Ratio Calculator
A ratio simplifies exactly as a fraction does — they are the same object written differently. Scaling one is cross multiplication, which is the arithmetic behind recipes, maps and mixing anything.
Simplified
2 : 3
- Simplified
- 2 : 3
- As a decimal
- 0.6667
- As a fraction
- 2/3
- First as a share
- 40.00%
- Scaling 20 gives
- 30.0000
- Solving 8 : 12 = 20 : x by cross multiplication gives 30.0000. This is the arithmetic behind scaling a recipe, reading a map and mixing anything to a proportion.
- A ratio simplifies by dividing both sides by their greatest common divisor, exactly as a fraction does — they are the same object written differently.
- A ratio of 2:3 means five parts in total, so the first is two fifths of the whole rather than two thirds. Reading a ratio as a fraction of the total is the most common mistake with them.
About ratios and proportions
Reducing a ratio means dividing both sides by their greatest common divisor. 8:12 becomes 2:3, because both share a factor of four. That is identical to reducing 8/12 to 2/3, and the two notations describe the same relationship.
The most common mistake with ratios is reading one side as a fraction of the whole. In a ratio of 2:3 there are five parts altogether, so the first side is two fifths of the total — forty per cent — not two thirds. Two thirds is the first side compared to the second, which is a different question.
Scaling is cross multiplication. If 8 is to 12 as 20 is to something, that something is 20 × 12 ÷ 8, which is 30. This is what you are doing when you double a recipe, read a map scale, mix concrete to a specification or work out a currency-free exchange between any two proportional quantities.
Ratios with more than two parts — 2:3:5 for a concrete mix, say — reduce the same way, by dividing every part by the greatest common divisor of all of them. This handles two at a time, which covers most of what people ask.
What it works out
- A ratio in its simplest form
- Solving a proportion for the missing value
- The fraction, decimal and percentage forms
The formula
Simplify: divide both by their GCD Solve a:b = c:x for x = c × b ÷ a
8 and 12 share a greatest common divisor of 4, so 8:12 reduces to 2:3.
For the proportion, write it as a fraction equation: 8/12 = 20/x. Cross multiplying gives 8x = 20 × 12, so x = 240 ÷ 8, which is 30. The ratio 8:12 scaled to start at 20 is 20:30 — the same relationship at a larger size.
As a share of the whole, the first part of 2:3 is 2 ÷ (2+3), which is 40%. That total-parts step is where most ratio errors happen; the instinct is to read 2:3 as two thirds, and two thirds is the first compared to the second rather than to the whole.
The decimal form, 0.667 here, is the first divided by the second. It is the form to use when comparing two ratios quickly — a higher decimal means a larger first part relative to the second.
- First, Second
- The two parts of the ratio.
- GCD
- Their greatest common divisor. What both divide by to reduce.
- Scale
- A new value for the first part. The second follows proportionally.
A worked example
The ratio 8:12, scaled so the first part becomes 20.
That works out to 2 : 3 .
- Simplified
- 2 : 3
- As a decimal
- 0.6667
- As a fraction
- 2/3
- First as a share
- 40.00%
- Scaling 20 gives
- 30.0000
Questions
How do I simplify a ratio?
Divide both sides by their greatest common divisor. 8:12 both divide by 4, giving 2:3 — exactly the same operation as reducing the fraction 8/12.
Is a ratio of 2:3 the same as two thirds?
No, and this is the most common ratio mistake. 2:3 means five parts in total, so the first is two fifths — forty per cent — of the whole. Two thirds is the first part compared to the second.
How do I scale a ratio?
Cross multiply. If 8:12 becomes 20:x, then x = 20 × 12 ÷ 8, which is 30. This is the arithmetic behind doubling a recipe or reading a map scale.
What is a proportion?
A statement that two ratios are equal — 8:12 = 20:30. Solving one means finding the missing value that makes it true, which is what cross multiplication does.
How do I handle a ratio with three parts?
Divide all three by the greatest common divisor of all of them. A 2:3:5 concrete mix is already reduced; 4:6:10 would become the same thing. This calculator handles two parts at a time.
What is the difference between a ratio and a rate?
A ratio compares two quantities of the same kind, so the units cancel and it has none. A rate compares different kinds — miles per hour, price per kilogram — and keeps its units.