5.6 Ratios and Rate

Write a Ratio as a Fraction

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. A ratio compares two quantities that are measured with the same unit. If we compare $a$ and $b$, the ratio is written as $a\text{to}b,\frac{a}{b},\text{or}\mathit{\text{a}}\text{:}\mathit{\text{b}}\text{.}$

In this section, we will use the fraction notation. 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 $\frac{4}{1}$ instead of simplifying it to $4$ so that we can see the two parts of the ratio.

Ratios Involving Decimals

We will often work with ratios of decimals, especially when we have ratios involving money. In these cases, we can eliminate the decimals by using the Equivalent Fractions Property to convert the ratio to a fraction with whole numbers in the numerator and denominator.

For example, consider the ratio $0.8\text{to}0.05.$ We can write it as a fraction with decimals and then multiply the numerator and denominator by $100$ to eliminate the decimals.

Do you see a shortcut to find the equivalent fraction? Notice that $0.8=\frac{8}{10}$ and $0.05=\frac{5}{100}.$ The least common denominator of $\frac{8}{10}$ and $\frac{5}{100}$ is $100.$ By multiplying the numerator and denominator of $\frac{0.8}{0.05}$ by $100,$ we ‘moved’ the decimal two places to the right to get the equivalent fraction with no decimals. Now that we understand the math behind the process, we can find the fraction with no decimals like this:

"Move" the decimal 2 places.	$\frac{80}{5}$
Simplify.	$\frac{16}{1}$

You do not have to write out every step when you multiply the numerator and denominator by powers of ten. As long as you move both decimal places the same number of places, the ratio will remain the same.

Example 5.59

Write each ratio as a fraction of whole numbers:

1. ⓐ $4.8\text{to}11.2$
2. ⓑ $2.7\text{to}0.54$

Solution

ⓐ $\text{4.8 to 11.2}$
Write as a fraction.	$\frac{4.8}{11.2}$
Rewrite as an equivalent fraction without decimals, by moving both decimal points 1 place to the right.	$\frac{48}{112}$
Simplify.	$\frac{3}{7}$

So $4.8\text{to}11.2$ is equivalent to $\frac{3}{7}.$

ⓑ The numerator has one decimal place and the denominator has $2.$ To clear both decimals we need to move the decimal $2$ places to the right. $2.7\text{to}0.54$
Write as a fraction.	$\frac{2.7}{0.54}$
Move both decimals right two places.	$\frac{270}{54}$
Simplify.	$\frac{5}{1}$

So $2.7\text{to}0.54$ is equivalent to $\frac{5}{1}.$

Try It 5.117

Write each ratio as a fraction: ⓐ $4.6\text{to}11.5$ⓑ $2.3\text{to}0.69.$

Try It 5.118

Write each ratio as a fraction: ⓐ $3.4\text{to}15.3$ⓑ $3.4\text{to}0.68.$

Some ratios compare two mixed numbers. Remember that to divide mixed numbers, you first rewrite them as improper fractions.

Example 5.60

Write the ratio of $1\frac{1}{4}\text{to}2\frac{3}{8}$ as a fraction.

Solution

	$1\frac{1}{4}\text{to}2\frac{3}{8}$
Write as a fraction.	$\frac{1\frac{1}{4}}{2\frac{3}{8}}$
Convert the numerator and denominator to improper fractions.	$\frac{\frac{5}{4}}{\frac{19}{8}}$
Rewrite as a division of fractions.	$\frac{5}{4}÷\frac{19}{8}$
Invert the divisor and multiply.	$\frac{5}{4}·\frac{8}{19}$
Simplify.	$\frac{10}{19}$

Try It 5.119

Write each ratio as a fraction: $1\frac{3}{4}\text{to}2\frac{5}{8}.$

Try It 5.120

Write each ratio as a fraction: $1\frac{1}{8}\text{to}2\frac{3}{4}.$

Write a Rate as a Fraction

Frequently we want to compare two different types of measurements, such as miles to gallons. To make this comparison, we use a rate. Examples of rates are $120$ miles in $2$ hours, $160$ words in $4$ minutes, and $\text{\$5}$ dollars per $64$ ounces.

When writing a fraction as a rate, we put the first given amount with its units in the numerator and the second amount with its units in the denominator. When rates are simplified, the units remain in the numerator and denominator.

Find Unit Rates

In the last example, we calculated that Bob was driving at a rate of $\frac{\text{175 miles}}{\text{3 hours}}.$ This tells us that every three hours, Bob will travel $175$ miles. This is correct, but not very useful. We usually want the rate to reflect the number of miles in one hour. A rate that has a denominator of $1$ unit is referred to as a unit rate.

Unit rates are very common in our lives. For example, when we say that we are driving at a speed of $68$ miles per hour we mean that we travel $68$ miles in $1$ hour. We would write this rate as $68$ miles/hour (read $68$ miles per hour). The common abbreviation for this is $68$ mph. Note that when no number is written before a unit, it is assumed to be $1.$

So $68$ miles/hour really means $\text{68 miles/1 hour.}$

Two rates we often use when driving can be written in different forms, as shown:

Another example of unit rate that you may already know about is hourly pay rate. It is usually expressed as the amount of money earned for one hour of work. For example, if you are paid $\text{\$12.50}$ for each hour you work, you could write that your hourly (unit) pay rate is $\text{\$12.50/hour}$ (read $\text{\$12.50}$ per hour.)

To convert a rate to a unit rate, we divide the numerator by the denominator. This gives us a denominator of $1.$

Find Unit Price

Sometimes we buy common household items ‘in bulk’, where several items are packaged together and sold for one price. To compare the prices of different sized packages, we need to find the unit price. To find the unit price, divide the total price by the number of items. A unit price is a unit rate for one item.

Unit prices are very useful if you comparison shop. The better buy is the item with the lower unit price. Most grocery stores list the unit price of each item on the shelves.

Notice in Example 5.67 that we rounded the unit price to the nearest cent. Sometimes we may need to carry the division to one more place to see the difference between the unit prices.

Translate Phrases to Expressions with Fractions

Have you noticed that the examples in this section used the comparison words ratio of, to, per, in, for, on, and from? When you translate phrases that include these words, you should think either ratio or rate. If the units measure the same quantity (length, time, etc.), you have a ratio. If the units are different, you have a rate. In both cases, you write a fraction.