Cross Multiply

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:

Steps

Cross Multiplying with a Single Variable

  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. Now, multiply 2 * 13. 2 * 13 = 26.
  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.
  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.
  4. Solve for the variable. Now that you're working with 26 = 10x, you can start by finding a common denominator and dividing both 26 and 10 by a number that divides evenly into both numbers. Since they are both even, you can divide them by 2; 26/2 = 13 and 10/2 = 5. You're 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'd like the answer in decimal form, you can start by dividing both sides of the equation by 10 to get 26/10 = 10/10, or 2.6 = x.

Cross Multiplying with Multiple Variables

  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 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.
  2. Multiply the numerator of the right-hand fraction by the denominator of the left-hand fraction. Repeat the process on the other side. (x +1) x 2 = 2(x +1). Distribute the 2 and you get 2x + 2.
  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 and the constant terms 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 you have 2x remaining.
    • Now, combine 12 and 2 by subtracting 12 from both sides of the equation. Subtract 12 from 12 on the left and you'll have 0, and subtract 12 from 2 on the right side to get 2-12 = -10.
    • You're left with 2x = -10.
  4. Solve. All you have to do is divide both sides of the equation by 2. 2x/2 = -10/2 = x = -5. After cross multiplying, you have found that x = -5. You can go back and check your work by plugging in -5 for x to make sure that both sides of the equation are equal. They are. If you plug -5 back in to the original equation, you'll get -1 = -1.

Tips

  • Note that if you substituted a different number (say, 5) into the same proportion, you'd have 2/5 = 10/13. Even if you multiply the left-side equation by 5/5 again, you get 10/25 = 10/13, which is clearly incorrect. The latter case signals that you made an error in your cross-multiplication technique.
  • You can check your work by substituting the result you got directly into the original proportion. If the proportion simplifies down to a valid statement, such as 1 = 1, your work was correct. If the proportion simplifies down to an invalid statement, such as 0 = 1, you made an error. For example, substituting 2.6 into the proportion gives you 2/(2.6) = 10/13. Multiply the left-side proportion by 5/5 and you have 10/13 = 10/13, a valid statement that cancels down to 1 = 1. So 2.6 is correct.

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Sources and Citations

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