**5.6 Ratios and Rate**

The topics covered in this section are:

- Write a ratio as a fraction
- Write a rate as a fraction
- Find unit rates
- Find unit price
- Translate phrases to expressions with fractions

**5.6.1 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 $a$ to $b$, $\frac{a}{b}$, or $a:b$.

**RATIOS**

A **ratio** compares two numbers or two quantities that are measured with the same unit. The ratio of $a$ to $b$ is written $a$ to $b$, $\frac{a}{b}$, or $a:b$.

**Example 1**

Write each ratio as a fraction:

- $15$ to $27$
- $45$ to $18$.

**Solution**

Part 1. | |

$15$ to $27$ | |

Write as a fraction with the first number in the numerator and the second in the denominator. | $\frac{15}{27}$ |

Simplify the fraction. | $\frac{5}{9}$ |

Part 2. | |

$45$ to $18$ | |

Write as a fraction with the first number in the numerator and the second in the denominator. | $\frac{45}{18}$ |

Simplify the fraction. | $\frac{5}{2}$ |

We leave the ratio in **Part 2.** as an improper fraction.

**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 $0.8$ to $0.5$. 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 2**

Write each ratio as a fraction of whole numbers:

- $4.8$ to $11.2$
- $2.7$ to $0.54$

**Solution**

Part 1. $4.8$ 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 $4.8$ to $11.2$ is equivalent to $\frac{3}{7}$.

Part 2. 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.$2.7$ to $0.54$ | |

Write as a fraction. | $\frac{2.7}{0.57}$ |

Move both decimals right two places. | $\frac{270}{54}$ |

Simplify. | $\frac{5}{1}$ |

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

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

**Example 3**

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

**Solution**

$1 \frac{1}{4}$ 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. | $\frac{5}{4} \div \frac{19}{8}$ |

Invert the divisor and multiply. | $\frac{5}{4} \cdot \frac{8}{19}$ |

Simplify. | $\frac{10}{19}$ |

**Applications of Ratios**

One real-world application of ratios that affects many people involves measuring cholesterol in blood. The ratio of total cholesterol to HDL cholesterol is one way doctors assess a person’s overall health. A ratio of less than $5$ to $1$ is considered good.

**Example 4**

Hector’s total cholesterol is $249$ mg/dl and his HDL cholesterol is $39$ mg/dl. **Part 1.** Find the ratio of his total cholesterol to his HDL cholesterol. **Part 2.** Assuming that a ratio less than $5$ to $1$ is considered good, what would you suggest to Hector?

**Solution**

**Part 1.** First, write the words that express the ratio. We want to know the ratio of Hector’s total cholesterol to his HDL cholesterol.

Write as a fraction. | $\frac{\mathrm{total\ cholesterol}}{\mathrm{HDL\ cholesterol}}$ |

Substitute the values. | $\frac{249}{39}$ |

Simplify. | $\frac{83}{13}$ |

**Part 2.** Is Hector’s cholesterol ratio ok? If we divide $83$ by $13$ we obtain approximately $6.4$, so $\frac{83}{13} \approx \frac{6.4}{1}$. Hector’s cholesterol ratio is high! Hector should either lower his total cholesterol or raise his HDL cholesterol.

**Ratios of Two Measurements in Different Units**

To find the ratio of two measurements, we must make sure the quantities have been measured with the same unit. If the measurements are not in the same units, we must first convert them to the same units.

We know that to simplify a fraction, we divide out common factors. Similarly in a ratio of measurements, we divide out the common unit.

**Example 5**

The Americans with Disabilities Act (ADA) Guidelines for wheel chair ramps require a maximum vertical rise of $1$ inch for every $1$ foot of horizontal run. What is the ratio of the rise to the run?

**Solution**

In a ratio, the measurements must be in the same units. We can change feet to inches, or inches to feet. It is usually easier to convert to the smaller unit, since this avoids introducing more fractions into the problem.

Write the words that express the ratio.

Ratio of the rise to the run | |

Write the ratio as a fraction. | $\frac{\mathrm{rise} }{\mathrm{run} }$ |

Substitute in the given values. | $\frac{1\ \mathrm{inch} }{1\ \mathrm{foot}}$ |

Convert $1$ foot to inches. | $\frac{1\ \mathrm{inch}}{12\ \mathrm{inches}}$ |

Simplify, dividing out common factors and units. | $\frac{1}{12}$ |

So the ratio of rise to run is $1$ to $12$. This means that the ramp should rise $1$ inch for every $12$ inches of horizontal run to comply with the guidelines.

**5.6.2 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 $ \$ 5$ dollars per $64$ ounces.

**RATE**

A **rate** compares two quantities of different units. A rate is usually written as a fraction.

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.

**Example 6**

Bob drove his car $525$ miles in $9$ hours. Write this rate as a fraction.

**Solution**

$525$ miles in $9$ hours | |

Write as a fraction, with $525$ miles in the numerator and $9$ hours in the denominator. | $\frac{525\ \mathrm{miles}}{9\ \mathrm{hours}}$ |

$\frac{175\ \mathrm{miles}}{3\ \mathrm{hours}}$ |

So $525$ miles in $9$ hours is equivalent to $\frac{175 \mathrm{miles}}{3 \mathrm{hours}}$.

**5.6.3 Find Unit Rates**

In the last example, we calculated that Bob was driving at a rate of $\frac{175\ \mathrm{miles}}{3\ \mathrm{hours}}$. 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 RATE**

A **unit rate** is a rate with denominator of $1$ unit.

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 $68$ miles/$1$ hour.

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

Example | Rate | Write | Abbreviate | Read |
---|---|---|---|---|

$68$ miles in $1$ hours | $\frac{68\ \mathrm{miles}}{1 \mathrm{hour}}$ | $68$ miles/hour | $68$ mph | $68$ miles per hour |

$36$ miles to $1$ gallon | $\frac{36\ \mathrm{miles}}{1 \mathrm{gallon}}$ | $36$ miles/gallon | $36$ mpg | $36$ miles per gallon |

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 $ \$ 12.50$ for each hour you work, you could write that your hourly (unit) pay rate is $ \$ 12.50$/hour (read $ \$ 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 $1$.

**Example 7**

Anita was paid $ \$ 384$ last week for working $32$ hours. What is Anita’s hourly pay rate?

**Solution**

Start with a rate of dollars to hours. Then divide. | $ \$ 384$ last week for $32$ hours |

Write as a rate. | $\frac{\$ 384}{32\ \mathrm{hours}}$ |

Divide the numerator by the denominator. | $\frac{\$ 12}{1\ \mathrm{hour}}$ |

Rewrite as a rate. | $ \$ 12/ \mathrm{hour}$ |

Anita’s hourly pay rate is $ \$ 12$ per hour.

**Example 8**

Sven drives his car $455$ miles, using $14$ gallons of gasoline. How many miles per gallon does his car get?

**Solution**

Start with a rate of miles to gallons. Then divide.

$455$ miles to $14$ gallons of gas | |

Write as a rate. | $\frac{455\ \mathrm{miles}}{14\ \mathrm{gallons}}$ |

Divide $455$ by $14$ to get the unit rate. | $\frac{32.5\ \mathrm{miles}}{1\ \mathrm{gallon}}$ |

Sven’s car gets $32.5$ miles/gallon, or $32.5$ mpg.

**5.6.4 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 PRICE**

A **unit price** is a unit rate that gives the price of one item.

**Example 9**

The grocery store charges $ \$ 3.99$ for a case of $24$ bottles of water. What is the unit price?

**Solution**

What are we asked to find? We are asked to find the unit price, which is the price per bottle.

Write as a rate. | $\frac{\$ 3.99}{24\ \mathrm{bottles}}$ |

Divide to find the unit price. | $\frac{\$ 0.16625}{1\ \mathrm{bottle}}$ |

Round the result to the nearest penny. | $\frac{\$ 0.17}{1\ \mathrm{bottle}}$ |

The unit price is approximately $ \$ 0.17$ per bottle. Each bottle costs about $ \$ 0.17$.

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.

**Example 10**

Paul is shopping for laundry detergent. At the grocery store, the liquid detergent is priced at $ \$ 14.99$ for $64$ loads of laundry and the same brand of powder detergent is priced at $ \$ 15.99$ for $80$ loads.

Which is the better buy, the liquid or the powder detergent?

**Solution**

To compare the prices, we first find the unit price for each type of detergent.

Liquid | Powder | |

Write as a rate. | $\frac{\$ 14.99}{64\ \mathrm{loads}}$ | $\frac{\$ 15.99}{80\ \mathrm{loads}}$ |

Find the unit price. | $\frac{\$ 0.234…}{1\ \mathrm{load}}$ | $\frac{\$ 0.199…}{1\ \mathrm{load}}$ |

Round to the nearest cent. | $ \$ 0.23$/load ($23$ cents per load.) | $ \$ 0.20$/load ($20$ cents per load) |

Now we compare the unit prices. The unit price of the liquid detergent is about $ \$ 0.23$ per load and the unit price of the powder detergent is about $ \$ 0.20$ per load. The powder is the better buy.

Notice in Example 10 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.

**5.6.5 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.

**Example 11**

Translate the word phrase into an algebraic expression:

- $427$ miles per $h$ hours
- $x$ students to $3$ teachers
- $y$ dollars for $18$ hours

**Solution**

Part 1. | |

$427$ miles per $h$ hours | |

Write as a rate. | $\frac{427\ \mathrm{miles}}{h\ \mathrm{hours}}$ |

Part 2. | |

$x$ students to $3$ teachers | |

Write as a rate. | $\frac{x\ \mathrm{students}}{3\ \mathrm{teachers}}$ |

Part 3. | |

$y$ dollars for $18$ hours | |

Write as a rate. | $\frac{\$ y}{18\ \mathrm{hours}}$ |

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*Marecek, L., Anthony-Smith, M., & Mathis, A. H. (2020). Use the Language of Algebra. In Prealgebra 2e. OpenStax. https://openstax.org/books/prealgebra-2e/pages/5-6-ratios-and-rate*.*License: CC BY 4.0. Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction*