Cross Multiplying

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:

Cross Multiplying with a Single Variable

Image titled Cross Multiply Step 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. Multiply 2 * 13.
2 * 13 = 26.
Image titled Cross Multiply Step 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.

Image titled Cross Multiply Step 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.

  • So, if you’re trying to solve 2/x = 10/13 for x, you’d have 2 * 13 = x * 10, or 26 = 10x.
Solve for the variable. Divide both sides by the number multiplied by the missing number, the one you are trying to figure out. Here 10 is being multiplied by the unknown number. Divide both sides by ten. You get 2.6 or 26/10 which reduces to 2 3/5.

 

(source)