The cross product or cross multiplication is a mathematical process that allows you to solve a proportion consisting of two fractional members that both have a variable. A variable is an alphabetic character that indicates an unknown arbitrary value. The cross product allows you to reduce the proportion to a simple equation which, if solved, will result in the value of the variable in question. The cross product is very useful in case you need to solve a proportion. Read on to find out how to use it.
Steps
Method 1 of 2: Cross Product with One Variable Only
Step 1. Multiply the numerator of the fraction on the left side of the proportion by the denominator of the fraction occupying the right side
Assume you need to solve the following equation 2 / x = 10/13. Following the directions you will have to perform these calculations 2 * 13, resulting in 26.
Step 2. Now multiply the numerator of the fraction on the right side of the proportion by the denominator of the fraction occupying the left side
Continuing with the previous example and following the directions, you will have to perform these calculations x * 10 resulting in 10. If you prefer, you can start from this step instead of the previous one. It doesn't matter the order in which you cross-product the numerators and denominators of the equation.
Step 3. Now match the two products you got in order to solve the resulting equation
At this point, you need to solve the following simple equation: 26 = 10x. Again, it doesn't matter which value you put first in the equation. You can choose to solve the equation 26 = 10x or 10x = 26. The important thing is that both terms of the equation are treated as integers.
Trying to solve the equation 2 / x = 10/13 based on the variable x you will get that 2 * 13 = x * 10 that is 26 = 10x
Step 4. Now solve the equation obtained on the basis of the variable under consideration
At this point you need to work on the following equation 26 = 10x. Start by finding a common denominator that can be used as a divisor for both 26 and 10 and that allows you to get an integer quotient in both cases. Since both the values involved are even numbers, you can divide them both by 2 to get 26/2 = 13 and 10/2 = 5. At this point the aspect of the starting equation will be 13 = 5x. Now, to isolate the variable x, it is necessary to divide both sides of the equation by 5 obtaining 13/5 = 5x / 5, that is 13/5 = x. If you want to express the final result in the form of a decimal number, you can divide both sides of the starting equation by 10 to get 26/10 = 10x / 10 that is 2, 6 = x.
Method 2 of 2: Cross Product with Two Equal Variables
Step 1. Multiply the numerator of the left side of the proportion by the denominator of the right side
Assume you need to solve the following equation: (x + 3) / 2 = (x + 1) / 4. Start by multiplying (x + 3) by 4 to get 4 (x + 3). Perform the calculations to simplify the expression by getting 4x + 12.
Step 2. Now multiply the numerator of the right side of the proportion by the denominator of the left side
Continuing with the previous example you will get (x +1) x 2 = 2 (x +1). By doing the calculations you will get 2x + 2.
Step 3. Set up a new equation using the two products you just calculated and combine similar terms together
At this point you will have to work on the equation 4x + 12 = 2x + 2. Rearrange the terms of the equation so as to isolate all those with the variable x on the one hand and all constants on the other.
- To handle terms with the variable x, i.e. 4x and 2x, subtract the 2x value from both sides of the equation so that the variable x disappears from the right side because 2x - 2x results in 0. Instead inside the member left you will get 4x - 2x i.e. 2x.
- Now move all integer values to the right side of the equation by subtracting the number 12 from both sides. In this way the integer value of the left member will be eliminated because 12 - 12 is equal to 0. While inside the right member you will get 2 - 12 that is -10.
- After performing the above calculations you will have obtained the following equation 2x = -10.
Step 4. Solve the new equation based on x
All you have to do is divide both sides of the equation by the number 2 to get 2x / 2 = -10/2 i.e. x = -5. After applying the cross product you found that the value of x is equal to -5. You can verify the correctness of your work by substituting the value -5 in the starting equation for the variable x and performing the calculations. In this case you will get a valid equation, that is -1 = -1, so it means that you have worked correctly.
Advice
- You can easily verify the correctness of your work by substituting the result obtained in place of the variable present in the original proportion. If by carrying out the calculations and the necessary simplifications, the equation turns out to be valid, for example 1 = 1, it means that the result you have obtained is correct. If after performing the calculations and simplifications you get an invalid equation, for example 0 = 1, it means that you have made some mistake. In the example shown in the article, substituting the value 2, 6 for the variable x you would get the following equation: 2 / (2.6) = 10/13. Multiplying the left limb by the fraction 5/5 you would get 10/13 = 10/13 which by simplifying it becomes 1 = 1. In this case it means that the value of x equal to 2, 6 turns out to be correct.
- Note that replacing the variable with any value other than the correct one, for example 5, would result in the following equation 2/5 = 10/13. In this case, even multiplying the left side of the equation again by 5/5, you would get 10/25 = 10/13, which is clearly incorrect. This is a clear and obvious sign that you have made a mistake in applying the cross product technique.
