How to Cross Multiply Fractions

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Plus, how to compare fractions & why cross multiplication works

Co-authored by David Jia and Luke Smith, MFA

Last Updated: January 20, 2025 Fact Checked

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With a Single Variable

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With Variables in Both Fractions

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Comparing Fractions

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Why It Works

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Practice Problems (PDF)

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Video

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Q&A

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. We talked to math tutor David Jia to help you learn how to cross multiply with a single variable, a variable on both sides of the equation, and also to compare two fractions, plus how and why cross multiplication even works.

Cross Multiplying at a Glance

To cross multiply 2 fractions, multiply the numerator of the fraction on the left side of the equal sign by the denominator of the fraction on the right side. Then multiply the denominator of the left fraction by the numerator of the right fraction. Set the 2 products equal to one another and solve for the unknown variable.

Section 1 of 4:

Cross Multiplying with a Single Variable

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1
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. $2*13=26$ .
- Jia reminds us that you can only cross-multiply 2 fractions if there’s an equals sign between them. If they’re on the same side of the equals sign, you can’t cross-multiply them.
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). $x*10=10x$ . 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. ^{[2]
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4
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; $26/2=13$ and $10/2=5$ . You're left with $13=5x$ . Then, to get x by itself and find its value, divide both sides of the equation by 5. So, $13/5=5x/5$ , or $13/5=x$ . ^{[3]
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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 $2.6=x$ .

Section 2 of 4:

Cross Multiplying with Variables in Both Fractions

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1
Multiply the numerator of the left fraction by the denominator of the right fraction. Let's say you're working with the following equation: $(x+3)/2=(x+1)/4$ . Multiply (x + 3) by 4 to get $4(x+3)$ . Then, distribute the 4 to get $4x+12$ . ^{[4]
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- To distribute the 4 into the equation (x + 3), multiply each part of the equation by 4. So that would be $(4*x)+(4*3)=4x+12$ .
Repeat the process using the remaining numerator and denominator. $(x+1)*2=2(x+1)$ . Distribute the 2 to get $2x+2$ ^{[5]
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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 with each other on one side of the equation and the constants (the numbers without x) on the other side. ^{[6]
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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 $2x/2=-10/2$ , which yields $x=-5$ . 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: ^{[7]
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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.

Section 3 of 4:

Cross Multiplying to Compare Fractions

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1
Multiply the numerator of the left fraction by the denominator of the right fraction. Cross multiplication can be used as a shortcut to see if two fractions are equal, or “proportional.” Let’s say we have the fractions $24/100$ and $12/50$ . To start, multiply the numerator of the first fraction by the denominator of the second fraction. Remember that the numerator is the top number, and the denominator is the bottom number.
- This gives us $24*50=1,200$
- Alternatively, multiply the denominators together, then multiply the numerator of each fraction by the denominator of the other fraction, and see if the products are the same.
2
Multiply the numerator of the right fraction by the denominator of the left fraction. Now, we cross multiply the other parts of the fractions. Multiply the numerator of the second fraction by the denominator of the first fraction. ^{[8]
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- This gives us $12*100=1,200$
Now, look at the two products we’re left with. 1,200 is equal to 1,200, so we can say that $24/100$ and $12/50$ are proportional and have equal values. ^{[9]
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Research source} If they weren’t the same number, the fractions wouldn’t be proportional.
4
Identify the bigger fraction if the products aren’t equal. When comparing the products, remember that the product belongs to the fraction whose numerator was multiplied. If that fraction’s product is larger than the other when you compare them with cross multiplication, then we can say the value of that fraction is more than the other fraction. ^{[10]
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- For example, if we were comparing the fractions $3/6$ and $6/10$ , then we’d cross multiply like this: $3*10=30$ and $6*6=36$ .
- In this case, since 36 is larger than 30, and since the 36 product belongs to the fraction with 6 in the numerator, we can say that $6/10$ is greater than $3/6$ .

Section 4 of 4:

Why does cross multiplying work?

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1
Cross multiplying works because it’s a quick way to find common denominators. When we compare two fractions with different denominators, we usually need to find a common denominator so that we’re looking at fractions that share the same number of total parts (for example, $2/5$ has fewer total parts than $2/10$ since 5 is less than 10). ^{[11]
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- By cross multiplying, we’re basically doing the same process as finding a common denominator , except we’re only looking at the numerators of the resulting fractions.
2
Example: Say we wanted to find a common denominator between $2/5$ and $2/10$ . First, we multiply the denominators by each other ( $5*10$ ) to get a common denominator of 50.
- Then, to make sure that each fraction still has the same value, we multiply the numerator of each fraction by whatever number we multiply the denominator by. So, multiply $2*10$ and $2*5$ to get the fractions $20/50$ and $10/50$ , which are much easier to compare.
- Notice how if we were simply to cross multiply the original fractions ( $2/5$ and $2/10$ ), we’d be doing the same operation as the step above, and looking only at the final numerators, 20 and 10. That’s cross multiplication!

Cross Multiplication Calculator, Practice Problems, and Answers

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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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How to Cross Multiply Fractions

Cross Multiplying at a Glance

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Cross Multiplying with a Single Variable

Cross Multiplying with Variables in Both Fractions

Cross Multiplying to Compare Fractions

Why does cross multiplying work?

Cross Multiplication Calculator, Practice Problems, and Answers

Community Q&A

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