## Solving Proportions

To solve a proportion containing a variable, we remember that the proportion is an equation. All of the techniques we have used so far to solve equations still apply. In the next example, we will solve a proportion by multiplying by the **Least Common Denominator** (LCD) using the **Multiplication Property of Equality**.

## Example

Solve: \(\frac{x}{63}=\frac{4}{7}.\)

### Solution

To isolate \(x\), multiply both sides by the LCD, 63. | ||

Simplify. | ||

Divide the common factors. | ||

Check: To check our answer, we substitute into the original proportion. | ||

Show common factors. | ||

Simplify. |

When the variable is in a denominator, we’ll use the fact that the **cross products** of a proportion are equal to solve the proportions.

We can find the cross products of the proportion and then set them equal. Then we solve the resulting equation using our familiar techniques.

## Example

Solve: \(\frac{144}{a}=\frac{9}{4}.\)

### Solution

Notice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.

Find the cross products and set them equal. | ||

Simplify. | ||

Divide both sides by 9. | ||

Simplify. | ||

Check your answer. | ||

Show common factors.. | ||

Simplify. |

Another method to solve this would be to multiply both sides by the LCD, \(4a.\) Try it and verify that you get the same solution.

## Example

Solve: \(\frac{52}{91}=\frac{-4}{y}.\)

### Solution

Find the cross products and set them equal. | ||

Simplify. | ||

Divide both sides by 52. | ||

Simplify. | ||

Check: | ||

Show common factors. | ||

Simplify. |