## Solving Proportions

Contents

When two rational expressions are equal, the equation relating them is called a *proportion*.

### Proportion

A **proportion** is an equation of the form \(\cfrac{a}{b}=\cfrac{c}{d}\), where \(b\ne 0,d\ne 0\).

The proportion is read “\(a\) is to \(b\), as \(c\) is to \(d\).”

The equation \(\frac{1}{2}=\frac{4}{8}\) is a proportion because the two fractions are equal. The proportion \(\frac{1}{2}=\frac{4}{8}\) is read “1 is to 2 as 4 is to 8.”

Proportions are used in many applications to ‘scale up’ quantities. We’ll start with a very simple example so you can see how proportions work. Even if you can figure out the answer to the example right away, make sure you also learn to solve it using proportions.

Suppose a school principal wants to have 1 teacher for 20 students. She could use proportions to find the number of teachers for 60 students. We let *x* be the number of teachers for 60 students and then set up the proportion:

\(\cfrac{1\phantom{\rule{0.2em}{0ex}}\text{teacher}}{20\phantom{\rule{0.2em}{0ex}}\text{students}}=\cfrac{x\phantom{\rule{0.2em}{0ex}}\text{teachers}}{60\phantom{\rule{0.2em}{0ex}}\text{students}}\)

We are careful to match the units of the numerators and the units of the denominators—teachers in the numerators, students in the denominators.

Since a proportion is an equation with rational expressions, we will solve proportions the same way we solved equations in Solving Rational Equations. We’ll multiply both sides of the equation by the LCD to clear the fractions and then solve the resulting equation.

So let’s finish solving the principal’s problem now. We will omit writing the units until the last step.

Multiply both sides by the LCD, 60. | |

Simplify. | |

The principal needs 3 teachers for 60 students. |

Now we’ll do a few examples of solving numerical proportions without any units. Then we will solve applications using proportions.

## Example

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

### 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 we work with **proportion**s, we exclude values that would make either denominator zero, just like we do for all rational expressions. What value(s) should be excluded for the proportion in the next example?

## Example

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

### Solution

Multiply both sides by the LCD. | ||

Remove common factors on each side. | ||

Simplify. | ||

Divide both sides by 9. | ||

Simplify. | ||

Check. | ||

Show common factors. | ||

Simplify. |

## Example

Solve the proportion: \(\frac{n}{n+14}=\frac{5}{7}.\)

### Solution

Multiply both sides by the LCD. | ||

Remove common factors on each side. | ||

Simplify. | ||

Solve for \(n\). | ||

Check. | ||

Simplify. | ||

Show common factors. | ||

Simplify. |

## Example

Solve: \(\frac{p+12}{9}=\frac{p-12}{6}.\)

### Solution

Multiply both sides by the LCD, 18. | ||

Simplify. | ||

Distribute. | ||

Solve for \(p\). | ||

Check. | ||

Simplify. | ||

Divide. |

To solve applications with proportions, we will follow our usual strategy for solving applications. But when we set up the proportion, we must make sure to have the units correct—the units in the numerators must match and the units in the denominators must match.

## Example

When pediatricians prescribe acetaminophen to children, they prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of the child’s weight. If Zoe weighs 80 pounds, how many milliliters of acetaminophen will her doctor prescribe?

### Solution

Identify what we are asked to find, and choose a variable to represent it. | How many ml of acetaminophen will the doctor prescribe? | |

Let \(a=\mathrm{ml}\) of acetaminophen. | ||

Write a sentence that gives the information to find it. | If 5 ml is prescribed for every 25 pounds, how much will be prescribed for 80 pounds? | |

Translate into a proportion–be careful of the units. \(\frac{\text{ml}}{\text{pounds}}=\frac{\text{ml}}{\text{pounds}}\) | ||

Multiply both sides by the LCD, 400. | ||

Remove common factors on each side. | ||

Simplify, but don’t multiply on the left. Notice what the next step will be. | ||

Solve for \(a\). | ||

Check. | ||

Is the answer reasonable? | ||

Yes, since 80 is about 3 times 25, the medicine should be about 3 times 5. So 16 ml makes sense. | ||

Write a complete sentence. | The pediatrician would prescribe 16 ml of acetaminophen to Zoe. |

## Example

A 16-ounce iced caramel macchiato has 230 calories. How many calories are there in a 24-ounce iced caramel macchiato?

### Solution

Identify what we are asked to find, and choose a variable to represent it. | How many calories are in a 24 ounce iced caramel macchiato? | |

Let \(c=\text{calories}\) in 24 ounces. | ||

Write a sentence that gives the information to find it. | If there are 230 calories in 16 ounces, then how many calories are in 24 ounces? | |

Translate into a proportion–be careful of the units. \(\frac{\text{calories}}{\text{ounce}}=\frac{\text{calories}}{\text{ounce}}\) | ||

Multiply both sides by the LCD, 48. | ||

Remove common factors on each side. | ||

Simplify. | ||

Solve for \(c\). | ||

Check. | ||

Is the answer reasonable? | ||

Yes, 345 calories for 24 ounces is more than 290 calories for 16 ounces, but not too much more. | ||

Write a complete sentence. | There are 345 calories in a 24-ounce iced caramel macchiato. |

## Example

Josiah went to Mexico for spring break and changed $325 dollars into Mexican pesos. At that time, the exchange rate had $1 US is equal to 12.54 Mexican pesos. How many Mexican pesos did he get for his trip?

### Solution

What are you asked to find? | How many Mexican pesos did Josiah get? | |

Assign a variable. | Let \(p=\text{the number of Mexican pesos.}\) | |

Write a sentence that gives the information to find it. | If $1 US is equal to 12.54 Mexican pesos, then $325 is how many pesos? | |

Translate into a proportion–be careful of the units. | ||

\(\frac{\text{\$}}{\text{pesos}}=\frac{\text{\$}}{\text{pesos}}\) | ||

Multiply both sides by the LCD, \(12.54p\). | ||

Remove common factors on each side. | ||

Simplify. | ||

Check. | ||

Is the answer reasonable? | ||

Yes, $100 would be 1,254 pesos. $325 is a little more than 3 times this amount, so our answer of 4075.5 pesos makes sense. | ||

Write a complete sentence. | Josiah got 4075.5 pesos for his spring break trip. |

In the example above, we related the number of pesos to the number of dollars by using a proportion. We could say the number of pesos *is proportional to* the number of dollars. If two quantities are related by a proportion, we say that they are proportional.