# How to do the rule of three: 8 Steps (with pictures)

## Table of contents:

The rule of three is a way to solve an equation that has a variable within two equivalent fractions. The variable is a symbol for an unknown number, and the rule of three reduces the proportion to a simple equation, thus solving the problem. The rule of three is especially useful if you are solving a proportion. You can apply this multiplication method as follows:

## Steps

### Method 1 of 2: Use the rule of three with a single variable

#### Step 1. Multiply the numerator of the fraction on the left by the denominator of the fraction on the right

Imagine that you are going to solve the equation 2 / x = 10/13. Multiply 2 by 13 to find 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, that is, x * 10 = 10x. You can multiply the fractions in this order first; you won't have any problem as long as you cross-multiply the nominators by the denominators.

#### Step 3. Match the products of the two operations

Match 26 to 10x, that is, 26 = 10x. It does not matter which number you put first, since being equal, you can interchange them within the equation without problems, as long as you treat each figure as a whole.

### In the case of the equation 2 / x = 10/13, the exercise can be carried out in 2 * 13 = x * 10 or 26 = 10x

#### Step 4. Solve for the variable

When you solve the equation to the point that you have 26 = 10x, you can find a common denominator to exactly divide 26 and 10 into a single number. Since both figures are even, you can divide them by 2 like this: 26/2 = 13 and 10/2 = 5. The result of the entire operation will be 13 = 5x. To solve for "x," divide both parts of the equation by 5, that is, 13/5 = 5/5 to calculate 13/5 = x. If you want to put the answer in decimal form, divide both parts of the equation by 10 like this: 10/26 = 10/10 to find the answer 2,6 = x.

### Method 2 of 2: Use the rule of three with multiple variables

#### Step 1. Multiply the numerator of the fraction on the left by the denominator of the fraction on the right

Imagine that you are going to solve the equation (x + 3) / 2 = (x + 1) / 4. Multiply (x + 3) by 4 to find 4 (x +3). Then distribute the number 4 to arrive at 4x + 12.

#### Step 2. Multiply the numerator of the fraction on the right by the denominator of the fraction on the left

Repeat the process on the other side of the equation like this: (x +1) x 2 = 2 (x +1). Then distribute the 2 to find 2x + 2.

#### Step 3. Set the products of the two operations equal and combine like terms

When you solve the equation until you have 4x + 12 = 2x + 2, combine the variables and constants on both sides of the equation together.

• You can combine 4x with 2x by subtracting 2x from both numbers. Subtract 2x from 2x on the right side of the equation to find 0. On the left side, perform the operation 4x - 2x to find 2x, which is the number that remains.
• Combine 12 with 2 by subtracting 12 from both sides of the equation. Subtract 12 from 12 on the left side of the equation to find 0. On the right side, subtract 12 from 2 to get to -10.
• The equation should now be 2x = -10.

#### Step 4. Solve the problem

All you have to do now is divide both sides of the equation by 2 like this: 2x / 2 = -10/2 = x = -5. By applying the rule of three in this problem, it is possible to find that x = -5. You can go back to the original equation and replace the "x" with -5 to make sure both sides of the problem are the same. You will find out that they really are! If you replace the variable in the original equation by -5, you will get: -1 = -1.

## Advice

• Keep in mind that if you substitute a different number (like 5) in the same proportion, the equation will be 2/5 = 10/13. Even if you multiply the left side of the equation by 5/5 again, you will get 10/25 = 10/13, which is incorrect. This means that there was an error when cross multiplying.
• You can replace the result of the problem in the original equation to check the exercise. If the equation simplifies to a valid answer like 1 = 1, it means that the answer is correct. On the other hand, if when simplifying the equation a result appears as 0 = 1, it means that the answer is wrong. For example, if you replace the variable in the original equation with the answer previously developed so that it is 2 / (2, 6) = 10/13 and then multiply the left side of that problem by 5/5, you will calculate 10/13 = 10/13. This answer is valid as it simplifies to 1 = 1. Therefore, the previously calculated answer (2, 6) is correct.