How to Cross Multiply Fractions

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Make cross multiplication fun and easy with these tips and tricks

Co-authored by David Jia and Ali Garbacz, B.A.

Last Updated: August 6, 2024 Fact Checked

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Cross Multiplying with a Single Variable

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Cross Multiplying with Variables in Both Ratios

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Cross-multiplication is a nifty method to use when you have to solve for an unknown variable in an equation where two fractions are set equal to one another. Cross-multiplying reduces these two fractions to one simple equation, allowing you to easily solve for the variable in question. It’s also a useful method to know when you’re adding and subtracting unlike fractions and comparing ratios and proportions. Keep reading and follow along as we take you through the steps of cross-multiplication.

Things You Should Know

Use the formula $ad=cb$ to cross-multiply the fractions in the equation $a/b=c/d$
Multiply the top number (numerator) of the fraction on the left side of the equal sign by the bottom number (denominator) of the fraction on the right side.
Multiply the denominator of the left-hand fraction by the numerator of the right-hand fraction.
Set the two products equal to one another and solve for the unknown variable.

Method 1

Method 1 of 2:

Cross Multiplying with a Single Variable

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Multiply the numerator of the left fraction by the denominator of the right fraction. The numerator is the top number in a fraction, and the denominator is at the bottom. Let's say you're working with the equation $2/x=10/13$ . You would then multiply the left numerator 2 by the right denominator 13. $2x13=26$ .
$Step 2 Multiply the numerator of the right fraction by the denominator of the left fraction.$

2
Multiply the numerator of the right fraction by the denominator of the left fraction. Now multiply 10 (the right numerator) by x (the left denominator). $xx10=10x$ $xx10=10x$ . It doesn’t really matter which pair of numerators and denominators you start multiplying first as long as you multiply both numerators by the denominators diagonal from them.^{[1]
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- Another way to think of cross-multiplication is by using this formula: if you have the equation $a/b=c/d$ , you would multiply them out to get the equation $ad=cb$ .
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Set 26 equal to 10x, written as $26=10x$ . Since the two products are equal to one another, it doesn’t matter which number goes on which side. So both $26=10x$ and $10x=26$ are acceptable ways to write your equation.
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Solve for the variable x. Start by finding the greatest common factor between 26 and 10x. This is the largest number that divides exactly into both numbers. In this case, since both 26 and 10x are even numbers, they can both be divided by 2; $26/2=13$ $26/2=13$ and $10/2=5$ $10/2=5$ . You're left with $13=5x$ $13=5x$ . Then, to get x by itself and find its value, divide both sides of the equation by 5. So, $13/5=5x/5$ $13/5=5x/5$ , or $13/5=x$ $13/5=x$ .^{[2]
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- If you'd like the convert the fraction to a decimal, start by dividing both sides of the equation by 10 to get $26/10=10x/10$ , or $2.6=x$ .

Method 2

Method 2 of 2:

Cross Multiplying with Variables in Both Ratios

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1
Multiply the numerator of the left ratio by the denominator of the right ratio. Let's say you're working with the following equation: $(x+3)/2=(x+1)/4$ $(x+3)/2=(x+1)/4$ . Multiply (x + 3) by 4 to get $4(x+3)$ $4(x+3)$ . Then, distribute the 4 to get $4x+12$ $4x+12$ .
- To distribute the 4 into the equation (x + 3), multiply each part of the equation by 4. So that would be $4xx+4x3=4x+12$ .
Repeat the process using the remaining numerator and denominator. $(x+1)x2=2(x+1)$ . Distribute the 2 to get $2x+2$ . $(2xx+2x1)$
3
Set the two products equal to each other and combine the like terms. Now, you'll have $4x+12=2x+2$ $4x+12=2x+2$ . Combine the x terms with each other on one side of the equation and the constants (the numbers without x) on the other side.^{[3]
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- 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 2x remains on the left-hand side of the equal sign. $2x+12=2$
- Then, combine 12 and 2 by subtracting 12 from both sides of the equation so that 2x is left by itself. Subtract 12 from 12 on the left to get 0, and subtract 12 from 2 on the right side to get -10 (since $2-12=-10$ ).
- You're left with $2x=-10$ .
4
Solve for x by dividing each side by 2. This would be $2x/2=-10/2$ $2x/2=-10/2$ , which yields $x=-5$ $x=-5$ . Go back and check your work by plugging in -5 for x in the original equation to make sure that both sides of the equation are equal. This would look like:^{[4]
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- $(-5+3)/2=(-5+1)/4$ , which simplifies to $-1=-1$ .
- Since this equation is true, it’s confirmed that x’s value is -5.

Cross Multiplication Calculator, Practice Problems, and Answers

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Sample Cross Multiplication Practice Problems

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How would I solve 30% of 65

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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.)

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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

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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.)

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What fractions do not work?

Donagan

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

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