How to Cross Multiply

Last Updated: May 5, 2022 References

Cross multiplying is a way to solve an equation that involves a variable as part of two fractions set equal to each other. The variable is a placeholder for an unknown number or quantity, and cross-multiplying reduces the proportion to one simple equation, allowing you to solve for the variable in question. Cross multiplying is especially useful when you're trying to solve a ratio. Here's how to do it:

Method 1

Method 1 of 2:

Cross Multiplying with a Single Variable

1
Multiply the numerator of the left-hand fraction by the denominator of the right-hand fraction. Let's say you're working with the equation 2/x = 10/13. Now, multiply 2 * 13. 2 * 13 = 26.^{[1]
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2
Multiply the numerator of the right-hand fraction by the denominator of the left-hand fraction. Now multiply x by 10. x * 10 = 10x. You can cross multiply in this direction first; it really doesn't matter as long as you multiply both numerators by the denominators diagonal from them.^{[2]
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3
Set the two products equal to each other. Just set 26 equal to 10x. 26 = 10x. It doesn't matter which number you list first; since they're equal, you can swap them from one side of the equation to the other with impunity, as long as you treat each term as a whole.^{[3]
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- So, if you're trying to solve 2/x = 10/13 for x, you'd have 2 * 13 = x * 10, or 26 = 10x.
4
Solve for the variable. Now that you're working with 26 = 10x, you can start by finding a common denominator and dividing both 26 and 10 by a number that divides evenly into both numbers. Since they are both even, you can divide them by 2; 26/2 = 13 and 10/2 = 5. You're left with 13 = 5x. Now, to isolate x, divide both sides of the equation by 5. So, 13/5 = 5x/5, or 13/5 = x. If you'd like the answer in decimal form, you can start by dividing both sides of the equation by 10 to get 26/10 = 10x/10, or 2.6 = x.^{[4]
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Method 2

Method 2 of 2:

Cross Multiplying with Two of the Same Variable

1
Multiply the numerator of the left-hand fraction by the denominator of the right-hand fraction.^{[5]
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Research source} Let's say you're working with the following equation: (x + 3)/2 = (x + 1)/4. Multiply (x + 3) by 4 to get 4(x +3). Distribute the 4 to get 4x + 12.
2
Multiply the numerator of the right-hand fraction by the denominator of the left-hand fraction.^{[6]
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Research source} Repeat the process on the other side. (x +1) x 2 = 2(x +1). Distribute the 2 and you get 2x + 2.
3
Set the two products equal to each other and combine the like terms. Now, you'll have 4x + 12 = 2x + 2. Combine the x terms and the constant terms on opposite sides of the equation.
- So, combine 4x and 2x by subtracting 2x from both sides. Subtracting 2x from 2x on the right side will leave you with 0. On the left side, 4x - 2x = 2x, so you have 2x remaining.
- Now, combine 12 and 2 by subtracting 12 from both sides of the equation. Subtract 12 from 12 on the left and you'll have 0, and subtract 12 from 2 on the right side to get 2-12 = -10.
- You're left with 2x = -10.
4
Solve. All you have to do is divide both sides of the equation by 2. 2x/2 = -10/2 = x = -5. After cross multiplying, you have found that x = -5. You can go back and check your work by plugging in -5 for x to make sure that both sides of the equation are equal. They are. If you plug -5 back into the original equation, you'll get -1 = -1.

Community Q&A

Question

How would I solve 30% of 65

Community Answer

Multiply 65 by 0.3. (You could think of it as multiplying 65 by 10% by just moving the decimal point to the left one digit to get 6.5 and then multiplying by 3 to get 30% of 65.)
Question

One day 176 people visited a small art museum. The ratio of members to nonmembers that day was 5 to 11. How many people who visited the museum that day were nonmembers?

Donagan

Top Answerer

Add together the two numbers in the ratio: 5 + 11 = 16. Divide that sum into the total number of visitors: 176 ÷ 16 = 11. Multiply that quotient by the ratio number representing the non-members (11): (11)(11) = 121 non-members. (If you wanted to find the number of members, you'd multiply 11 by the 5 in the ratio: (11)(5) = 55 members.)
Question

What fractions do not work?

Donagan

Top Answerer

The above technique works with all fractions. Some fractions are just more complicated than others, as in Method 2 above.