Content

• To step
• Method 1 of 2: Cross Multiply by One Variable
• Method 2 of 2: Cross Multiply with Multiple Variables
• Tips

Cross multiplication is one way of solving an equation, using a variable as part of two fractions that are made equal. The variable is an unknown number or quantity, and cross multiplication makes this equation with fractions a simple equation, allowing you to solve the variable in question. Cross multiplication is especially useful when trying to solve a ratio. You can read how to do that here.

To step

Method 1 of 2: Cross Multiply by One Variable

1. Multiply the numerator of the left fraction by the denominator of the right fraction. Let's say you are working on the equation 2 / x = 10/13. Now multiply 2 by 13.2 x 13 = 26.
2. Multiply the numerator of the right fraction by the denominator of the left fraction. Multiply x by 10. x * 10 = 10x. You can first cross multiply in this direction; it doesn't matter in the end as long as you multiply both numerators by the diagonal denominators of the other fraction.
3. Make the two products equal to each other. Make 26 equal to 10x. 26 = 10x. It doesn't matter which number you take first; because they are equivalent, you can move them from one side of the equation to the other without any consequences; as long as you treat each term as a whole.
   • So if you try to solve for 2 / x = 10/13 for x, you get 2 * 13 = x * 10, or 26 = 10x.
4. Solve for the variable. Now that you're working on 26 = 10x, you can start to find the common denominator by dividing both 26 and 10 by a number that both denominators are divisible by. Since they are both even numbers, it is possible to divide them by 2; 26/2 = 13 and 10/2 = 5. Now you are left with 13 = 5x as equation. Then, to isolate x, divide both sides of the equation by 5. So 13/5 = 5/5, or 13/5 = x. If you'd like the answer as a decimal fraction or a decimal point, you can divide both sides of the equation by 10 to get 26/10 = 10/10, or 2.6 = x.

Method 2 of 2: Cross Multiply with Multiple Variables

1. Multiply the numerator of the left fraction by the denominator of the right fraction. Let's say you are working on the following equation: (x + 3) / 2 = (x + 1) / 4. Multiply (x + 3) with 4 to 4 (x +3). This is worked out 4x + 12.
2. Multiply the numerator of the right fraction by the denominator of the left fraction. Repeat this procedure on the other side. (x +1) x 2 = 2 (x +1). We work out 2 (x +1) then 2x + 2.
3. Make the two products equal and combine like terms. Now you have got it 4x + 12 = 2x + 2. Combine the X terms and the constants on either side of the equation.
   • So, combine 4x and 2x through 2x subtract on both sides of the equation. Elaborated, this yields the following equation 2x + 12 = 2.
   • Combine now 12 and 2 through 12 subtract on both sides of the equation. Elaborated it looks like this: 2x + 12-12 = 2-12.
   • So the equation becomes: 2x = -10.
4. Solve. All you have to do now is divide both sides of the equation 2. 2x / 2 = -10/2 = x = -5. After cross multiplication you will see that x = -5. You can go back and check that everything is correct by entering -5 for x to make sure that both sides of the equation are equal. The result of this check is -1 = -1, and this is correct because both sides of the equation are equal. Would the control eg. 0 = -1 return the equation, so something went wrong.

Tips

• Note that if you enter another number (say 5) into the same equation, you will get the following result: 2/5 = 10/13. Even if you multiply the left side of the equation by 5/5 again, you get 10/25 = 10/13, which is clearly incorrect. The latter case clearly shows that you have made a mistake when multiplying crosswise.