Introduction to Writing Ratios and Calculating Rates

What you’ll learn to do: Write ratios as fractions and calculate unit rates

Several different kinds of Tillamook cheese in packages of different sizes

How can you compare prices on packages of different sizes?

When you go grocery shopping, do you sometimes find it difficult to compare prices when you’re choosing between packages of different sizes or quantities of an item? For example, imagine you want to buy a block of cheese. One brand is $4.99 for a 16-ounce brick. Another brand is on sale for $6.99 for a 24-ounce brick. Which one is the better deal? To make sure you get the most for your money, you’ll need to figure out the price of cheese per ounce so that you can compare equal quantities. In this section, we’ll explore ratios and rates, which will help you calculate unit rates and unit prices.

Before you get started, take this readiness quiz.

readiness quiz

1)

If you missed this problem, review this video.

2)

Divide: [latex]2.76\div 11.5[/latex]

Solution: [latex]0.24[/latex]

3)

If you missed this problem, review the video below.

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

Translate the word phrase into an algebraic expression:
ⓐ [latex]427[/latex] miles per [latex]h[/latex] hours
ⓑ [latex]x[/latex] students to [latex]3[/latex] teachers
ⓒ [latex]y[/latex] dollars for [latex]18[/latex] hours

Solution

ⓐ
[latex]\text{427 miles per }h\text{ hours}[/latex]
Write as a rate. [latex]\Large\frac{\text{427 miles }}{h\text{ hours}}[/latex]
ⓑ
[latex]x\text{ students to 3 teachers}[/latex]
Write as a rate. [latex]\Large\frac{x\text{ students}}{\text{3 teachers}}[/latex]
ⓒ
[latex]y\text{ dollars for 18 hours}[/latex]
Write as a rate. [latex]\Large\frac{y\text{ dollars}}{\text{18 hours}}[/latex]

 

 

TRY IT

Translate the word phrase into an algebraic expression.

ⓐ [latex]689[/latex] miles per [latex]h[/latex] hours ⓑ [latex]y[/latex] parents to [latex]22[/latex] students ⓒ [latex]d[/latex] dollars for [latex]9[/latex] minutes

ⓐ [latex]689[/latex] mi/h hours
ⓑ y parents/[latex]22[/latex] students
ⓒ $d/[latex]9[/latex] min

Translate the word phrase into an algebraic expression.
ⓐ [latex]m[/latex] miles per [latex]9[/latex] hours ⓑ [latex]x[/latex] students to [latex]8[/latex] buses ⓒ [latex]y[/latex] dollars for [latex]40[/latex] hours

ⓐ m mi/[latex]9[/latex] h
ⓑ x students/[latex]8[/latex] buses
ⓒ $y/[latex]40[/latex] h

 

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 [latex]5[/latex] to [latex]1[/latex] is considered good.

Example

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

Solution
ⓐ 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. [latex]\Large\frac{\text{total cholesterol}}{\text{HDL cholesterol}}[/latex]
Substitute the values. [latex]\Large\frac{249}{39}[/latex]
Simplify. [latex]\Large\frac{83}{13}[/latex]

ⓑ Is Hector’s cholesterol ratio ok? If we divide [latex]83[/latex] by [latex]13[/latex] we obtain approximately [latex]6.4[/latex], so [latex]\Large\frac{83}{13}\normalsize\approx\Large\frac{6.4}{1}[/latex]. Hector’s cholesterol ratio is high! Hector should either lower his total cholesterol or raise his HDL cholesterol.

 

TRY IT

Find the patient’s ratio of total cholesterol to HDL cholesterol using the given information.
Total cholesterol is [latex]185[/latex] mg/dL and HDL cholesterol is [latex]40[/latex] mg/dL.

[latex]\Large\frac{37}{8}[/latex]

Find the patient’s ratio of total cholesterol to HDL cholesterol using the given information.
Total cholesterol is [latex]204[/latex] mg/dL and HDL cholesterol is [latex]38[/latex] mg/dL.

[latex]\Large\frac{102}{19}[/latex]

 

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

The Americans with Disabilities Act (ADA) Guidelines for wheel chair ramps require a maximum vertical rise of [latex]1[/latex] inch for every [latex]1[/latex] 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. [latex]\Large\frac{\text{rise}}{\text{run}}[/latex]
Substitute in the given values. [latex]\Large\frac{\text{1 inch}}{\text{1 foot}}[/latex]
Convert [latex]1[/latex] foot to inches. [latex]\Large\frac{\text{1 inch}}{\text{12 inches}}[/latex]
Simplify, dividing out common factors and units. [latex]\Large\frac{1}{12}[/latex]

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

 

TRY IT

1. Find the ratio of the first length to the second length: [latex]32[/latex] inches to [latex]1[/latex] foot.

[latex]\Large\frac{8}{3}[/latex]

2. Find the ratio of the first length to the second length: [latex]1[/latex] foot to [latex]54[/latex] inches.

[latex]\Large\frac{2}{9}[/latex]