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CHAPTER 4 Ratio, Proportion, and Percent

4.1 Ratios and Rate

Learning Objectives

By the end of this section, you will be able to:

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

Write a Ratio as a Fraction

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\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}b,\phantom{\rule{0.2em}{0ex}}\frac{a}{b},\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\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.

EXAMPLE 1

Write each ratio as a fraction: a)\phantom{\rule{0.2em}{0ex}}15\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}27\phantom{\rule{0.2em}{0ex}}b)\phantom{\rule{0.2em}{0ex}}45\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}18.

Solution
a)
\text{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}


We leave the ratio in b) as an improper fraction.

b)
\text{45 to 18}
Write as a fraction with the first number in the numerator and the second in the denominator. \frac{45}{18}
Simplify. \frac{5}{2}

TRY IT 1.1

Write each ratio as a fraction: a) \phantom{\rule{0.2em}{0ex}}21\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}56\phantom{\rule{0.2em}{0ex}} b) \phantom{\rule{0.2em}{0ex}}48\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}32.

Show answer
  1. \phantom{\rule{0.2em}{0ex}}\frac{3}{8}
  2. \phantom{\rule{0.2em}{0ex}}\frac{3}{2}

TRY IT 1.2

Write each ratio as a fraction: a)\phantom{\rule{0.2em}{0ex}}27\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}72\phantom{\rule{0.2em}{0ex}} b) \phantom{\rule{0.2em}{0ex}}51\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}34.

Show answer
  1. \phantom{\rule{0.2em}{0ex}}\frac{3}{8}
  2. \phantom{\rule{0.2em}{0ex}}\frac{3}{2}

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\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.05. We can write it as a fraction with decimals and then multiply the numerator and denominator by 100 to eliminate the decimals.

A fraction is shown with 0.8 in the numerator and 0.05 in the denominator. Below it is the same fraction with both the numerator and denominator multiplied by 100. Below that is a fraction with 80 in the numerator and 5 in the denominator.

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:

The top line says 0.80 over 0.05. There are blue arrows moving the decimal points over 2 places to the right.
“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:

a) \phantom{\rule{0.2em}{0ex}}4.8\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}11.2

b) \phantom{\rule{0.2em}{0ex}}2.7\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.54

Solution
a)

\phantom{\rule{0.2em}{0ex}}\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\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}11.2 is equivalent to \frac{3}{7}.

b)
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\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}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\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.54 is equivalent to \frac{5}{1}.

TRY IT 2.1

Write each ratio as a fraction: a) \phantom{\rule{0.2em}{0ex}}4.6\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}11.5\phantom{\rule{0.2em}{0ex}} b) \phantom{\rule{0.2em}{0ex}}2.3\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.69.

Show answer
  1. \phantom{\rule{0.2em}{0ex}}\frac{2}{5}\phantom{\rule{0.2em}{0ex}}
  2. \phantom{\rule{0.2em}{0ex}}\frac{10}{3}

TRY IT 2.2

Write each ratio as a fraction: a) \phantom{\rule{0.2em}{0ex}}3.4\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}15.3\phantom{\rule{0.2em}{0ex}} b) \phantom{\rule{0.2em}{0ex}}3.4\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}0.68.

Show answer
  1. \phantom{\rule{0.2em}{0ex}}\frac{2}{9}
  2. \phantom{\rule{0.2em}{0ex}}\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}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}2\frac{3}{8} as a fraction.

Solution
1\frac{1}{4}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}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}

TRY IT 3.1

Write each ratio as a fraction: 1\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}2\frac{5}{8}.

Show answer

\frac{2}{3}

TRY IT 3.2

Write each ratio as a fraction: 1\frac{1}{8}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}2\frac{3}{4}.

Show answer

\frac{9}{22}

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. a) Find the ratio of his total cholesterol to his HDL cholesterol. b) Assuming that a ratio less than 5 to 1 is considered good, what would you suggest to Hector?

Solution

a) 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{\text{total cholesterol}}{\text{HDL cholesterol}}
Substitute the values. \frac{249}{39}
Simplify. \frac{83}{13}

b) 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.

TRY IT 4.1

Find the patient’s ratio of total cholesterol to HDL cholesterol using the given information.

Total cholesterol is 185 mg/dL and HDL cholesterol is 40 mg/dL.

Show answer

\frac{37}{8}

TRY IT 4.2

Find the patient’s ratio of total cholesterol to HDL cholesterol using the given information.

Total cholesterol is 204 mg/dL and HDL cholesterol is 38 mg/dL.

Show answer

\frac{102}{19}

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 Canadian National Building Code (CNBC) 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{\text{rise}}{\text{run}}
Substitute in the given values. \frac{\text{1 inch}}{\text{1 foot}}
Convert 1 foot to inches. \frac{\text{1 inch}}{\text{12 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.

TRY IT 5.1

Find the ratio of the first length to the second length: 32 inches to 1 foot.

Show answer

\frac{8}{3}

TRY IT 5.2

Find the ratio of the first length to the second length: 1 foot to 54 inches.

Show answer

\frac{2}{9}

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.

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
\text{525 miles in 9 hours}
Write as a fraction, with 525 miles in the numerator and 9 hours in the denominator. \frac{\text{525 miles}}{\text{9 hours}}
\frac{\text{175 miles}}{\text{3 hours}}

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

TRY IT 6.1

Write the rate as a fraction: 492 miles in 8 hours.

Show answer

\frac{\text{123 miles}}{\text{2 hours}}

TRY IT 6.2

Write the rate as a fraction: 242 miles in 6 hours.

Show answer

\frac{\text{121 miles}}{\text{3 hours}}

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 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 \text{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 hour \frac{\text{68 miles}}{\text{1 hour}} 68 miles/hour 68 mph \text{68 miles per hour}
36 miles to 1 gallon \frac{\text{36 miles}}{\text{1 gallon}} 36 miles/gallon 36 mpg \text{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 \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.

EXAMPLE 7

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

Solution
Start with a rate of dollars to hours. Then divide. \text{\$384 last week for 32 hours}
Write as a rate. \frac{\$384}{\text{32 hours}}
Divide the numerator by the denominator. \frac{\$12}{\text{1 hour}}
Rewrite as a rate. \$12/\text{hour}

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

TRY IT 7.1

Find the unit rate: \text{\$630} for 35 hours.

Show answer

$18.00/hour

TRY IT 7.2

Find the unit rate: \text{\$684} for 36 hours.

Show answer

$19.00/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.

\text{455 miles to 14 gallons of gas}
Write as a rate. \frac{\text{455 miles}}{\text{14 gallons}}
Divide 455 by 14 to get the unit rate. \frac{\text{32.5 miles}}{\text{1 gallon}}

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

TRY IT 8.1

Find the unit rate: 423 miles to 18 gallons of gas.

Show answer

23.5 mpg

TRY IT 8.2

Find the unit rate: 406 miles to 14.5 gallons of gas.

Show answer

28 mpg

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 \text{\$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}{\text{24 bottles}}
Divide to find the unit price. \frac{\$0.16625}{\text{1 bottle}}
Round the result to the nearest penny. \frac{\$0.17}{\text{1 bottle}}

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

TRY IT 9.1

Find the unit price. Round your answer to the nearest cent if necessary.

\text{24-pack} of juice boxes for \text{\$6.99}

Show answer

$0.29/box

TRY IT 9.2

Find the unit price. Round your answer to the nearest cent if necessary.

\text{24-pack} of bottles of ice tea for \text{\$12.72}

Show answer

$0.53/bottle

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 \text{\$14.99} for 64 loads of laundry and the same brand of powder detergent is priced at \text{\$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{\text{\$14.99}}{\text{64 loads}} \frac{\text{\$15.99}}{\text{80 loads}}
Find the unit price. \frac{\text{\$0.234…}}{\text{1 load}} \frac{\text{\$0.199…}}{\text{1 load}}
Round to the nearest cent. \begin{array}{c}\text{\$0.23/load}\hfill \\ \text{(23 cents per load.)}\hfill \end{array} \begin{array}{c}\text{\$0.20/load}\hfill \\ \text{(20 cents per load)}\hfill \end{array}

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

TRY IT 10.1

Find each unit price and then determine the better buy. Round to the nearest cent if necessary.

Brand A Storage Bags, \text{\$4.59} for 40 count, or Brand B Storage Bags, \text{\$3.99} for 30 count

Show answer

Brand A costs $0.12 per bag. Brand B costs $0.13 per bag. Brand A is the better buy.

TRY IT 10.2

Find each unit price and then determine the better buy. Round to the nearest cent if necessary.

Brand C Chicken Noodle Soup, \text{\$1.89} for 26 ounces, or Brand D Chicken Noodle Soup, \text{\$0.95} for 10.75 ounces

Show answer

Brand C costs $0.07 per ounce. Brand D costs $0.09 per ounce. Brand C is the better buy.

Notice in the above example 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 11

Translate the word phrase into an algebraic expression:

a) \phantom{\rule{0.2em}{0ex}}427 miles per h hours

b) \phantom{\rule{0.2em}{0ex}}x students to 3 teachers

c) \phantom{\rule{0.2em}{0ex}}y dollars for 18 hours

Solution
a)
\text{427 miles per}\phantom{\rule{0.2em}{0ex}}h\phantom{\rule{0.2em}{0ex}}\text{hours}
Write as a rate. \frac{\text{427 miles}}{h\phantom{\rule{0.2em}{0ex}}\text{hours}}
b)
x\phantom{\rule{0.2em}{0ex}}\text{students to 3 teachers}
Write as a rate. \frac{x\phantom{\rule{0.2em}{0ex}}\text{students}}{\text{3 teachers}}
c)
y\phantom{\rule{0.2em}{0ex}}\text{dollars for 18 hours}
Write as a rate. \frac{\$y}{\text{18 hours}}

TRY IT 11.1

Translate the word phrase into an algebraic expression.

a) \phantom{\rule{0.2em}{0ex}}689 miles per h hours b) y parents to 22 students c) d dollars for 9 minutes

Show answer
  1. 689 mi/h hours
  2. y parents/22 students
  3. $d/9 min

TRY IT 11.2

Translate the word phrase into an algebraic expression.

a)m miles per 9 hours b) x students to 8 buses c) y dollars for 40 hours

Show answer
  1. m mi/9 h
  2. x students/8 buses
  3. $y/40 h

Access to Additional Online R

Glossary

ratio
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.
rate
A rate compares two quantities of different units. A rate is usually written as a fraction.
unit rate
A unit rate is a rate with denominator of 1 unit.
unit price
A unit price is a unit rate that gives the price of one item.

Practice Makes Perfect

Write a Ratio as a Fraction

In the following exercises, write each ratio as a fraction.

1. 20 to 36 2. 20 to 32
3. 42 to 48 4. 45 to 54
5. 49 to 21 6. 56 to 16
7. 84 to 36 8. 6.4 to 0.8
9. 0.56 to 2.8 10. 1.26 to 4.2
11. 1\frac{2}{3} to 2\frac{5}{6} 12. 1\frac{3}{4} to 2\frac{5}{8}
13. 4\frac{1}{6} to 3\frac{1}{3} 14. 5\frac{3}{5} to 3\frac{3}{5}
15. \text{\$18} to \text{\$63} 16. \text{\$16} to \text{\$72}
17. \text{\$1.21} to \text{\$0.44} 18. \text{\$1.38} to \text{\$0.69}
19. 28 ounces to 84 ounces 20. 32 ounces to 128 ounces
21. 12 feet to 46 feet 22. 15 feet to 57 feet
23. 246 milligrams to 45 milligrams 24. 304 milligrams to 48 milligrams
25. total cholesterol of 175 to HDL cholesterol of 45 26. total cholesterol of 215 to HDL cholesterol of 55
27. 27 inches to 1 foot 28. 28 inches to 1 foot

Write a Rate as a Fraction

In the following exercises, write each rate as a fraction.

29. 140 calories per 12 ounces 30. 180 calories per 16 ounces
31. 8.2 pounds per 3 square inches 32. 9.5 pounds per 4 square inches
33. 488 miles in 7 hours 34. 527 miles in 9 hours
35. \text{\$595} for 40 hours 36. \text{\$798} for 40 hours

Find Unit Rates

In the following exercises, find the unit rate. Round to two decimal places, if necessary.

37. 140 calories per 12 ounces 38. 180 calories per 16 ounces
39. 8.2 pounds per 3 square inches 40. 9.5 pounds per 4 square inches
41. 488 miles in 7 hours 42. 527 miles in 9 hours
43. \text{\$595} for 40 hours 44. \text{\$798} for 40 hours
45. 576 miles on 18 gallons of gas 46. 435 miles on 15 gallons of gas
47. 43 pounds in 16 weeks 48. 57 pounds in 24 weeks
49. 46 beats in 0.5 minute 50. 54 beats in 0.5 minute
51. The bindery at a printing plant assembles 96,000 magazines in 12 hours. How many magazines are assembled in one hour? 52. The pressroom at a printing plant prints 540,000 sections in 12 hours. How many sections are printed per hour?

Find Unit Price

In the following exercises, find the unit price. Round to the nearest cent.

53. Soap bars at 8 for \text{\$8.69} 54. Soap bars at 4 for \text{\$3.39}
55. Women’s sports socks at 6 pairs for \text{\$7.99} 56. Men’s dress socks at 3 pairs for \text{\$8.49}
57. Snack packs of cookies at 12 for \text{\$5.79} 58. Granola bars at 5 for \text{\$3.69}
59. CD-RW discs at 25 for \text{\$14.99} 60. CDs at 50 for \text{\$4.49}
61. The grocery store has a special on macaroni and cheese. The price is \text{\$3.87} for 3 boxes. How much does each box cost? 62. The pet store has a special on cat food. The price is \text{\$4.32} for 12 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.

63. Mouthwash, \text{50.7-ounce} size for \text{\$6.99} or \text{33.8-ounce} size for \text{\$4.79} 64. Toothpaste, 6 ounce size for \text{\$3.19} or 7.8-ounce size for \text{\$5.19}
65. Breakfast cereal, 18 ounces for \text{\$3.99} or 14 ounces for \text{\$3.29} 66. Breakfast Cereal, 10.7 ounces for \text{\$2.69} or 14.8 ounces for \text{\$3.69}
67. Ketchup, \text{40-ounce} regular bottle for \text{\$2.99} or \text{64-ounce} squeeze bottle for \text{\$4.39} 68. Mayonnaise \text{15-ounce} regular bottle for \text{\$3.49} or \text{22-ounce} squeeze bottle for \text{\$4.99}
69. Cheese \text{\$6.49} for 1 lb. block or \text{\$3.39} for \frac{1}{2} lb. block 70. Candy \text{\$10.99} for a 1 lb. bag or \text{\$2.89} for \frac{1}{4} lb. of loose candy

Translate Phrases to Expressions with Fractions

In the following exercises, translate the English phrase into an algebraic expression.

71. 793 miles per p hours 72. 78 feet per r seconds
73. \text{\$3} for 0.5 lbs. 74. j beats in 0.5 minutes
75. 105 calories in x ounces 76. 400 minutes for m dollars
77. the ratio of y and 5x 78. the ratio of 12x and y

Everyday Math

79. One elementary school in Saskatchewan has 684 students and 45 teachers. Write the student-to-teacher ratio as a unit rate. 80. The average Canadian produces about 350 pounds of paper trash per year \text{(365 days).} How many pounds of paper trash does the average Canadian produce each day? (Round to the nearest tenth of a pound.)
81. A popular fast food burger weighs 7.5 ounces and contains 540 calories, 29 grams of fat, 43 grams of carbohydrates, and 25 grams of protein. Find the unit rate of a) calories per ounce b) grams of fat per ounce c) grams of carbohydrates per ounce d) grams of protein per ounce. Round to two decimal places. 82. A 16-ounce chocolate mocha coffee with whipped cream contains 470 calories, 18 grams of fat, 63 grams of carbohydrates, and 15 grams of protein. Find the unit rate of a) calories per ounce b) grams of fat per ounce c) grams of carbohydrates per ounce d) grams of protein per ounce.

Writing Exercises

83. Would you prefer the ratio of your income to your friend’s income to be \text{3/1} or 1/3? Explain your reasoning. 84. The parking lot at the airport charges \text{\$0.75} for every 15 minutes. a) How much does it cost to park for 1 hour? b) Explain how you got your answer to part a). Was your reasoning based on the unit cost or did you use another method?
85. Kathryn ate a 4-ounce cup of frozen yogurt and then went for a swim. The frozen yogurt had 115 calories. Swimming burns 422 calories per hour. For how many minutes should Kathryn swim to burn off the calories in the frozen yogurt? Explain your reasoning. 86. Arjun had a 16-ounce cappuccino at his neighbourhood coffee shop. The cappuccino had 110 calories. If Arjun walks for one hour, he burns 246 calories. For how many minutes must Arjun walk to burn off the calories in the cappuccino? Explain your reasoning.

Answers

1. \frac{5}{9} 3. \frac{7}{8} 5. \frac{7}{3}
7. \frac{7}{3} 9. \frac{1}{5} 11. \frac{10}{17}
13. \frac{5}{4} 15. \frac{2}{7} 17. \frac{11}{4}
19. \frac{1}{3} 21. \frac{6}{23} 23. \frac{82}{15}
25. \frac{35}{9} 27. \frac{9}{4} 29. \frac{\text{35 calories}}{\text{3 ounces}}
31. \frac{\text{41 lbs}}{\text{15 sq. in}.} 33. \frac{\text{488 miles}}{\text{7 hours}} 35. \frac{\text{\$119}}{\text{8 hours}}
37. 11.67 calories/ounce 39. 2.73 lbs./sq. in. 41. 69.71 mph
43. $14.88/hour 45. 32 mpg 47. 2.69 lbs./week
49. 92 beats/minute 51. 8,000 53. $1.09/bar
55. $1.33/pair 57. $0.48/pack 59. $0.60/disc
61. $1.29/box 63. The 50.7-ounce size costs $0.138 per ounce. The 33.8-ounce size costs $0.142 per ounce. The 50.7-ounce size is the better buy. 65. The 18-ounce size costs $0.222 per ounce. The 14-ounce size costs $0.235 per ounce. The 18-ounce size is a better buy.
67. The regular bottle costs $0.075 per ounce. The squeeze bottle costs $0.069 per ounce. The squeeze bottle is a better buy. 69. The half-pound block costs $6.78/lb, so the 1-lb. block is a better buy. 71. \frac{\text{793 miles}}{p\phantom{\rule{0.2em}{0ex}}\text{hours}}
73. \frac{\text{?3}}{\text{0.5 lbs}.} 75. \frac{\text{105 calories}}{x\phantom{\rule{0.2em}{0ex}}\text{ounces}} 77. \frac{y}{5x}
79. 15.2 students per teacher 81. a) 72 calories/ounce

      b) 3.87 grams of fat/ounce

c) 5.73 grams carbs/once

d) 3.33 grams protein/ounce

 83. Answers will vary.
85. Answers will vary.

Attributions

This chapter has been adapted from “Ratios and Rate” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

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Introductory Algebra Copyright © 2021 by Izabela Mazur is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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