Proportion Calculator with Fractions

Each of the four terms in the proportion gets its own numerator and denominator, so fractions and mixed numbers go in exactly as written. Leave one term blank and the answer comes back as an exact fraction — reduced, with the mixed number and decimal alongside.

Formula a⁄b : c⁄d = e⁄f : x

Leave both boxes of one term blank to solve for it. A term that is a plain whole number just needs its top box — the bottom defaults to 1. Mixed numbers like 1 1/2 can go straight in the top box.

Term A
Term B
Term C
Term D

Reads as A ÷ B = C ÷ D, the same shape as A/B = C/D with each term allowed to be a fraction of its own.

Try an example
Method

Solving a proportion in fractions

The structure is identical to the whole-number case. What changes is that each arithmetic step is fraction arithmetic.

Make every term a single fraction

A mixed number becomes improper: 1 1/2 → 3/2. A whole number becomes itself over one: 4 → 4/1. A decimal becomes a fraction over a power of ten: 0.75 → 75/100 → 3/4. After this step every term has the same shape.

Cross multiply

From A/B = C/D you get A × D = B × C. Multiplying fractions means numerator times numerator over denominator times denominator, so (2/3) × (1/2) = 2/6 = 1/3.

Divide by multiplying by the reciprocal

The unknown is multiplied by a fraction, so undo it by multiplying by that fraction's reciprocal. Dividing by 4/5 is the same as multiplying by 5/4. This is the step where most hand-worked errors creep in.

Reduce with the greatest common divisor

Find the GCD of numerator and denominator and divide both by it. 10/24 has GCD 2, giving 5/12. If the numerator is now larger than the denominator, you can also write it as a mixed number.

Substitute and verify

Put the answer back in and evaluate both sides as decimals. They should agree to every digit shown — if they do not, the reciprocal step is the first place to look.

a/b ÷ c/d = a/b × d/c

Division as multiplication. Flip the divisor and multiply. This one rule is what makes fractional proportions no harder than whole-number ones.

(a/b) × (c/d) = ac / bd

Multiplication. Straight across, top by top and bottom by bottom. Reducing before multiplying keeps the numbers small.

Why decimals lose information

1/3 as a decimal is 0.333333… forever. Round it and you have introduced an error before the calculation even starts. Keeping numerator and denominator separate means 3 × (1/3) comes back as exactly 1, not 0.999999.

Worked examples

Fractional proportions, worked out

Each of these keeps the answer exact all the way through.

Both sides fractional

Solve (2/3) ÷ (4/5) = x ÷ (1/2)

Cross multiply
(2/3) × (1/2) = (4/5) × x
Left product
2/6 = 1/3
Divide by 4/5
x = (1/3) × (5/4)
Multiply across
x = 5/12
Check
0.6667 ÷ 0.8 = 0.8333; 0.4167 ÷ 0.5 = 0.8333 ✓

x = 5/12 ≈ 0.4167

Recipe scaling

A recipe needs ¾ cup of sugar for every 2 cups of flour. How much sugar for 5 cups of flour?

Proportion
(3/4) ÷ 2 = x ÷ 5
Cross multiply
(3/4) × 5 = 2x
Left product
15/4 = 2x
Divide by 2
x = 15/8
As a mixed number
1 7/8

x = 15/8 cups = 1⅞ cups

Mixed number input

Solve (1½) ÷ 2 = x ÷ 6

Improper form
(3/2) ÷ 2 = x ÷ 6
Cross multiply
(3/2) × 6 = 2x
Left product
18/2 = 9, so 9 = 2x
Divide by 2
x = 9/2
As a mixed number
4 1/2

x = 9/2 = 4½

Unknown in the second position

Solve (5/6) ÷ x = (10/3) ÷ 4

Cross multiply
(5/6) × 4 = x × (10/3)
Left product
20/6 = 10/3
Divide by 10/3
x = (10/3) × (3/10)
Simplify
x = 30/30 = 1
Check
0.8333 ÷ 1 = 0.8333; 3.3333 ÷ 4 = 0.8333 ✓

x = 1

Reference

Fraction arithmetic you will need

Four operations, each with the form the calculator uses internally.

Operation Rule Example Result
Multiply a/b × c/d = ac/bd 2/3 × 1/2 2/6 = 1/3
Divide a/b ÷ c/d = ad/bc 1/3 ÷ 4/5 5/12
Add a/b + c/d = (ad+bc)/bd 1/2 + 1/3 5/6
Subtract a/b − c/d = (ad−bc)/bd 3/4 − 1/6 14/24 = 7/12
Mixed → improper w n/d = (wd+n)/d 2 3/5 13/5
Improper → mixed n/d = ⌊n/d⌋ + (n mod d)/d 15/8 1 7/8
Reduce divide both by gcd(n, d) 10/24 5/12
Reciprocal 1 ÷ (a/b) = b/a 1 ÷ (4/5) 5/4
Background

Where fractional proportions come up

Fractions appear in proportions wherever the underlying quantities are naturally measured in parts rather than units. Cooking is the obvious case — measuring cups come in halves, thirds and quarters, so scaling a recipe from four servings to six means working with 3/2 of everything. Sheet music divides beats into halves and quarters. Imperial measurement is fractional throughout: a drawing at 1/4 inch to the foot is a proportion before you have written anything down.

In each case the input data is exact. Three-quarters of a cup is not approximately 0.75 cups, it is exactly that. Turning it into a decimal is harmless here, but a third of a cup is 0.3333…, and multiplying that by 12 gives 3.9999996 rather than 4. The error is small, but it makes the answer look wrong even when the method was right, and it prevents you recognising a clean result when you get one.

Reading the answer back as a measurement

An exact fraction is also more useful at the other end. If a calculation says you need 15/8 cups, the mixed-number form 1 7/8 tells you immediately to reach for the one-cup measure and then the seven-eighths — whereas 1.875 needs converting back before you can act on it. This is why the calculator shows the mixed number whenever the fraction is improper.

When a decimal is genuinely better

Not always, though. If the quantities came from a measuring instrument they were approximate to begin with, and an exact fraction implies a precision that is not really there. Money is the clearest example: a price of 2/3 of a dollar is not payable, so you want 0.67. Percentages behave the same way — see the ratio to percentage calculator for that conversion, or the ratio to decimal and fraction calculator to move between the two forms directly.

If your terms are whole numbers

Then this layout is more machinery than you need. The cross multiplication calculator takes four single fields instead of eight and is quicker for the ordinary 3/4 = x/12 case. Come back here when a term is itself a fraction.

Questions

Fractional proportions — questions

Fraction arithmetic is where most of the difficulty lives. These cover the parts that matter.

How do you solve a proportion when the terms are fractions?
Exactly the same way as with whole numbers — cross multiply. The only difference is that multiplying and dividing fractions replaces multiplying and dividing integers. For (2/3)/(4/5) = x/(1/2), cross multiplying gives (2/3) × (1/2) = (4/5) × x, so x = (2/3 × 1/2) ÷ (4/5) = (1/3) × (5/4) = 5/12. Working in fractions throughout keeps the answer exact.
Why not just convert the fractions to decimals first?
Because many fractions have no exact decimal form. 1/3 becomes 0.333333…, and once you truncate it, every later step inherits that error. A result that should be exactly 5/12 comes out as 0.4166667, which is neither exact nor easy to recognise. This calculator keeps every value as a numerator and denominator internally and only produces a decimal at the very end, next to the exact fraction.
Can I enter mixed numbers like 1½?
Yes. Type mixed numbers as a whole number, a space, then the fraction — 1 1/2. You can also type them straight into the numerator box on their own, or split them across the numerator and denominator boxes. Improper fractions such as 3/2 work identically and are what the mixed number converts to internally.
What does "simplest form" mean for the answer?
A fraction is in simplest form when the numerator and denominator share no common factor other than 1. 10/24 is not simplified; dividing both by their greatest common divisor of 2 gives 5/12, which is. Every answer here is reduced automatically, and the calculator also shows the mixed-number form when the fraction is improper.
How do I divide one fraction by another?
Multiply by its reciprocal — flip the second fraction and multiply. (1/3) ÷ (4/5) becomes (1/3) × (5/4) = 5/12. This is the step that appears most often in fractional proportions, because isolating the unknown means dividing by whatever fraction it was multiplied by.
Do complex fractions work — a fraction inside a fraction?
Yes, that is precisely what this layout handles. Each of the four terms in the proportion gets its own numerator and denominator field, so a term like (2/3) sitting in the numerator position of a larger fraction is entered directly rather than having to be simplified by hand first. You can also type 2/3 into a single field and leave its denominator as 1.
What if my answer is a whole number?
It is displayed as a whole number rather than as something over 1. The calculator only shows a denominator when there is one to show — so an answer of 12/1 is reported as 12, and the decimal line is dropped because it would add nothing.
Can the terms be negative?
Yes. Put the minus sign in front of the numerator, or in front of a whole term. Cross multiplication is unaffected by sign, and the calculator normalises the answer so the minus sign ends up in the numerator rather than the denominator.