Cross Multiplication Calculator
Enter any 3 of the 4 values in the proportion a/b = c/d and leave the one you want to find empty.
Updated
Cross Multiplication Calculator
Our Cross Multiplication Calculator solves proportions of the form a/b = c/d. Fill in any 3 of the 4 boxes, leave the one you want to find empty, and the missing value appears straight away. The answer is worked out exactly, shown as a decimal and as a fraction, with the cross products that prove it. Fill in all four boxes and it tells you whether the two fractions are equal.
Quick answer
Cross multiplication turns a/b = c/d into a × d = b × c, so one division finds the missing value. For example, in 2/5 = x/20, x = (2 × 20) ÷ 5 = 8. Enter any three values above and the calculator finds the fourth instantly.
- Rule: a/b = c/d means a × d = b × c
- Missing value: multiply the diagonal pair, divide by the number opposite x
- Check: two fractions are equal when their cross products are equal
- Limit: b and d cannot be 0
What Is Cross Multiplication?
A proportion says that two fractions or ratios are equal, such as 2/5 = 8/20. Cross multiplication multiplies each numerator by the denominator of the other fraction, along the two diagonals of an X:
- = - → a × d = b × c
b d
The two products are always equal when the proportion is true (2 × 20 = 5 × 8 = 40). That turns a fraction equation into a simple multiplication, which is why it is the standard way to solve for an unknown in a proportion.
Cross Multiplication Formula
Start from a × d = b × c and divide by the number that sits opposite the unknown:
| Missing | Formula |
|---|---|
| a (numerator 1) | a = (b × c) ÷ d |
| b (denominator 1) | b = (a × d) ÷ c |
| c (numerator 2) | c = (a × d) ÷ b |
| d (denominator 2) | d = (b × c) ÷ a |
A quick way to remember it: multiply the two numbers on the diagonal you know, then divide by the number left over.
Why Cross Multiplication Works
Cross multiplication is not a separate rule. It is what you get when you multiply both sides of a/b = c/d by b × d to clear the fractions:
a × d = c × b
This only works when b and d are not 0, since you cannot divide by 0 in the first place.
Worked Examples
Missing numerator. Solve 2/5 = x/20:
= 40 ÷ 5 = 8
Missing denominator. Solve 3/x = 9/12: x = (3 × 12) ÷ 9 = 36 ÷ 9 = 4, so 3/4 = 9/12.
Scaling a recipe. 2 cups of flour make 12 cookies. For 30 cookies, 2/12 = x/30, so x = (2 × 30) ÷ 12 = 5 cups.
Map scale. 1 cm on a map is 25 km. A 7.4 cm road is 1/25 = 7.4/x, so x = 25 × 7.4 = 185 km.
An answer that is not whole. x/4 = 7/3 gives x = 28/3 = 9 1/3 ≈ 9.3333333333. The calculator shows both the fraction and the decimal.
Checking and Comparing Fractions
Cross products also tell you whether two fractions are equal, without finding a common denominator:
- 3/4 and 9/12: 3 × 12 = 36 and 4 × 9 = 36, so they are equal.
- 2/3 and 5/8: 2 × 8 = 16 and 3 × 5 = 15, so they are not equal. With positive denominators, the bigger cross product sits with the bigger fraction: 2/3 > 5/8.
Type all four numbers into the calculator and it says whether the proportion is true.
Cross Multiplication With Variables
The same step works when the unknown is part of an expression. To solve (x + 1)/3 = x/2:
2x + 2 = 3x
x = 2
Check: 3/3 = 1 and 2/2 = 1. This calculator handles numbers in all four places; for expressions like this, cross multiply by hand and then solve the linear equation.
Common Mistakes
- Multiplying straight across. a × c and b × d is multiplying fractions, a different operation.
- Dividing by the wrong number. Divide by the number opposite x, not the one next to it.
- Mixing units. Keep the same units in both numerators and in both denominators.
- Cross multiplying a sum. It only works when there is exactly one fraction on each side, as in a/b = c/d.
- A zero denominator. b and d can never be 0.
Key Terms
- Proportion
- An equation saying two ratios are equal, a/b = c/d.
- Numerator
- The top number of a fraction.
- Denominator
- The bottom number of a fraction; never 0.
- Cross product
- a × d or b × c, a numerator times the other denominator.
- Ratio
- A comparison of two amounts, such as 2 : 5.
How to Use the Cross Multiplication Calculator
- Fill in three values. Type three of the four numbers in a/b = c/d: the numerators go in the top boxes and the denominators in the bottom boxes. Whole numbers, decimals, negative numbers and fractions such as 3/4 all work.
- Leave the unknown empty. Leave the box you want to find empty. It gets a dashed border and an x so you can see which value is being solved for.
- Read and copy the answer. The missing value appears as you type, with the working, the completed proportion, the cross products, the answer as a fraction and the value of both fractions. Press Copy to copy it, or Clear or Esc to start again.
Example: enter a = 2, b = 5 and d = 20, leaving c empty. The result is c = 8, with the working c = (2 × 20) ÷ 5 = 40 ÷ 5 = 8 and the cross products 2 × 20 = 40 and 5 × 8 = 40.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is cross multiplication?
Cross multiplication is a way to solve a proportion a/b = c/d. You multiply each numerator by the other fraction's denominator, which gives a × d = b × c, and then divide to find the missing value.
How do you solve for x using cross multiplication?
Multiply the two known numbers on one diagonal, then divide by the number opposite x. For 2/5 = x/20, x = (2 × 20) ÷ 5 = 40 ÷ 5 = 8.
What is the cross multiplication formula?
If a/b = c/d, then a × d = b × c. Rearranged: a = bc ÷ d, b = ad ÷ c, c = ad ÷ b and d = bc ÷ a.
Why does cross multiplication work?
Multiplying both sides of a/b = c/d by b × d clears the fractions and leaves a × d = c × b. It is ordinary algebra, valid whenever b and d are not 0.
How do I check if two fractions are equal?
Compare the cross products. 3/4 and 9/12 are equal because 3 × 12 = 36 and 4 × 9 = 36. 2/3 and 5/8 are not, because 2 × 8 = 16 but 3 × 5 = 15. Enter all four numbers in the calculator and it tells you which.
Is cross multiplication the same as multiplying fractions?
No. Multiplying fractions multiplies straight across: a/b × c/d = ac/bd. Cross multiplication multiplies along the diagonals of an equation, a × d and b × c, to solve a proportion.
Can I use decimals, negative numbers or fractions?
Yes. Each box accepts whole numbers, decimals such as 2.5, negative numbers and fractions such as 3/4. The answer is worked out exactly and shown as a decimal and as a fraction, for example 28/3 = 9 1/3 ≈ 9.3333333333.
What happens if a denominator is 0?
A fraction with denominator 0 is undefined, so b and d cannot be 0. If solving would need a denominator of 0, or the proportion has no single answer, the calculator explains why instead of giving a number.
How is cross multiplication used in real life?
It scales recipes (2 cups for 12 cookies means 5 cups for 30), converts map and model scales, works out unit prices, and solves percentage problems such as 15/x = 30/100, which gives x = 50.
Is the Cross Multiplication Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Cross Multiplication Calculator to solve any proportion a/b = c/d instantly.