Cross multiplication is a way of solving an equation involving a variable as part of two fractions equal to each other. The variable is where a number or quantity of unknown value is found, and cross-multiplication reduces the ratio to a simple equation, allowing you to unravel the variable in question. Cross multiplication is especially useful when trying to solve a reason. Learn here how to do it.
Method 1 of 2: Multiplying Cross with a Single Variable
Step 1. Multiply the numerator of the fraction on the left by the denominator of the fraction on the right
Let's say you're working with the equation 2/x = 10/13. Now multiply 2 by 13: 2 × 13 = 26.
Step 2. Multiply the numerator of the fraction on the right by the denominator of the fraction on the left
Now multiply x by 10: x × 10 = 10x. You can cross-multiply initially in this direction - this is not important as you will multiply both numerators by diagonally opposite denominators.
Step 3. Match the two resulting products
Equate 26 to 10x: 26 = 10x. It doesn't matter which number comes first - since they're the same, you can switch them from one side of the equation to the other without worry, as long as you treat them as a whole unit.
So if you're trying to solve 2/x = 10/13 for x, we're going to have 2 × 13 = x × 10, or 26 = 10x
Step 4. Solve the variable
Now that you're working with 26 = 10x, we can start by finding a common denominator and dividing both 26 and 10 by a common divisor between both numbers. Since they are both pairs, you can divide them by 2: 26/2 = 13 and 10/2 = 5. You will be left with 13 = 5x. Now, to isolate x, divide both sides of the equation by 5. So 13/5 = 5/5, or 13/5 = x. If you would like to give the answer in decimal format, start by dividing both sides by 10 to get 26/10 = 10/10, or 2, 6 = x.
Method 2 of 2: Multiplying Cross with Multiple Variables
Step 1. Multiply the numerator on the left by the denominator on the right
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). Distribute the 4 to get 4x + 12.
Step 2. Multiply the numerator on the right by the denominator on the left
Repeat the process on the other side: (x + 1) × 2 = 2(x + 1). Distribute the 2 and you get 2x + 2.
Step 3. Match both products and combine similar terms
Now you will have 4x + 12 = 2x + 2. Combine the x terms and the constants on opposite sides of the equation.
- So 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 will be left.
- Now combine 12 and 2 by subtracting 12 from both sides of the equation. Subtract 12 from 12 on the left side and you get 0 - subtract 12 from 2 on the right side, and you get 2-12 = -10.
- You will be left with 2x = -10.
Step 4. Fix the problem
All you have to do is divide both sides of the equation by 2. 2x/2 = -10/2 = x = -5. After cross-multiplication, you found that x = -5. You can go back and check your work by setting x equal to -5 to make sure both sides of the equation are equal. If you enter -5 again into the original equation, you'll notice that -1 = -1.
- Note that if you substitute a different number (eg 5) in an equal proportion, we would find that 2/5 = 10/13. Even if you were to multiply the left equation again by 5/5, you would have 10/25 = 10/13, which is clearly incorrect. The latter case indicates that you made a mistake in the cross-multiplication technique.
- You can check the work itself by substituting the result obtained directly in the original proportion. If she simplifies the result to a correct statement, such as 1 = 1, the calculations are correct. If the proportion simplifies the result to an incorrect statement, such as 0 = 1, some error has been made. For example, substituting 2, 6 proportionally will give you 2/(2, 6) = 10/13. Multiply the ratio on the left by 5/5 and you get 10/13 = 10/13, a correct statement that synthesizes to the result 1 = 1. So 2, 6 is the correct answer.