Ratio Calculator

Two things in one tool: solve A : B = C : D for whichever term is missing, or take a single ratio and scale it up or down by a factor. Both modes report the simplified form, the decimal, the percentage and the 1 : n form.

Formula A : B = C : D

A
B
C
D
Try an example
Method

Solving and scaling

Two different operations that people often confuse. Solving finds a missing value; scaling rewrites a ratio you already have.

Solving for a missing term

A : B = C : D, one term unknown

  • You have two ratios and three of the four numbers
  • Cross multiply: A × D = B × C
  • Divide to isolate the unknown
  • Answer is a single number
  • "3 : 5 = x : 20 — what is x?"

Scaling a ratio

A : B → kA : kB

  • You have one ratio and a factor
  • Multiply or divide every term by the factor
  • Nothing to solve — it is a rewrite
  • Answer is a new ratio
  • "3 : 5 scaled by 4 — what is it?"

Keep the order consistent

If the left ratio is cats to dogs, the right ratio must be cats to dogs too. Swapping the two quantities on one side and not the other is the most common error in ratio problems, and it produces an answer that looks reasonable.

Cross multiply to clear the ratios

A : B = C : D is the same statement as A/B = C/D, so cross multiplication applies: A × D = B × C. This removes both divisions and leaves a straightforward equation.

Divide by whatever multiplies the unknown

For 3 : 5 = x : 20 you get 60 = 5x, so divide by 5. The divisor is always the term diagonally opposite the unknown.

Simplify to check

Reduce both ratios to lowest terms. 3 : 5 is already reduced and 12 : 20 reduces to 3 : 5, so the answer is right. Two equivalent ratios always reduce to the same pair.

For a whole-of-total question, add the parts

If 3 : 5 describes parts of a whole, the total is 8 parts, so the first share is 3/8 = 37.5%. Reading it as 3/5 = 60% is a different question entirely — that is A as a fraction of B, not of the total.

A : B = C : D  ⟺  AD = BC

The proportion. Ratios written with colons behave exactly like fractions, so every fraction technique carries over.

A : B = kA : kB

Scaling. Multiplying every term by the same non-zero k leaves the ratio unchanged. This is why a ratio has infinitely many equivalent forms and only one simplest form.

A : B is not always A/B

As a comparison, 3 : 5 means 3/5 of B. As parts of a whole, the first share is 3/8 of the total. Both readings are correct for different questions — the deciding factor is whether B is the other part or the whole.

Worked examples

Ratios in practice

Missing term

A class has boys and girls in the ratio 3 : 5. If there are 20 girls, how many boys?

Proportion
3 : 5 = x : 20
Cross multiply
3 × 20 = 5 × x
Known side
60 = 5x
Divide by 5
x = 12
Check
12 : 20 reduces to 3 : 5 ✓

12 boys — 32 students in total

Scaling up

A concrete mix is 1 : 2 : 4 cement to sand to gravel. For 2 buckets of cement, what about the cement-to-sand part?

Ratio
1 : 2
Factor
× 2
Both terms
1×2 : 2×2
New ratio
2 : 4
Check
2/4 = 1/2 ✓ unchanged

2 : 4 — for the full three-part mix use the 3-number tool

Scaling down

Reduce the ratio 12 : 18 by dividing by 6.

Ratio
12 : 18
Divide both
12÷6 : 18÷6
New ratio
2 : 3
Decimal
2/3 ≈ 0.6667 — same as 12/18
Note
6 is the GCD, so this is simplest form

2 : 3

Fractional terms

Solve ½ : ¾ = x : 9

Cross multiply
(1/2) × 9 = (3/4) × x
Left side
9/2 = (3/4)x
Multiply by 4/3
x = (9/2) × (4/3)
Simplify
x = 36/6 = 6
Check
0.5/0.75 = 0.667, 6/9 = 0.667 ✓

x = 6

Reference

The same ratio in every form

A ratio can be written several ways without changing what it says. These are the conversions worth knowing.

Ratio Simplest As A/B Decimal First part of total 1 : n
1 : 21 : 21/20.533.33%1 : 2
2 : 32 : 32/30.666740%1 : 1.5
3 : 43 : 43/40.7542.86%1 : 1.3333
3 : 53 : 53/50.637.5%1 : 1.6667
12 : 203 : 53/50.637.5%1 : 1.6667
16 : 916 : 916/91.777864%1 : 0.5625
45 : 603 : 43/40.7542.86%1 : 1.3333
1.5 : 2.53 : 53/50.637.5%1 : 1.6667
100 : 254 : 14/1480%1 : 0.25
Background

What a ratio does and does not tell you

A ratio carries relative information only. Told that a recipe uses flour and butter in the ratio 5 : 2, you know the proportions exactly but nothing about the quantity — it could be 500 g and 200 g, or 5 kg and 2 kg. That is the point: the ratio is the part of the recipe that stays the same however much you make, which is what makes it useful for scaling.

The flip side is that a ratio alone cannot answer a question about amounts. "The ratio of staff to students is 1 : 15" does not tell you how many students there are. To get an absolute figure you need one more piece of information: either one of the actual quantities, or the total. With a total you are in share-out territory — see the 3-number ratio calculator, which is built for that.

The two-fraction ambiguity

The single most persistent confusion in ratio work is what fraction a ratio corresponds to. Given 3 : 5, both of these are correct statements about different things: the first quantity is 3/5 of the second, and the first quantity is 3/8 of the total. Which one you need depends entirely on the question. "How much bigger is B than A?" wants 3/5. "What share of the whole is A?" wants 3/8. This calculator reports both so you can pick.

Why ratios are usually written as whole numbers

Nothing forbids a ratio like 1.5 : 2.5, but it is harder to act on. Whole numbers tell you directly how many equal parts to count out — 3 scoops to 5 scoops. That is why the convention is to scale to integers and then reduce, which is what the simplify ratio calculator does. The exception is the 1 : n form, where a decimal second term is deliberate because n is being used as a direct multiplier.

Where to go from here

For a three-part ratio like a concrete mix, use the 3-number ratio calculator; for four to six terms, the longer-ratio version. To convert a ratio into percentages, the ratio to percentage calculator handles both readings described above. And to generate a whole series of equivalent ratios rather than one, use the equivalent ratio calculator.

Questions

Ratios — questions

Including the fraction ambiguity, which accounts for more wrong answers than the arithmetic does.

What is a ratio?
A ratio compares two quantities of the same kind, written A : B and read "A to B". It tells you the relative size of the two, not their absolute amounts — a ratio of 3 : 5 could describe 3 and 5, or 30 and 50, or 1.5 and 2.5. What stays fixed is the quotient A ÷ B, which is why 3 : 5 and 6 : 10 are the same ratio.
How do you find a missing number in a ratio?
Set the two ratios equal as a proportion and cross multiply. For 3 : 5 = x : 20, cross multiplying gives 3 × 20 = 5 × x, so 60 = 5x and x = 12. Which division you finish with depends on which of the four positions is blank; this calculator picks the right rearrangement and shows it.
What is the difference between a ratio and a proportion?
A ratio is a single comparison — 3 : 5. A proportion is a statement that two ratios are equal — 3 : 5 = 6 : 10. So a ratio is a quantity and a proportion is an equation. You solve a proportion; you simplify or scale a ratio.
Does the order of a ratio matter?
Yes, always. 3 : 5 and 5 : 3 describe opposite situations. If the ratio of cats to dogs is 3 : 5 then there are more dogs; reversing it says there are more cats. Whenever you write a ratio down, note which quantity is which — most wrong answers in ratio problems come from swapping the order partway through.
How do you write a ratio as a fraction?
The ratio A : B becomes the fraction A/B, which gives you the relative size of the first quantity to the second. Be careful with a different question though: if a ratio of 3 : 5 describes parts of a whole, then the first part is 3/8 of the total, not 3/5. The 8 comes from the total number of parts, 3 + 5.
How do you scale a ratio?
Multiply or divide every term by the same number. Doubling 3 : 5 gives 6 : 10; halving it gives 1.5 : 2.5. Because every term changes by the same factor, the quotient is unchanged and the new ratio is equivalent to the original. The scale mode on this calculator does this and shows the working.
Can a ratio have decimals or fractions in it?
Yes, and this calculator accepts them. In practice ratios are usually converted to whole numbers because they are easier to read and to work with — 1.5 : 2.5 is normally written 3 : 5. The simplify ratio calculator does that conversion, scaling to integers before reducing.
What does a ratio of 1 : n mean?
It is the ratio rewritten so the first term is exactly 1, which makes the second term a direct multiplier. 1 : 2.5 tells you the second quantity is two and a half times the first. This form is used for map and model scales, gear ratios and mixing instructions because it removes any mental arithmetic. To convert, divide both terms by the first.