## Find Unit Rates

In the last example, we calculated that Bob was driving at a rate of (dfrac{175; miles}{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**.

Definition: 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 hour | $$dfrac{68; miles}{1; hour}$$ | 68 miles/hour | 68 mph | 68 miles per hour |

36 miles to 1 gallon | $$dfrac{36; miles}{1; gallon}$$ | 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 (PageIndex{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. | $$dfrac{$384}{32; hours}$$ |

Divide the numerator by the denominator. | $$dfrac{$12}{1; hour}$$ |

Rewrite as a rate. | $12 / hour |

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

Exercise (PageIndex{13}):

Find the unit rate: $630 for 35 hours.

- Answer
$18 / hour

Exercise (PageIndex{14}):

Find the unit rate: $684 for 36 hours.

- Answer
$19 / hour

Example (PageIndex{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.

Write as a rate. | $$dfrac{455; miles}{14; gallons}$$ |

Divide 455 by 14 to get the unit rate. | $$dfrac{32.5; miles}{1; gallon}$$ |

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

Exercise (PageIndex{15}):

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

- Answer
23.5 mpg

Exercise (PageIndex{16}):

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

- 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.

Definition: unit price

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

Example (PageIndex{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. | $$dfrac{$3.99}{24; bottles}$$ |

Divide to find the unit price. | $$dfrac{$0.16625}{1; bottle}$$ |

Round the result to the nearest penny. | $$dfrac{$0.17}{1; bottle}$$ |

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

Exercise (PageIndex{17}):

Find the unit price. Round your answer to the nearest cent if necessary: 24-pack of juice boxes for $6.99

- Answer
(dfrac{$0.29}{1; box})

Exercise (PageIndex{18}):

Find the unit price. Round your answer to the nearest cent if necessary: 24-pack of bottles of ice tea for $12.72

- Answer
(dfrac{$0.53}{1; 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 (PageIndex{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. | $$dfrac{$14.99}{64; loads}$$ | $$dfrac{$15.99}{80; loads}$$ |

Find the unit price. | $$dfrac{$0.234 ldots}{1; load}$$ | $$dfrac{$0.199 ldots}{1; load}$$ |

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.

Exercise (PageIndex{19}):

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

Brand A Storage Bags, $4.59 for 40 count, or Brand B Storage Bags, $3.99 for 30 count

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

Exercise (PageIndex{20}):

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

Brand C Chicken Noodle Soup, $1.89 for 26 ounces, or Brand D Chicken Noodle Soup, $0.95 for 10.75 ounces

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

Notice in Example (PageIndex{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.

## 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 (PageIndex{11}):

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

**Solution**

(a) 427 miles per h hours

Write as a rate. | $$dfrac{427; miles}{h; hours}$$ |

(b) x students to 3 teachers

Write as a rate. | $$dfrac{x; students}{3; teachers}$$ |

(c) y dollars for 18 hours

Write as a rate. | $$dfrac{$y}{18; hours}$$ |

Exercise (PageIndex{21}):

Translate the word phrase into an algebraic expression. (a) 689 miles per h hours (b) y parents to 22 students (c) d dollars for 9 minutes

- Answer a
(dfrac{689; mi}{h; hours})

- Answer b
(dfrac{y; parents}{22; students})

- Answer c
(dfrac{$d}{9; min})

Exercise (PageIndex{22}):

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

- Answer a
(dfrac{m; mi}{9; h})

- Answer b
(dfrac{x; students}{8; buses})

- Answer c
(dfrac{$y}{40; h})

## Practice Makes Perfect

### Write a Ratio as a Fraction

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

- 20 to 36
- 20 to 32
- 42 to 48
- 45 to 54
- 49 to 21
- 56 to 16
- 84 to 36
- 6.4 to 0.8
- 0.56 to 2.8
- 1.26 to 4.2
- (1 dfrac{2}{3}) to (2 dfrac{5}{6})
- (1 dfrac{3}{4}) to (2 dfrac{5}{8})
- (4 dfrac{1}{6}) to (3 dfrac{1}{3})
- (5 dfrac{3}{5}) to (3 dfrac{3}{5})
- $18 to $63
- $16 to $72
- $1.21 to $0.44
- $1.38 to $0.69
- 28 ounces to 84 ounces
- 32 ounces to 128 ounces
- 12 feet to 46 feet
- 15 feet to 57 feet
- 246 milligrams to 45 milligrams
- 304 milligrams to 48 milligrams
- total cholesterol of 175 to HDL cholesterol of 45
- total cholesterol of 215 to HDL cholesterol of 55
- 27 inches to 1 foot 430. 28 inches to 1 foot

### Write a Rate as a Fraction

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

- 140 calories per 12 ounces
- 180 calories per 16 ounces
- 8.2 pounds per 3 square inches
- 9.5 pounds per 4 square inches
- 488 miles in 7 hours
- 527 miles in 9 hours
- $595 for 40 hours
- $798 for 40 hours

### Find Unit Rates

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

- 140 calories per 12 ounces
- 180 calories per 16 ounces
- 8.2 pounds per 3 square inches
- 9.5 pounds per 4 square inches
- 488 miles in 7 hours
- 527 miles in 9 hours
- $595 for 40 hours
- $798 for 40 hours
- 576 miles on 18 gallons of gas
- 435 miles on 15 gallons of gas
- 43 pounds in 16 weeks
- 57 pounds in 24 weeks
- 46 beats in 0.5 minute
- 54 beats in 0.5 minute
- The bindery at a printing plant assembles 96,000 magazines in 12 hours. How many magazines are assembled in one hour?
- 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.

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

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

### Translate Phrases to Expressions with Fractions

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

- 793 miles per p hours
- 78 feet per r seconds
- $3 for 0.5 lbs.
- j beats in 0.5 minutes
- 105 calories in x ounces
- 400 minutes for m dollars
- the ratio of y and 5x
- the ratio of 12x and y

## Everyday Math

- One elementary school in Ohio has 684 students and 45 teachers. Write the student-to-teacher ratio as a unit rate.
- The average American produces about 1,600 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 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.
- 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

- Would you prefer the ratio of your income to your friend’s income to be 3/1 or 1/3? Explain your reasoning.
- The parking lot at the airport charges $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?
- 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.
- Mollie had a 16-ounce cappuccino at her neighborhood coffee shop. The cappuccino had 110 calories. If Mollie walks for one hour, she burns 246 calories. For how many minutes must Mollie walk to burn off the calories in the cappuccino? Explain your reasoning.

## Self Check

(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

(b) After reviewing this checklist, what will you do to become confident for all objectives?