Problem Set: Ratios and Rates

Write a Ratio as a Fraction

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

[latex]20[/latex] to [latex]36[/latex]
Show Solution

[latex]\Large\frac{5}{9}[/latex]
[latex]20[/latex] to [latex]32[/latex]
[latex]42[/latex] to [latex]48[/latex]
Show Solution

[latex]\Large\frac{7}{8}[/latex]
[latex]45[/latex] to [latex]54[/latex]
[latex]49[/latex] to [latex]21[/latex]
Show Solution

[latex]\Large\frac{7}{3}[/latex]
[latex]56[/latex] to [latex]16[/latex]
[latex]84[/latex] to [latex]36[/latex]
Show Solution

[latex]\Large\frac{7}{3}[/latex]
[latex]6.4[/latex] to [latex]0.8[/latex]
[latex]0.56[/latex] to [latex]2.8[/latex]
Show Solution

[latex]\Large\frac{1}{5}[/latex]
[latex]1.26[/latex] to [latex]4.2[/latex]
[latex]1\Large\frac{2}{3}[/latex] to [latex]2\Large\frac{5}{6}[/latex]
Show Solution

[latex]\Large\frac{10}{17}[/latex]
[latex]1\Large\frac{3}{4}[/latex] to [latex]2\Large\frac{5}{8}[/latex]
[latex]4\Large\frac{1}{6}[/latex] to [latex]3\Large\frac{1}{3}[/latex]
Show Solution

[latex]\Large\frac{5}{4}[/latex]
[latex]5\Large\frac{3}{5}[/latex] to [latex]3\Large\frac{3}{5}[/latex]
[latex]{$18}[/latex] to [latex]{$63}[/latex]
Show Solution

[latex]\Large\frac{2}{7}[/latex]
[latex]{$16}[/latex] to [latex]{$72}[/latex]
[latex]{$1.21}[/latex] to [latex]{$0.44}[/latex]
Show Solution

[latex]\Large\frac{11}{4}[/latex]
[latex]{$1.38}[/latex] to [latex]{$0.69}[/latex]
[latex]28[/latex] ounces to [latex]84[/latex] ounces
Show Solution

[latex]\Large\frac{1}{3}[/latex]
[latex]32[/latex] ounces to [latex]128[/latex] ounces
[latex]12[/latex] feet to [latex]46[/latex] feet
Show Solution

[latex]\Large\frac{6}{23}[/latex]
[latex]15[/latex] feet to [latex]57[/latex] feet
[latex]246[/latex] milligrams to [latex]45[/latex] milligrams
Show Solution

[latex]\Large\frac{82}{15}[/latex]
[latex]304[/latex] milligrams to [latex]48[/latex] milligrams
total cholesterol of [latex]175[/latex] to HDL cholesterol of [latex]45[/latex]
Show Solution

[latex]\Large\frac{35}{9}[/latex]
total cholesterol of [latex]215[/latex] to HDL cholesterol of [latex]55[/latex]
[latex]27[/latex] inches to [latex]1[/latex] foot
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[latex]\Large\frac{9}{4}[/latex]
[latex]28[/latex] inches to [latex]1[/latex] foot
[latex]28[/latex] to [latex]40[/latex][latex]56[/latex] to [latex]32[/latex]
Show Solution

[latex]\Large\frac{7}{4}[/latex]
[latex]3.5[/latex] to [latex]0.5[/latex]
[latex]1.2[/latex] to [latex]1.8[/latex]
Show Solution

[latex]\Large\frac{2}{3}[/latex]
[latex]1\Large\frac{3}{4}\normalsize\text{to}1\Large\frac{5}{8}[/latex]
[latex]2\Large\frac{1}{3}\normalsize\text{to}5\Large\frac{1}{4}[/latex]
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[latex]\Large\frac{4}{9}[/latex]
[latex]64[/latex] ounces to [latex]30[/latex] ounces
[latex]28[/latex] inches to [latex]3[/latex] feet
Show Solution

[latex]\Large\frac{7}{9}[/latex]

Write a Rate as a Fraction

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

[latex]140[/latex] calories per [latex]12[/latex] ounces
Show Solution

[latex]\Large\frac{\text{35 calories}}{\text{3 ounces}}[/latex]
[latex]180[/latex] calories per [latex]16[/latex] ounces
[latex]8.2[/latex] pounds per [latex]3[/latex] square inches
Show Solution

[latex]\Large\frac{\text{41 lbs}}{\text{15 sq. in}.}[/latex]
[latex]9.5[/latex] pounds per [latex]4[/latex] square inches
[latex]488[/latex] miles in [latex]7[/latex] hours
Show Solution

[latex]\Large\frac{\text{488 miles}}{\text{7 hours}}[/latex]
[latex]527[/latex] miles in [latex]9[/latex] hours
[latex]{$595}[/latex] for [latex]40[/latex] hours
Show Solution

[latex]\Large\frac{{$119}}{{8 hours}}[/latex]
[latex]{$798}[/latex] for [latex]40[/latex] hours
[latex]180[/latex] calories per [latex]8[/latex] ounces
[latex]90[/latex] pounds per [latex]7.5[/latex] square inches
Show Solution

[latex]\Large\frac{90\text{pounds}}{7.5\text{square inches}}[/latex]
[latex]126[/latex] miles in [latex]4[/latex] hours
[latex]{$612.50}[/latex] for [latex]35[/latex] hours
Show Solution

[latex]\Large\frac{$612.50}{35\text{hours}}[/latex]

Find Unit Rates

Exercise 1

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

1. [latex]140[/latex] calories per [latex]12[/latex] ounces
  Show Solution
  
  11.67 calories/ounce
2. [latex]180[/latex] calories per [latex]16[/latex] ounces
3. [latex]8.2[/latex] pounds per [latex]3[/latex] square inches
  Show Solution
  
  2.73 lbs./sq. in.
4. [latex]9.5[/latex] pounds per [latex]4[/latex] square inches
5. [latex]488[/latex] miles in [latex]7[/latex] hours
  Show Solution
  
  69.71 mph
6. [latex]527[/latex] miles in [latex]9[/latex] hours
7. [latex]{$595}[/latex] for [latex]40[/latex] hours
  Show Solution
  
  $14.88/hour
8. [latex]{$798}[/latex] for [latex]40[/latex] hours
9. [latex]576[/latex] miles on [latex]18[/latex] gallons of gas
  Show Solution
  
  32 mpg
10. [latex]435[/latex] miles on [latex]15[/latex] gallons of gas
11. [latex]43[/latex] pounds in [latex]16[/latex] weeks
  Show Solution
  
  2.69 lbs./week
12. [latex]57[/latex] pounds in [latex]24[/latex] weeks
13. [latex]46[/latex] beats in [latex]0.5[/latex] minute
  Show Solution
  
  92 beats/minute
14. [latex]54[/latex] beats in [latex]0.5[/latex] minute
15. [latex]180[/latex] calories per [latex]8[/latex] ounces
16. [latex]90[/latex] pounds per [latex]7.5[/latex] square inches
  Show Solution
  
  12 pounds/sq.in.
17. [latex]126[/latex] miles in [latex]4[/latex] hours
18. [latex]{$612.50}[/latex] for [latex]35[/latex] hours
  Show Solution
  
  $17.50/hour

Exercise 2

1. The bindery at a printing plant assembles [latex]96,000[/latex] magazines in [latex]12[/latex] hours. How many magazines are assembled in one hour?
  Show Solution
  
  8,000
2. The pressroom at a printing plant prints [latex]540,000[/latex] sections in [latex]12[/latex] hours. How many sections are printed per hour?

Find Unit Price

Exercise 1

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

1. Soap bars at [latex]8[/latex] for [latex]{$8.69}[/latex]
  Show Solution
  
  $1.09/bar
2. Soap bars at [latex]4[/latex] for [latex]{$3.39}[/latex]
3. Women’s sports socks at [latex]6[/latex] pairs for [latex]{$7.99}[/latex]
  Show Solution
  
  $1.33/pair
4. Men’s dress socks at [latex]3[/latex] pairs for [latex]{$8.49}[/latex]
5. Snack packs of cookies at [latex]12[/latex] for [latex]{$5.79}[/latex]
  Show Solution
  
  $0.48/pack
6. Granola bars at [latex]5[/latex] for [latex]{$3.69}[/latex]
7. CD-RW discs at [latex]25[/latex] for [latex]{$14.99}[/latex]
  Show Solution
  
  $0.60/disc
8. CDs at [latex]50[/latex] for [latex]{$4.49}[/latex]
9. t-shirts: [latex]3[/latex] for [latex]{$8.97}[/latex]
10. Highlighters: [latex]6[/latex] for [latex]{$2.52}[/latex]
  Show Solution
  
  $0.42
11. An office supply store sells a box of pens for [latex]{$11}[/latex]. The box contains [latex]12[/latex] pens. How much does each pen cost?
12. Anna bought a pack of [latex]8[/latex] kitchen towels for [latex]{$13.20}[/latex]. How much did each towel cost? Round to the nearest cent if necessary.
  Show Solution
  
  $1.65

Exercise 2

1. The grocery store has a special on macaroni and cheese. The price is [latex]{$3.87}[/latex] for [latex]3[/latex] boxes. How much does each box cost?
  Show Solution
  
  $1.29/box
2. The pet store has a special on cat food. The price is [latex]{$4.32}[/latex] for [latex]12[/latex] cans. How much does each can cost?

Exercise 3

In the following exercises, find each unit price and then identify the better buy. Round to three decimal places.

1. Mouthwash, [latex]{50.7-ounce}[/latex] size for [latex]{$6.99}[/latex] or [latex]{33.8-ounce}[/latex] size for [latex]{$4.79}[/latex]
  Show Solution
  
  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.
2. Toothpaste, [latex]6[/latex] ounce size for [latex]{$3.19}[/latex] or [latex]7.8-ounce[/latex] size for [latex]{$5.19}[/latex]
3. Breakfast cereal, [latex]18[/latex] ounces for [latex]{$3.99}[/latex] or [latex]14[/latex] ounces for [latex]{$3.29}[/latex]
  Show Solution
  
  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.
4. Breakfast Cereal, [latex]10.7[/latex] ounces for [latex]{$2.69}[/latex] or [latex]14.8[/latex] ounces for [latex]{$3.69}[/latex]
5. Ketchup, [latex]{40-ounce}[/latex] regular bottle for [latex]{$2.99}[/latex] or [latex]{64-ounce}[/latex] squeeze bottle for [latex]{$4.39}[/latex]
  Show Solution
  
  The regular bottle costs $0.075 per ounce. The squeeze bottle costs $0.069 per ounce. The squeeze bottle is a better buy.
6. Mayonnaise [latex]{15-ounce}[/latex] regular bottle for [latex]{$3.49}[/latex] or [latex]{22-ounce}[/latex] squeeze bottle for [latex]{$4.99}[/latex]
7. Cheese [latex]{$6.49}[/latex] for [latex]1[/latex] lb. block or [latex]{$3.39}[/latex] for [latex]\Large\frac{1}{2}[/latex] lb. block
  Show Solution
  
  The half-pound block costs $6.78/lb, so the 1-lb. block is a better buy.
8. Candy [latex]{$10.99}[/latex] for a [latex]1[/latex] lb. bag or [latex]{$2.89}[/latex] for [latex]\Large\frac{1}{4}[/latex] lb. of loose candy
9. Shampoo: [latex]12[/latex] ounces for [latex]{$4.29}[/latex] or [latex]22[/latex] ounces for [latex]{$7.29}?[/latex]
10. Vitamins: [latex]60[/latex] tablets for [latex]{$6.49}[/latex] or [latex]100[/latex] for [latex]{$11.99}?[/latex]
  Show Solution
  
  $0.11, $0.12; 60 tablets for $6.49

Translate Phrases to Expressions with Fractions

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

1. [latex]793[/latex] miles per [latex]p[/latex] hours
  Show Solution
  
  [latex]\Large\frac{\text{793 miles}}{p \text{hours}}[/latex]
2. [latex]78[/latex] feet per [latex]r[/latex] seconds
3. [latex]{$3}[/latex] for [latex]0.5[/latex] lbs.
  Show Solution
  
  [latex]\Large\frac{{$3}}{{0.5 lbs}.}[/latex]
4. [latex]j[/latex] beats in [latex]0.5[/latex] minutes
5. [latex]105[/latex] calories in [latex]x[/latex] ounces
  Show Solution
  
  [latex]\Large\frac{\text{105 calories}} {x \text {ounces}}[/latex]
6. [latex]400[/latex] minutes for [latex]m[/latex] dollars
7. the ratio of [latex]y[/latex] and [latex]5x[/latex]
  Show Solution
  
  [latex]\Large\frac{y}{5x}[/latex]
8. the ratio of [latex]12x[/latex] and [latex]y[/latex]
9. [latex]535[/latex] miles per [latex]h\text{hours}[/latex]
10. [latex]a[/latex] adults to [latex]45[/latex] children
  Show Solution
  
  [latex]\Large\frac{a\text{adults}}{45\text{children}}[/latex]
11. the ratio of [latex]4y[/latex] and the difference of [latex]x[/latex] and [latex]10[/latex]
12. the ratio of [latex]19[/latex] and the sum of [latex]3[/latex] and [latex]n[/latex]
  Show Solution
  
  [latex]\Large\frac{19}{3+n}[/latex]

Everyday Math

Everyday math

One elementary school in Ohio has [latex]684[/latex] students and [latex]45[/latex] teachers. Write the student-to-teacher ratio as a unit rate.
Show Solution

15.2 students per teacher
The average American produces about [latex]1,600[/latex] pounds of paper trash per year (365 days). 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 [latex]7.5[/latex] ounces and contains [latex]540[/latex] calories, [latex]29[/latex] grams of fat, [latex]43[/latex] grams of carbohydrates, and [latex]25[/latex] 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.
Show Solution

ⓐ 72 calories/ounce ⓑ 3.87 grams of fat/ounce ⓒ 5.73 grams carbs/once ⓓ 3.33 grams protein/ounce
A [latex]16-ounce[/latex] chocolate mocha coffee with whipped cream contains [latex]470[/latex] calories, [latex]18[/latex] grams of fat, [latex]63[/latex] grams of carbohydrates, and [latex]15[/latex] 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 [latex]\text{3/1}[/latex] or [latex]1/3?[/latex] Explain your reasoning.
Answers will vary.
The parking lot at the airport charges [latex]{$0.75}[/latex] for every [latex]15[/latex] minutes. ⓐ How much does it cost to park for [latex]1[/latex] 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 [latex]4-ounce[/latex] cup of frozen yogurt and then went for a swim. The frozen yogurt had [latex]115[/latex] calories. Swimming burns [latex]422[/latex] calories per hour. For how many minutes should Kathryn swim to burn off the calories in the frozen yogurt? Explain your reasoning.
Answers will vary.
Mollie had a [latex]16-ounce[/latex] cappuccino at her neighborhood coffee shop. The cappuccino had [latex]110[/latex] calories. If Mollie walks for one hour, she burns [latex]246[/latex] calories. For how many minutes must Mollie walk to burn off the calories in the cappuccino? Explain your reasoning.

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Module 3: Rates, Ratios and Proportions

Write a Ratio as a Fraction

Write a Rate as a Fraction

Find Unit Rates

Exercise 1

Exercise 2

Find Unit Price

Exercise 1

Exercise 2

Exercise 3

Translate Phrases to Expressions with Fractions

Everyday Math

Everyday math

Writing Exercises

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