acalculator

How do I solve a proportion?

Type three of the four numbers in a/b = c/d and leave the one you want empty. The proportion calculator cross-multiplies and shows each step.

Your numbers

Missing number
7.5

The missing number in the proportion is 7.5 (15/2).

Missing number as a fraction
15/2
Value of a/b
3/4
Cross-multiplication
Cross-multiply: 3 × 10 = 4 × c; 4 × c = 30; c = 30 ÷ 4 = 15/2; As a decimal: c = 7.5

Missing number: 7.5. The missing number in the proportion is 7.5 (15/2).

How is the missing number found?

How to calculate

Solves the proportion a/b = c/d for the missing number by cross-multiplication, as an exact fraction and a decimal, or checks whether four numbers form a proportion.

Example with the default inputs (a 3, b 4, d 10): The missing number in the proportion is 7.5 (15/2).

Method: a/b = c/d means a × d = b × c (cross-multiplication); divide by the number next to the unknown.

  • b and d are the bottoms of the fractions and cannot be 0.
  • Every number is read exactly as typed (0.1 is 1/10), and the arithmetic is exact before rounding for display.
  • With all four numbers typed, the calculator checks whether they form a proportion instead.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. a 3, b 4, d 10 gives Missing number 7.5, Missing number as a fraction 15/2, Value of a/b 3/4, Cross-multiplication Cross-multiply: 3 × 10 = 4 × c; 4 × c = 30; c = 30 ÷ 4 = 15/2; As a decimal: c = 7.5.Source: OpenStax, Prealgebra 2e, §6.5, cross products (https://openstax.org/books/prealgebra-2e/pages/6-5-solve-proportions-and-their-applications)
  2. a 5, c 20, d 36 gives Missing number 9, Missing number as a fraction 9, Value of a/b 5/9.
  3. b 3, c 2, d 7 gives Missing number 0.857143, Missing number as a fraction 6/7.
  4. a 0.1, b 0.3, c 5 gives Missing number 15, Missing number as a fraction 15, Value of a/b 1/3.
  5. a -2, b 5, d 8 gives Missing number -3.2, Missing number as a fraction -16/5.
  6. a 3, b 4, c 9, d 12 gives Do the four numbers form a proportion? yes, Value of a/b 3/4.
  7. a 2, b 3, c 5, d 7 gives Do the four numbers form a proportion? no.

How it works

A proportion says two fractions are equal: a/b = c/d. Multiplying both sides by b × d gives the cross products, which are equal too:

a × d = b × c

Solve for the missing number. Type three numbers and leave one box empty. The missing number is the other cross product divided by the number it is multiplied by:

  • a = b × c ÷ d
  • b = a × d ÷ c
  • c = a × d ÷ b
  • d = b × c ÷ a

Check a proportion. Type all four numbers. The four form a proportion when a × d = b × c exactly (Yes), and not otherwise (No).

The calculator also shows the value of a/b as a fraction in lowest terms.

Rules

  • b and d are the bottoms of the fractions, so they cannot be 0 ("b is the bottom of a fraction, so it cannot be 0.").
  • When the number that multiplies the unknown is 0, there is no single answer. If the other cross product is 0 too, any number fits (for example 0/5 = 0/d); otherwise none does (0/5 = 3/d).
  • A bottom b or d cannot be solved as 0: 3/b = 0/7 has no answer, because 3/b is never 0.
  • Fewer than three numbers give no answer ("Enter three of the four numbers and leave the one to find empty.").
  • An answer beyond what a 64-bit float can hold (larger than about 1.8 × 10³⁰⁸, or not 0 but smaller than about 2.2 × 10⁻³⁰⁸ in size) is not shown.

Exact arithmetic and display

  • Each number is read exactly as its decimal (0.1 is 1/10, 0.3 is 3/10), and the cross products and the division are exact fractions. The missing number is shown as a fraction in lowest terms (for example 15/2, or -16/5; a whole number shows with no bottom), and as a number rounded for display: at most 6 decimals, or 6 significant figures below 0.0001, rounded half up on the decimal value.

The working, step by step

The steps are written like this, with the letter of the missing number in its place:

  1. Cross-multiply: <a> × <d> = <b> × <c>
  2. <number next to the unknown> × <letter> = <other cross product>
  3. <letter> = <other cross product> ÷ <number next to the unknown> = <answer as a fraction>
  4. Only when the answer is not whole: As a decimal: <letter> = <answer as a decimal>

With all four numbers typed, the steps are the cross-multiplication, then <a × d> = <b × c> (or ≠), then "The cross products are equal, so a/b = c/d." or "The cross products differ, so a/b ≠ c/d.".

Products and answers in the steps are exact fractions in lowest terms. Typed numbers and the decimal in step 4 are decimals rounded half up to 10 significant figures, written like a JavaScript number (plain from 0.000001 up to below 10²¹, else e notation such as 1.5e-7). A negative number that is multiplied or divided is written in brackets, for example (-2) × 8.

Worked examples by hand

3/4 = c/10. Cross-multiply: 3 × 10 = 4 × c, so 4c = 30 and c = 30 ÷ 4 = 15/2 = 7.5. The first fraction is 3/4.

5/b = 20/36. Cross-multiply: 5 × 36 = b × 20, so 20b = 180 and b = 180 ÷ 20 = 9. The first fraction is 5/9.

a/3 = 2/7. Cross-multiply: a × 7 = 3 × 2, so 7a = 6 and a = 6/7 = 0.857143.

0.1/0.3 = 5/d. Read exactly, 0.1/0.3 = 1/3. Cross-multiply: 0.1 × d = 0.3 × 5 = 3/2, so d = 3/2 ÷ 0.1 = 15.

−2/5 = c/8. Cross-multiply: 5 × c = −2 × 8 = −16, so c = −16/5 = −3.2.

Check 3/4 and 9/12. 3 × 12 = 36 and 4 × 9 = 36: equal, so they form a proportion (Yes).

Check 2/3 and 5/7. 2 × 7 = 14 and 3 × 5 = 15: different, so they do not (No).

Other questions people ask

How do I solve a proportion?

Cross-multiply, then divide. In 3/4 = c/10, multiply across the equals sign: 4 × c = 3 × 10 = 30. Divide both sides by 4: c = 30 ÷ 4 = 7.5.

What is a proportion?

A proportion is an equation that says two ratios (fractions) are equal, such as 3/4 = 9/12. Both sides have the same value, 0.75, so the four numbers are in proportion.

Why does cross-multiplication work?

Multiplying both sides of a/b = c/d by b × d clears the fractions and leaves a × d = b × c. It is the same equation, so any number that makes one true makes the other true, as long as b and d are not 0.

How do I check if two ratios form a proportion?

Type all four numbers. The calculator compares the cross products exactly: 3/4 and 9/12 give 3 × 12 = 36 and 4 × 9 = 36, so they form a proportion; 2/3 and 5/7 give 14 and 15, so they do not.

Can the numbers be decimals, fractions, or negative?

Yes. Decimals are read exactly (0.1 is one tenth, not a rounded binary value), negatives work, and the answer is shown as an exact fraction in lowest terms and as a decimal. Only the bottoms b and d cannot be 0.

Why does it say there is no single answer?

When the number next to the unknown in its cross product is 0, the equation does not fix the unknown. In 0/5 = 0/d, any d works; in 0/5 = 3/d, no d works, because 0/5 is 0 and 3/d never is.

Is a proportion the same as a ratio?

A ratio compares two numbers (3 to 4). A proportion says two ratios are equal (3 : 4 = 9 : 12). For ratios written with a colon, the ratio calculator solves the same equation.