Opening the book…
By the end of this section, you will be able to:
- Write a ratio as a fraction
- Write a rate as a fraction
- Find unit rates
- Find unit price
- Translate phrases to expressions with fractions
Be Prepared 5.16
Before you get started, take this readiness quiz.
Simplify:
If you missed this problem, review Example 4.19.
Be Prepared 5.17
Divide:
If you missed this problem, review Example 5.19.
Be Prepared 5.18
Simplify:
If you missed this problem, review Example 4.43.
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 and , the ratio is written as
Ratios
A ratio compares two numbers or two quantities that are measured with the same unit. The ratio of to is written
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 instead of simplifying it to so that we can see the two parts of the ratio.
Example 5.58
Write each ratio as a fraction: ⓐⓑ
- Answer
ⓐ Write as a fraction with the first number in the numerator and the second in the denominator. Simplify the fraction. ⓑ Write as a fraction with the first number in the numerator and the second in the denominator. Simplify.
We leave the ratio in ⓑ as an improper fraction.
Try It 5.115
Write each ratio as a fraction: ⓐⓑ
Try It 5.116
Write each ratio as a 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 We can write it as a fraction with decimals and then multiply the numerator and denominator by to eliminate the decimals.
Do you see a shortcut to find the equivalent fraction? Notice that and The least common denominator of and is By multiplying the numerator and denominator of by 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. | |
| Simplify. |
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:
- ⓐ
- ⓑ
- Answer
ⓐ Write as a fraction. Rewrite as an equivalent fraction without decimals, by moving both decimal points 1 place to the right. Simplify. So is equivalent to
ⓑ
The numerator has one decimal place and the denominator has To clear both decimals we need to move the decimal places to the right.
Write as a fraction. Move both decimals right two places. Simplify.
So is equivalent to
Try It 5.117
Write each ratio as a fraction: ⓐⓑ
Try It 5.118
Write each ratio as a fraction: ⓐⓑ
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 as a fraction.
- Answer
Write as a fraction. Convert the numerator and denominator to improper fractions. Rewrite as a division of fractions. Invert the divisor and multiply. Simplify.
Try It 5.119
Write each ratio as a fraction:
Try It 5.120
Write each ratio as a fraction:
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 to is considered good.
Example 5.61
Hector's total cholesterol is mg/dl and his HDL cholesterol is mg/dl. ⓐ Find the ratio of his total cholesterol to his HDL cholesterol. ⓑ Assuming that a ratio less than to is considered good, what would you suggest to Hector?
- Answer
ⓐ 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. Substitute the values. Simplify. ⓑ Is Hector's cholesterol ratio ok? If we divide by we obtain approximately so Hector's cholesterol ratio is high! Hector should either lower his total cholesterol or raise his HDL cholesterol.
Try It 5.121
Find the patient's ratio of total cholesterol to HDL cholesterol using the given information.
Total cholesterol is mg/dL and HDL cholesterol is mg/dL.
Try It 5.122
Find the patient’s ratio of total cholesterol to HDL cholesterol using the given information.
Total cholesterol is mg/dL and HDL cholesterol is mg/dL.
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 5.62
The Americans with Disabilities Act (ADA) Guidelines for wheel chair ramps require a maximum vertical rise of inch for every foot of horizontal run. What is the ratio of the rise to the run?
- Answer
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. Substitute in the given values. Convert 1 foot to inches. Simplify, dividing out common factors and units. So the ratio of rise to run is to This means that the ramp should rise inch for every inches of horizontal run to comply with the guidelines.
Try It 5.123
Find the ratio of the first length to the second length: inches to foot.
Try It 5.124
Find the ratio of the first length to the second length: foot to inches.
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 miles in hours, words in minutes, and dollars per 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 5.63
Bob drove his car miles in hours. Write this rate as a fraction.
- Answer
Write as a fraction, with 525 miles in the numerator and 9 hours in the denominator. So miles in hours is equivalent to
Try It 5.125
Write the rate as a fraction: miles in hours.
Try It 5.126
Write the rate as a fraction: miles in hours.
Find Unit Rates
In the last example, we calculated that Bob was driving at a rate of This tells us that every three hours, Bob will travel 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 unit is referred to as a unit rate.
Unit Rate
A unit rate is a rate with denominator of unit.
Unit rates are very common in our lives. For example, when we say that we are driving at a speed of miles per hour we mean that we travel miles in hour. We would write this rate as miles/hour (read miles per hour). The common abbreviation for this is mph. Note that when no number is written before a unit, it is assumed to be
So miles/hour really means
Two rates we often use when driving can be written in different forms, as shown:
| Example | Rate | Write | Abbreviate | Read |
|---|---|---|---|---|
| miles in hour | miles/hour | mph | ||
| miles to gallon | miles/gallon | mpg |
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 for each hour you work, you could write that your hourly (unit) pay rate is (read per hour.)
To convert a rate to a unit rate, we divide the numerator by the denominator. This gives us a denominator of
Example 5.64
Anita was paid last week for working What is Anita’s hourly pay rate?
- Answer
Start with a rate of dollars to hours. Then divide. Write as a rate. Divide the numerator by the denominator. Rewrite as a rate. Anita’s hourly pay rate is per hour.
Try It 5.127
Find the unit rate: for hours.
Try It 5.128
Find the unit rate: for hours.
Example 5.65
Sven drives his car miles, using gallons of gasoline. How many miles per gallon does his car get?
- Answer
Start with a rate of miles to gallons. Then divide.
Write as a rate. Divide 455 by 14 to get the unit rate. Sven’s car gets miles/gallon, or mpg.
Try It 5.129
Find the unit rate: miles to gallons of gas.
Try It 5.130
Find the unit rate: miles to gallons of gas.
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 5.66
The grocery store charges for a case of bottles of water. What is the unit price?
- Answer
What are we asked to find? We are asked to find the unit price, which is the price per bottle.
Write as a rate. Divide to find the unit price. Round the result to the nearest penny. The unit price is approximately per bottle. Each bottle costs about
Try It 5.131
Find the unit price. Round your answer to the nearest cent if necessary.
of juice boxes for
Try It 5.132
Find the unit price. Round your answer to the nearest cent if necessary.
of bottles of ice tea for
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 5.67
Paul is shopping for laundry detergent. At the grocery store, the liquid detergent is priced at for loads of laundry and the same brand of powder detergent is priced at for loads.
Which detergent has the lowest cost per load?
- Answer
To compare the prices, we first find the unit price for each type of detergent.
Liquid Powder Write as a rate. Find the unit price. Round to the nearest cent. Now we compare the unit prices. The unit price of the liquid detergent is about per load and the unit price of the powder detergent is about per load. The powder is the better buy.
Try It 5.133
Find each unit price and then determine the better buy. Round to the nearest cent if necessary.
Brand A Storage Bags, for count, or Brand B Storage Bags, for count
Try It 5.134
Find each unit price and then determine the better buy. Round to the nearest cent if necessary.
Brand C Chicken Noodle Soup, for ounces, or Brand D Chicken Noodle Soup, for ounces
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.
Example 5.68
Translate the word phrase into an algebraic expression:
- ⓐ miles per hours
- ⓑ students to teachers
- ⓒ dollars for hours
- Answer
ⓐ Write as a rate. ⓑ Write as a rate.
| ⓒ | |
| Write as a rate. |
Try It 5.135
Translate the word phrase into an algebraic expression.
ⓐ miles per hours ⓑ parents to students ⓒ dollars for minutes
Try It 5.136
Translate the word phrase into an algebraic expression.
ⓐ miles per hours ⓑ students to buses ⓒ dollars for hours
Media
Section 5.6 Exercises
Practice Makes Perfect
Write a Ratio as a Fraction
In the following exercises, write each ratio as a fraction.
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
to
ounces to ounces
ounces to ounces
feet to feet
feet to feet
milligrams to milligrams
milligrams to milligrams
total cholesterol of to HDL cholesterol of
total cholesterol of to HDL cholesterol of
inches to foot
inches to foot
Write a Rate as a Fraction
In the following exercises, write each rate as a fraction.
calories per ounces
calories per ounces
pounds per square inches
pounds per square inches
miles in hours
miles in hours
for hours
for hours
Find Unit Rates
In the following exercises, find the unit rate. Round to two decimal places, if necessary.
calories per ounces
calories per ounces
pounds per square inches
pounds per square inches
miles in hours
miles in hours
for hours
for hours
miles on gallons of gas
miles on gallons of gas
pounds in weeks
pounds in weeks
beats in minute
beats in minute
The bindery at a printing plant assembles magazines in hours. How many magazines are assembled in one hour?
The pressroom at a printing plant prints sections in hours. How many sections are printed per hour?
Find Unit Price
In the following exercises, find the unit price. Round to the nearest cent.
Soap bars at for
Soap bars at for
Women’s sports socks at pairs for
Men’s dress socks at pairs for
Snack packs of cookies at for
Granola bars at for
CD-RW discs at for
CDs at for
The grocery store has a special on macaroni and cheese. The price is for boxes. How much does each box cost?
The pet store has a special on cat food. The price is for cans. How much does each can cost?
In the following exercises, find each unit price and then identify the better buy. Round to three decimal places.
Mouthwash, size for or size for
Toothpaste, ounce size for or size for
Breakfast cereal, ounces for or ounces for
Breakfast Cereal, ounces for or ounces for
Ketchup, regular bottle for or squeeze bottle for
Mayonnaise regular bottle for or squeeze bottle for
Cheese for lb. block or for lb. block
Candy for a lb. bag or for lb. of loose candy
Translate Phrases to Expressions with Fractions
In the following exercises, translate the English phrase into an algebraic expression.
miles per hours
feet per seconds
for lbs.
beats in minutes
calories in ounces
minutes for dollars
the ratio of and
the ratio of and
Everyday Math
One elementary school in Ohio has students and teachers. Write the student-to-teacher ratio as a unit rate.
The average American produces about pounds of paper trash per year How many pounds of paper trash does the average American produce each day? (Round to the nearest tenth of a pound.)
A popular fast food burger weighs ounces and contains calories, grams of fat, grams of carbohydrates, and grams of protein. Find the unit rate of ⓐ calories per ounce ⓑ grams of fat per ounce ⓒ grams of carbohydrates per ounce ⓓ grams of protein per ounce. Round to two decimal places.
A chocolate mocha coffee with whipped cream contains calories, grams of fat, grams of carbohydrates, and grams of protein. Find the unit rate of ⓐ calories per ounce ⓑ grams of fat per ounce ⓒ grams of carbohydrates per ounce ⓓ grams of protein per ounce.
Writing Exercises
Would you prefer the ratio of your income to your friend’s income to be or Explain your reasoning.
The parking lot at the airport charges for every minutes. ⓐ How much does it cost to park for hour? ⓑ Explain how you got your answer to part ⓐ. Was your reasoning based on the unit cost or did you use another method?
Kathryn ate a cup of frozen yogurt and then went for a swim. The frozen yogurt had calories. Swimming burns calories per hour. For how many minutes should Kathryn swim to burn off the calories in the frozen yogurt? Explain your reasoning.
Mollie had a cappuccino at her neighborhood coffee shop. The cappuccino had calories. If Mollie walks for one hour, she burns calories. For how many minutes must Mollie walk to burn off the calories in the cappuccino? Explain your reasoning.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?
This page titled 5.7: Ratios and Rate was last modified on Sun, 28 May 2023 07:43:01 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.
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