What is cross-multiplication?

Cross-multiplication is the shortcut for clearing the denominators from a/b = c/d in one move. Multiplying both sides by b and by d cancels both denominators and leaves a × d = b × c, which is why the rule is usually taught as 'multiply diagonally'. It works for any proportion whose denominators are not zero, and it is the reason a proportion with three known terms always has exactly one solution.

How do you solve 3/4 = 9/x?

Cross-multiply to get 3 × x = 4 × 9, which is 3x = 36. Divide both sides by 3 and x = 12, so the completed proportion is 3/4 = 9/12. Check it by comparing the cross products: 3 × 12 = 36 and 4 × 9 = 36, so the two ratios really are equal. You can also spot it without algebra — 9 is 3 tripled, so the bottom must be 4 tripled.

How do you know if two ratios are in proportion?

Multiply diagonally and compare. Two ratios a/b and c/d are in proportion exactly when a × d equals b × c. If the two cross products match, the ratios are equivalent and the proportion is true; if they differ, it is false, and the size of the gap tells you how far off it is. This test is safer than comparing decimals, because a ratio like one third has no exact decimal form to compare.

What is the difference between a ratio and a proportion?

A ratio is a single comparison of two quantities, written a : b or a/b — it states a relationship but claims nothing. A proportion is an equation saying two ratios are equal, a/b = c/d. That difference matters in practice: a ratio can be simplified but not solved, while a proportion can be solved, because setting three known terms equal to one unknown pins that unknown to exactly one value.

What is the difference between direct and inverse proportion?

In direct proportion the two quantities move together — double one and the other doubles, so the ratio y/x stays fixed at a constant k and the rule is y = kx. In inverse proportion they move opposite ways — double one and the other halves, so it is the product x × y that stays fixed and the rule is y = k/x. Direct problems are solved by cross-multiplying a/b = c/d; inverse problems are solved by setting the two products equal, a × b = c × d.

Can a proportion contain decimals or fractions?

Yes, and it changes nothing about the method. Cross-multiplication works on any numbers, so 0.75/2.5 = x/10 is solved exactly the same way as a proportion made only of whole numbers. If you prefer whole numbers, multiply every term in one ratio by a power of ten to clear the decimals, or multiply through by the denominators to clear the fractions — both leave the proportion equivalent because you are scaling a whole ratio, not one term of it.

What is the constant of proportionality?

It is the fixed number that links the two quantities in any proportional relationship. For a direct proportion it is the value of the ratio itself, k = y/x, so every pair of matching values multiplies out to the same k. For an inverse proportion it is the product, k = xy. Finding k once from a known pair is often quicker than cross-multiplying repeatedly, because once you have it every other value follows from a single multiplication or division.

Arithmetic

Proportion Calculator

A proportion says two ratios are equal — a/b = c/d — and it is solved by cross-multiplication: a × d = b × c. Leave any one of the four terms blank and this calculator fills it in, shows every line of the working, tells you whether the answer is exact, and handles inverse proportion too.

Your proportion

What do you want

Relationship

a / b = c / d — two equal ratios, solved by cross-multiplying

Solve for

d is the second denominator. Type the other three terms — whole numbers, decimals like 0.75, or fractions like 3/4 all work.

The missing term · direct proportion

d = 12

3/4 = 9/12

Exact — a whole number

Both ratios, drawn to scale

3 : 4

9 : 12

Cross product a × d

Cross product b × c

Scale factor c ÷ a

Within ratio b ÷ a

4/3

a ÷ b as a decimal

0.75

a ÷ b as a percent

75%

Try one

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The basics

Ratio, fraction, proportion — what is the difference?

The three words get used interchangeably and they are not the same thing. Only one of them is an equation, and only an equation can be solved — which is exactly why a proportion has a missing term to find and a ratio does not.

a : b

A ratio compares

A ratio sets one quantity against another. It says how the two sizes relate without fixing either of them, so it can be simplified or scaled, but there is nothing in it to solve.

3 : 4 — three parts to four parts

a / b

A fraction is one number

Written as a fraction, the same comparison becomes a single value you can put on a number line. It is the ratio evaluated — the answer to the division rather than the comparison itself.

3/4 = 0.75 = 75%

a / b = c / d

A proportion is an equation

A proportion asserts that two ratios are equal. Because it is an equation, three known terms force the fourth, and that is the whole job of this calculator.

3/4 = 9/x → x = 12

Method

How to solve a proportion by hand

Five steps, and the fifth is the one most people skip. The example runs the classic case — the second ratio has a missing denominator.

1
Set the two ratios equal, unknown as x
Put the same kind of quantity in the same place in both fractions — if a is a length, c must be a length too. Getting this right is most of the battle.
3/4 = 9/x
2
Cross-multiply
Multiply diagonally: a by d, and b by c. Both denominators vanish and you are left with an ordinary equation.
a × d = b × c → 3 × x = 4 × 9
3
Work out the side you know
One side has no unknown in it. Multiply it out to get a single number, so the equation reads as a coefficient times x equals that number.
3x = 36
4
Divide by the number beside x
Whatever multiplies x divides the other side. That single division is the answer.
x = 36 ÷ 3 = 12
5
Check the two cross products
Put the answer back in and multiply diagonally again. Equal products mean the proportion holds; unequal products mean an arithmetic slip somewhere above.
36 = 36 ✓

The property

Means and extremes

Cross-multiplication is not a trick — it is a named property of proportions, and knowing the names makes textbook wording readable. Written on one line as a : b = c : d, the two outer terms are the extremes and the two inner terms are the means.

a : b = c : d

extremes in accent · means in the middle

product of the extremes = product of the means

a × d = b × c

Why it works

Multiplying both sides of the equation by both denominators cancels them. Nothing is being moved diagonally in any magical sense — the diagonal is just what is left when the fractions clear.

3/4 = 9/12 → 3 × 12 = 36 and 4 × 9 = 36

equal products ⟺ true proportion

It runs both ways

The property is an if-and-only-if. Equal cross products prove the proportion; unequal cross products disprove it. That single test is what the check mode in the calculator runs.

3/5 vs 5/8 → 24 ≠ 25, so it is false

Rearrangements

The four ways to solve a / b = c / d

Whichever term is missing, the recipe is the same: multiply the two terms that are not beside the unknown, then divide by the one that is. These four rows are that rule written out, each solved on the same worked proportion.

Unknown	Proportion	Rearranged	Worked on 3 : 4 = 9 : 12
a	x/b = c/d	a = (b × c) ÷ d	a = 3
b	a/x = c/d	b = (a × d) ÷ c	b = 4
c	a/b = x/d	c = (a × d) ÷ b	c = 9
d	a/b = c/x	d = (b × c) ÷ a	d = 12

The same proportion, rewritten

A true proportion stays true under several standard rearrangements. Exam questions lean on these constantly, usually without naming them.

Name	Form	What it does
Original	a / b = c / d	The proportion as written: the first ratio equals the second.
Invertendo	b / a = d / c	Flip both fractions. Turning both ratios upside down keeps them equal.
Alternando	a / c = b / d	Swap the two means. Useful when the pairing you want compares like with like across the two ratios.
Cross form	a × d = b × c	The cross-multiplied form — no fractions at all, which is why it is the one you actually solve with.
Componendo	(a + b) / b = (c + d) / d	Add one to both sides. This is how a part-to-part ratio becomes a part-to-whole ratio.
Dividendo	(a − b) / b = (c − d) / d	Subtract one from both sides — the difference-based version, used for percentage-change problems.

Worked examples

Six proportions from real life

Recipes, maps, models, prices, geometry, and a genuinely inverse case. Switch tabs to watch the setup and the working change while the method stays identical.

Scaling a recipe up

A cookie recipe uses 3 cups of flour for 24 cookies. You want 60 cookies. How much flour do you need?

Set it up 3 / 24 = x / 60

a = 3 · cups of flour in the original recipe
b = 24 · cookies the original recipe makes
c = the unknown · cups of flour you need
d = 60 · cookies you want

Cross-multiply — the product of the means equals the product of the extremes: a × d = b × c

Substitute the numbers you know: 24 × x = 3 × 60

Work out the known product: 24x = 180

Divide both sides by 24: x = 180 ÷ 24

c = 7.5 cups of flour(15/2)

Keep the same quantity on top of both fractions — flour above, cookies below — and the units line up on their own.

Two kinds

Direct and inverse proportion

Not every pair of quantities that move together is directly proportional. Deciding which of the two relationships you are looking at is the single biggest source of wrong answers in proportion problems, because the arithmetic for each is completely different.

y = kx

Direct — they move together

Multiply one quantity and the other multiplies by the same factor, so their ratio never changes. The constant k is that ratio, and the graph is a straight line through the origin.

3 cups per 24 cookies → 7.5 cups for 60

xy = k

Inverse — one up, one down

Multiply one quantity and the other divides by the same factor, so their product never changes. The constant k is that product, and the graph is a curve that never touches either axis.

60 km/h for 4 h → 3 h at 80 km/h (k = 240)

x	Direct · y = 12x	Ratio y ÷ x	Inverse · y = 12 ÷ x	Product x × y
1	12	12	12	12
2	24	12	6	12
3	36	12	4	12
4	48	12	3	12
6	72	12	2	12
12	144	12	1	12

One column keeps its ratio constant, the other keeps its product constant — that is the entire difference, and it is the test to apply before writing anything down.

the sanity check

Ask which way the answer should move

More workers means fewer days, so an answer that came out larger when it should have shrunk is a sign the relationship was treated as direct when it is inverse.

6 workers take 10 days → 15 workers finish in 4 days

mixed cases exist

Some pairs are neither

Plenty of real relationships are proportional to a square, or only approximately proportional over a range. If doubling the input does not double or halve the output, a simple proportion is the wrong model.

Checking

Is this proportion true?

The second job of a proportion calculator: not solving for a missing term, but testing a pair of ratios you already have. Multiply diagonally and compare the two products — equal means true, unequal means false, and the gap tells you by how much.

Proportion	a × d	b × c	Verdict	Why
3/4 = 9/12	36	36	True	Both reduce to 3 : 4
2/5 = 6/15	30	30	True	The second is the first tripled
4/6 = 6/9	36	36	True	Both reduce to 2 : 3
8/12 = 10/15	120	120	True	Both reduce to 2 : 3
3/5 = 5/8	24	25	False	Close, but 24 is not 25
5/8 = 8/13	65	64	False	Consecutive Fibonacci numbers — near-miss by one
1/3 = 2/5	5	6	False	Adding one to each term is not scaling
16/9 = 4/3	48	36	False	16:9 and 4:3 are different screen shapes

Notice the near-misses. Consecutive terms of the Fibonacci sequence come within one of forming a proportion but never do, and adding the same amount to both terms of a ratio never preserves it — only multiplying does.

Where it shows up

Scaling things up and down

Almost every practical use of a proportion is a scaling problem: something works at one size and you need it at another, with the relationships intact.

ingredient : servings

Recipes and batches

Scaling a recipe means holding every ingredient in the same ratio to the yield. Cooking times are the classic exception — they do not scale proportionally, which is why a double batch is not twice the time in the oven.

3 cups per 24 cookies → 7.5 cups for 60 cookies

map : ground

Maps and scale drawings

A map scale is a ratio printed on the sheet. Because both sides are lengths in the same unit, the scale works in either direction and needs no unit conversion until the very last step.

1 : 25,000 → 8.4 cm on paper is 210,000 cm, or 2.1 km

model : real

Models and blueprints

Scale models, architectural plans, and technical drawings all fix one ratio and apply it to every dimension. Areas scale by the square of that factor and volumes by the cube, which is why a half-scale model is nowhere near half the material.

1 : 18 → a 4,410 mm car becomes a 245 mm model

part : part or part : whole

Mixing and concentration

Paint, fuel, feed, and concrete are specified as ratios so the mix stays right at any batch size. The trap is reading a part-to-part ratio as a part-to-whole one, which quietly changes every quantity.

Concrete at 1 : 2 : 3 → 6 parts total, so cement is one sixth

Quick chart

Equivalent ratios at a glance

Set a base ratio and read off every pair in proportion with it — the answer to “what is 3 to 4 the same as” and its thousand variations. The left table multiplies both terms together; the right one starts from a first number and gives the second.

First:Second3 : 4 = 0.75 = 75%

Equivalent pairs — multiply both terms
×	Ratio	Written as
1	3 : 4	3/4
2	6 : 8	6/8
3	9 : 12	9/12
4	12 : 16	12/16
5	15 : 20	15/20
6	18 : 24	18/24
7	21 : 28	21/28
8	24 : 32	24/32
9	27 : 36	27/36
10	30 : 40	30/40
11	33 : 44	33/44
12	36 : 48	36/48
13	39 : 52	39/52
14	42 : 56	42/56
15	45 : 60	45/60

Given the first number, the second follows
First	Second	As a decimal
1	4/3	1.3333
2	8/3	2.6667
3	4	4
4	16/3	5.3333
5	20/3	6.6667
6	8	8
7	28/3	9.3333
8	32/3	10.6667
9	12	12
10	40/3	13.3333
12	16	16
15	20	20
20	80/3	26.6667
25	100/3	33.3333
50	200/3	66.6667
100	400/3	133.3333

Every row is in proportion with every other row: the ratio 3 : 4 never changes, only the size of the numbers.

Worth knowing

Rules worth remembering

Six facts that turn proportion questions from fiddly into routine.

three knowns fix the fourth

A proportion has exactly one solution

Because it is a linear equation once the denominators clear, there is never more than one answer and never fewer, provided the term beside the unknown is not zero.

5/7 = 12/x → x = 84/5 = 16.8

units must match down each column

Set it up before you solve it

The same kind of quantity has to sit in the same position in both ratios. Almost every wrong answer in a word problem comes from the setup, not the arithmetic.

scale, do not add

Equivalence is multiplicative

Two ratios are equivalent when one is the other multiplied through by a single factor. Adding the same number to both terms always changes the relationship.

3 : 4 scaled by three is 9 : 12, but 3 + 3 : 4 + 3 is 6 : 7 — a different ratio

the exact answer is a fraction

Decimals can only approximate

Whenever the division does not terminate, the fraction is the only exact answer. This calculator keeps the exact value and shows the rounded decimal beside it rather than the other way round.

3/7 = x/5 gives exactly 15/7, which no decimal writes out in full

k = y ÷ x

Find the constant once, reuse it

If you need several answers from the same relationship, work out the constant of proportionality from one known pair and then multiply. It is faster and less error-prone than cross-multiplying each time.

check by cross-multiplying back

Verification is free

Substituting the answer back and comparing the two products takes seconds and catches every arithmetic slip. It is the step worth never skipping.

Pitfalls

Common mistakes

Four errors account for most lost marks, and none of them are arithmetic.

minutes against hours

Mismatched units

A proportion compares like with like. Mixing units in the same column silently scales one side, and the answer comes out wrong by exactly the conversion factor you skipped.

90 minutes must become 1.5 hours before it meets a rate in hours

the flipped setup

Building the proportion backwards

Putting the numerator of one ratio against the denominator of the other inverts the answer. Label each position with what it measures before you write a single number.

Cost over quantity on one side must be cost over quantity on the other

additive thinking

Adding instead of multiplying

The most common conceptual slip in the whole topic. Ratios are preserved by multiplication, never by adding a constant to both terms.

Scaling 2 : 3 up to 4 needs 4 : 6, not 4 : 5

wrong relationship

Cross-multiplying an inverse pair

If more of one quantity should mean less of the other, the products are equal, not the ratios. Cross-multiplying that case pushes the answer the wrong way entirely.

60 km/h for 4 h at 80 km/h gives 3 h, not the larger figure cross-multiplying returns

Glossary

The vocabulary

Textbook wording for proportions is older than most of the rest of school arithmetic, so the terms are worth having straight — particularly means, extremes, and the constant of proportionality.

Ratio: A comparison of two quantities, written a : b or a/b. On its own it is just a comparison — no equation, nothing to solve.
Proportion: A statement that two ratios are equal. It is an equation, which is why it can be solved for a missing term.
Rate: A ratio between quantities with different units — miles per hour, dollars per kilogram. Most real proportion problems are rate problems.
Unit rate: A rate whose second term is one: cost per single item, distance per single hour. Reducing to a unit rate is an alternative to cross-multiplying.
Means: In a : b = c : d, the two inner terms b and c. Their product is one side of the cross-multiplication.
Extremes: The two outer terms a and d. The means-and-extremes property says their product equals the product of the means.
Cross product: Either of the two products a × d and b × c. Equal cross products is the exact test for whether a proportion is true.
Constant of proportionality (k): The fixed number linking the two quantities: y = kx for direct proportion, y = k/x for inverse. Every proportion problem is really a hunt for k.
Direct proportion: As one quantity multiplies, so does the other, by the same factor. Its graph is a straight line through the origin.
Inverse proportion: As one quantity multiplies, the other divides by the same factor, so their product never changes. Its graph is a hyperbola.
Scale factor: The multiplier taking one ratio to its equivalent — the number you multiply both terms by to move from a : b to c : d.
Equivalent ratios: Two ratios that reduce to the same lowest terms. Every pair of equivalent ratios forms a true proportion.

FAQ

Common questions

Write the proportion as a/b = c/d with the unknown as x, then cross-multiply: the product of the extremes a × d equals the product of the means b × c. That turns the two fractions into a single multiplication statement with x in it. Multiply out the side with no unknown, then divide both sides by the number sitting next to x. Finish by checking that the two cross products are equal — if they are, the answer is right.

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