How to Cross Multiply

Cross multiplication is a way of solving equations where the variable is in two equal fractions. The variable represents an unknown value, and the cross-multiplication reduces the rule of triangles to a simple equation, allowing you to solve problems to find the variable. The cross-multiplication method is especially useful if you want to calculate proportions. Here’s how to do it:

Table of Contents

For an equation with one variable

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Multiply the numerator of the fraction on the left by the denominator of the fraction on the right. For example, we have the equation 2/x = 10/13. Proceed to multiply 2 by 13. We have 2 * 13 = 26. ^{[1] X Research Source}

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Multiply the numerator of the fraction on the right by the denominator of the fraction on the left. Performing multiplication with variables, we multiply x by 10. x * 10 = 10x. You can cross multiply either way first, as long as both the numerator and denominator of the two fractions are multiplied diagonally. ^{[2] X Research Source}

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Put the two results in the equation. 26 would be equivalent to 10x. We have 26 = 10x. The order of the two sides does not matter; since they are equal, you can swap both sides of the equation at the same time without affecting anything. ^{[3] X Research Sources}

So, to solve the equation 2/x = 10/13 and find x, we have 2 * 13 = x * 10, which is equivalent to 26 = 10x.

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Find x. With 26 = 10x, you can divide both 26 and 10 by the common denominator of both numbers. Since both are even, both are divisible by 2; 26/2 = 13 and 10/2 = 5. The remaining equation will be 13 = 5x. So, you need to divide both sides of the equation by 5 to find x. We have 13/5 = 5/5, which is equivalent to 13/5 = x. If you want the answer in decimal form, you can divide both sides by 10 to get 26/10 = 10/10, deducing x = 2.6. ^{[4] X Research Sources}

With an equation with the same two variables

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Multiply the numerator of the fraction on the left by the denominator of the fraction on the right. ^{[5] X Research Source} Example: the problem asks to find x in the equation: (x + 3)/2 = (x + 1)/4 . To start, you take (x + 3) * 4 = 4(x +3) = 4x + 12.

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Multiply the numerator of the fraction on the right by the denominator of the fraction on the left. ^{[6] X Research Sources} Doing the same as before, we have (x +1) x 2 = 2(x +1) = 2x + 2.

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Set both sides equal and combine the same terms. Now we have 4x + 12 = 2x + 2. Let’s put the terms containing x on one side and the term unchanged on the other side of the equation.

Combine 4x and 2x by moving 2x to the left side and changing the sign of the term. When you move 2x to the left, the right side is left with only 2 . On the left side, we have 4x – 2x = 2x, so 2x is left.
Do the same with 12 and 2 by bringing 12 from the left side to the right side and changing the sign of the term. The left side will be 2-12 = -10.
The other equation is 2x = -10.

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Find x. Now you just need to divide both sides of the equation by 2 . 2x/2 = -10/2 => x = -5. After cross multiplication, we find x = -5. You can double check by substituting x = -5 and trying to see if both sides of the equation are equal. After substituting -5 back into the original equation, we get -1 = -1.

Advice

You can check your work by substituting the answers you found in the original equation. If after simplifying the remaining equation is valid, such as 1 = 1, then you have calculated correctly. If the equation after simplifying is not valid, for example 0 = 1 then you are doing it wrong. For example, if we put 2.6 into the first equation, we get 2/(2,6) = 10/13. Multiplying the left side by 5/5 we get 10/13 = 10/13, this equation is valid because after simplifying it will become 1 = 1. So 2.6 is the correct result.
Note that when you put another number (e.g. 5) into the same equation, you get 2/5 = 10/13. Even if you multiply the left side by 5/5 again, the result will be 10/25 = 10/13 and obviously not. If this is the case, it means that you are wrong while performing the cross multiplication.

Steps

For an equation with one variable

With an equation with the same two variables

Advice

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