4. The panel does not always print the terms in the order they arrived, either. When a sign is involved the two terms are multiplied by the same number until the sign sits on the first one, so 3 : -4 is printed as -3 : 4 — the same ratio, spelled the way the rest of the site spells it.
Common ratios, simplified and read
| Ratio | Simplified | Decimal |
|---|---|---|
| 2 : 4 | 1 : 2 | 0.5 |
| 3 : 12 | 1 : 4 | 0.25 |
| 6 : 4 | 3 : 2 | 1.5 |
| 12 : 18 | 2 : 3 | 0.6667 |
| 8 : 12 | 2 : 3 | 0.6667 |
| 9 : 12 | 3 : 4 | 0.75 |
| 7 : 7 | 1 : 1 | 1 |
| 5 : 2 | 5 : 2 | 2.5 |
| 100 : 25 | 4 : 1 | 4 |
| 0.5 : 2 | 1 : 4 | 0.25 |
| 2.5 : 5 | 1 : 2 | 0.5 |
| 1 : 3 | 1 : 3 | 0.3333 |
Each row is a ratio as it might be written, the same pair in lowest terms, and the first term divided by the second. Read across and the three agree: 6 : 4 and 3 : 2 are the same relationship and both are 1.5, which is the whole point of simplifying. Two rows are worth knowing by heart because they are the ones people get wrong. The row pair 12 : 18 and 8 : 12 both simplify to 2 : 3 — different pairs, same ratio — and 7 : 7 simplifies to 1 : 1, which is the ratio of two equal quantities and the only row where all three columns say the same thing. The last two rows carry a decimal point in the original pair, and they are there to show that it makes no difference to the answer: 2.5 : 5 simplifies to 1 : 2, exactly as 25 : 50 would.
Formula
Simplified ratio = both terms ÷ their greatest common divisor 6 : 4 → 3 : 2 Decimal = first ÷ second → 1.5 Per one = 1 : second ÷ first → 1 : 0.6667
- First term
- The number before the colon. It is the term that gets divided by the second one to give the decimal reading, and it is the one that carries the sign.
- Second term
- The number after the colon. It may not be zero: a ratio with nothing in the second position is not a comparison, and the decimal reading would be a division by zero.
- Simplified ratio
- The same pair with the greatest common divisor of the two terms cancelled, so the terms are relatively prime. This is the form to use where only whole numbers can be written, and it is exact: 12.3457 : 1 comes back as 123457 : 10000 rather than as a rounded decimal.
- Decimal
- The first term divided by the second, which is the quotient the ratio stands for. 1.5 means the first term is one and a half times the second. It is printed to four decimal places, so a ratio such as 1 : 3 shows 0.3333.
- Per one
- The second term divided by the first, written as 1 : that figure. It answers how much of the second there is for each single unit of the first, and it needs the first term to be nonzero. Also printed to four decimal places.
Three uses, and the first is the one that needs explaining: to simplify a ratio is to put it in the form a scale or a recipe can actually be built from, because 6 : 4 is a pair of numbers while 3 : 2 is a recipe. The second is arithmetic that follows the ratio: converting a ratio to decimal form turns it into a number that can be multiplied, compared or plotted, which is what happens to a slope, a rate or an odds figure before it goes anywhere else. The third is explaining a ratio in words, where the per-one reading does the work, since for every one of the first term there are 0.6667 of the second is a sentence somebody can act on. All three come from the same pair, and none of them is more correct than the others: they are the same relationship written for three different kinds of reader.
Worked examples
6 : 4
- Common divisor: both 6 and 4 divide by 2
- Simplify: 6 ÷ 2 = 3 and 4 ÷ 2 = 2, so 3 : 2
- Decimal: 6 ÷ 4 = 1.5
- Per one: 4 ÷ 6 = 0.6667 to four decimal places
The default case. The two terms were both whole numbers to begin with, so the simplification is a single division by two, and the per-one reading is where the rounding shows up.
2.5 : 5, with a decimal term
- Scale both terms by ten: 25 : 50
- Simplify: 25 ÷ 25 = 1 and 50 ÷ 25 = 2, so 1 : 2
- Decimal: 2.5 ÷ 5 = 0.5
- Per one: 5 ÷ 2.5 = 2
A term with a decimal point is dealt with by multiplying both terms by ten first, which leaves the ratio unchanged. Skipping that step and cancelling anyway would leave a term that is still not a whole number.
3 : 4, already in lowest terms
- Common divisor: 3 and 4 have none above 1
- Simplify: the pair is already 3 : 4
- Decimal: 3 ÷ 4 = 0.75
- Per one: 4 ÷ 3 = 1.3333 to four decimal places
The panel prints the same pair back when there is nothing to cancel, which is the correct answer rather than a sign that nothing happened. The per-one reading is still there and is not a whole number.
3 : -4, where the sign moves
- Both terms are multiplied by -1: -3 : 4
- Nothing else cancels, so the simplified ratio is -3 : 4
- Decimal: -3 ÷ 4 = -0.75, the same as 3 ÷ -4
- Per one: 4 ÷ -3 = -1.3333
The ratio is unchanged by multiplying both terms by the same number, so -3 : 4 and 3 : -4 are the same relationship and the panel prints the first. Every reading confirms it: the decimal is -0.75 either way.
Limitations
The decimal and per-one readings are printed to four decimal places, so a ratio that does not divide evenly is rounded there: 1 : 3 shows 0.3333 rather than a third, and the exact figure is a repeating decimal that only a fraction can write down. The simplified ratio is not rounded and is not approximate: 12.3457 : 1 comes back as 123457 : 10000, which is the exact pair, not a decimal dressed up. A second term of zero is refused, because a : 0 is not a comparison and the decimal reading would be a division by zero. A first term of zero is also refused, even though 0 : 5 arguably simplifies to 0 : 1 — the per-one reading needs to divide by the first term, and printing two thirds of a panel would be worse than saying so. The sign is always printed on the first term, so a term entered as 3 : -4 comes back as -3 : 4; the ratio is the same, and a figure copied out of the panel may look different from the one that was typed in. Very large figures are refused rather than approximated: the simplification is done in whole numbers, and past the point where whole numbers stop being exact the common factor would be computed from noise. This page simplifies a ratio, reads it two ways and stops there. Filling in a missing term of a proportion, splitting a quantity in a given ratio, and working out a ratio between quantities with units are three different calculations.
Frequently asked questions
- How do I simplify a ratio?
- Find the largest number that divides both terms and divide each term by it. In 6 : 4 that number is 2, which gives 3 : 2. When the terms are not whole numbers, multiply both of them by ten as many times as needed until they are, and cancel after that.
- What does the decimal reading mean?
- It is the first term divided by the second, which is the quotient the ratio stands for: 3 : 2 is 1.5, meaning the first term is one and a half times the second. It is the reading to move into arithmetic, since a slope or a rate has to be a number before it can be multiplied.
- What does the per-one reading mean?
- It rewrites the ratio with the first term as 1: the second term divided by the first, printed as 1 : that figure. For 6 : 4 it is 1 : 0.6667, which reads as for every one of the first term there are 0.6667 of the second. It is the reading to say out loud.
- Can a ratio have a decimal in it?
- Yes. 2.5 : 5 is a perfectly good ratio, and it simplifies to 1 : 2 by multiplying both terms by ten and then cancelling. Multiplying both terms by the same number never changes the ratio, which is what makes that step safe.
- Why does the panel print the same ratio back sometimes?
- Because the pair was already in lowest terms. 3 : 4 has no common divisor above 1, so the simplification has nothing to do; the panel prints 3 : 4 because that is the answer, and the other two readings still change.
- Why is 3 : -4 shown as -3 : 4?
- Because multiplying both terms by the same number leaves the ratio unchanged, and the site always writes the sign on the first term. The two forms are the same relationship, and every reading agrees: the decimal is -0.75 either way.
- What happens if one of the terms is zero?
- A second term of zero is refused, because a : 0 is not a comparison and the decimal reading would divide by zero. A first term of zero is refused too: the ratio itself could be simplified, but the per-one reading divides by the first term and there is no answer for it.
References
- Ratio — the ratio of two numbers written with a colon, where the first behaves as the numerator and the second as the denominator, so a ratio is equivalent to the quotient of the two and the colon form and the decimal form are two spellings of one relationship — Wolfram MathWorld (United States)
- Greatest Common Divisor — the largest divisor common to two integers, with two integers having a greatest common divisor of 1 described as relatively prime, which is exactly what cancelling to lowest terms leaves behind — Wolfram MathWorld (United States)
- Reduced Fraction — a pair written with the common factors cancelled, which is the same operation this page performs on the two terms of a ratio before printing them — Wolfram MathWorld (United States)