## Solving Problems Using Ratios

### Learning Outcome

• Use ratios to solve simple word problems

When you apply for a mortgage, the loan officer will compare your total debt to your total income to decide if you qualify for the loan. This comparison is called the debt-to-income ratio. The ratio must fall below a certain threshold for a lender to agree to loan you money. You can improve your debt to income ratio either by increasing your income, decreasing your debt, or both.

### Ratios

A ratio compares two numbers or two quantities and indicates their proportion to one another.

The ratio of $a$ to $b$ is written $a\text{ to }b,{\Large\frac{a}{b}},\text{ or }\mathit{\text{a}}\text{ :}\mathit{\text{b}}$

In this section, we will use the various notations interchangeably (i.e. $\large\frac{5}{7}$ is the same as $5$ : $7$, is the same as $5$ to $7$). When a ratio is written in fraction form, the fraction should be simplified. If it is an improper fraction, we do not change it to a mixed number. Because a ratio compares two quantities, we would leave a ratio as ${\large\frac{4}{1}}$ or $4$ : $1$ instead of simplifying it to $4$ so that we can see the two parts of the ratio.

### example

Write each ratio as a fraction:

1.  $15\text{ to }27$
2.   $45\text{ to }18$

Solution

 1. $\text{15 to 27}$ Write as a fraction with the first number in the numerator and the second in the denominator. ${\Large\frac{15}{27}}$ Simplify the fraction. ${\Large\frac{5}{9}}$
 2. $\text{45 to 18}$ Write as a fraction with the first number in the numerator and the second in the denominator. ${\Large\frac{45}{18}}$ Simplify. ${\Large\frac{5}{2}}$

Notice we leave the second ratio as an improper fraction.

### try it

In the following video you will see more examples of how to express a ratio as a fraction.

Let’s consider some word problems with ratios. We will need to remember how to create equivalent ratios. As long as you are dividing or multiplying both parts of the ratio (or the numerator and denominator, if you are using a faction representation), then you are creating an equivalent ratio.

Take the ratio $10$ : $4$

Let’s create an equivalent ratio by simplifying it. A common factor is $2$, so let’s divide both values by $2$. The result is a true statement of equivalent ratios.

$10$ : $4$ equals $5$ : $2$

or

$\frac{10}{4}$ = $\frac{5}{2}$

Let’s create another equivalent ratio by making it larger. Let’s multiply both values by $3$. The result is a true statement of equivalent ratios.

$10$ : $4$ equals $30$ : $12$

or

$\frac{10}{4}$ = $\frac{30}{12}$

### EXample

If Maria buys one dress for every three pairs of pants, and last year she bought nine pairs of pants, how many dresses did she buy last year?

Solution:

We will set up two equivalent ratio sentences like this:

_____ dress : _____ pants  (Maria’s purchase ratio)

_____ dress : _____ pants  (last year’s actual purchases)

Then we will fill them out with the values we know:

__$\color{red}{1}$__ dress : __$\color{red}{3}$__ pants

__$\color{red}{?}$__ dress : __$\color{red}{9}$__ pants

To make an equivalent ratio and to get from $3$ pants to $9$ pants, we know that we must have multiplied the initial value by $3$. Let’s do the same to the dresses to complete the equivalent ratio.

$3\cdot1=3$

Therefore, if Maria bought nine pairs of pants last year, based on her purchase ratio, she also bought three dresses.

### Try It

The ratio of Sophomores to Seniors at Mauve Hills High School is seven to ten. If there are 154 Sophomores enrolled this year, how many Seniors are there?

The next examples go one step further and take a little more thought as to what value you are looking for and what to do with it after you calculate it.

### ExAMPLE

The Pet Adoption Center in San Antonio houses a ratio of $4$ cats to $7$ dogs. If the shelter can house $49$ dogs, how many total animals can they keep at the shelter?

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