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Lesson
Multiplying and Dividing Rates 269
Lesson 5-
Rates can be multiplied or divided by other quantities. The units are multiplied as if they were numeric fractions. Keeping the units in the calculations can help you understand the meaning.
If you bought 2.5 pounds of fish at $8.49 a pound, it would cost
2.5 lb Ā· 8.49 ______dollars lb = 21.225 dollars, or $21.23.
Notice how the unit ālbā in ā2.5 lbā cancels the unit ālbā in the
denominator of ______dollars lb , so that the result is in dollars.
Conversion Rates
A conversion rate is a rate determined from an equality between two quantities with different units. For example, since 1 hour = 60 minutes, dividing one unit by the other equals 1. ______1 hour 60 min =^
______60 min 1 hour =^1
Both ______ 60 min1 hour and 60 min______ 1 hour are conversion rates. Multiplying a quantity
by a conversion rate does not change its value.
Example 1
On average, an adult human heart beats about 70 times per minute. At this rate, how many times does a heart beat in a 365-day year? Solution We begin with the given information and multiply by conversion rates until one minute is converted into one year.
70 _____ minutebeats = 70 _____ minutebeats Ā· ________60 minutes 1 hour Ā· 24 hours______ 1 day Ā· 365 days______ 1 year = 70 Ā· 60 Ā· 24 Ā· 365 beats_____ year
= 36,792,000 beats_____ year
Vocabulary conversion rate
BIG IDEA When rates are multiplied or divided, the unit of the answer follows rules of arithmetic on the units of the original rates.
Multiplying and
Dividing Rates
The Pike Place Market in Seattle, Washington, attracts 10 million visitors per year. Source: Pike Place Market
Find a pair of numbers x and y that satisfy the equation. a. 2 x + 3 y = 600 b. 2 x - 3y = 600 c. 20x + 30y = 600
Mental Math
270 Division and Proportions in Algebra
Rates are used in many formulas. One of the more important formulas is d = rt, which gives the distance d traveled by an object moving at a constant rate r during a time t. ( r is often called the speed.)
Example 2
Explain why a train traveling 68 mi__ hr is traveling about 100 ___ secft. Solution The goal is to convert miles to?^ and an hour to?^. Use conversion rates so that the units cancel in the numerator and denominator just like common factors. 68 mi_____ hr =^
68 mi_____ hr Ā·^
_____1 hr ?
______? 60 sec Ā·^
_____? 1 mi
=^68 Ā·^ ________? 60 Ā· 60
_____ft ?
=? So 68 mi__ hr is about 100 ___ secft.
QY
Using Reciprocal Rates
Sometimes a rate does not help to simplify the computation but its
reciprocal does. Guided Example 2 used ______ 60 min1 hr , while Example 1
used ______60 min 1 hr.
Remember the algebraic definition of division: __ a b = a Ā· 1 __ b. So
to divide by a rate, you can multiply by its reciprocal. For
example, if a book has 672 pages and Marta reads an average
of 35 pages per day, you can use division. ________672 pages 35 pages_______ day
= 672 pages Ā· __ 351 _____day pages =^ 19.2 days
So it will take her a little more than 19 days to finish reading
the book.
Not every reciprocal rate is meaningful in every situation.
For example, in a school with an average of 23 _______students class , the
equivalent reciprocal rate of __ 231 _______ studentsclass does not have much meaning.
Still, it might be useful in a computation.
Chapter 5
GUIDED
QY The International Sports Medicine Institute has a formula for daily water intake. A nonactive person should consume 1 __ 2 ounce per pound of body weight. An athletic person should consume 2 __ 3 ounce per pound of body weight. (Note: 1 cup = 8 ounces.) If a person weighs p pounds, how many cups of water should the person drink each day if he or she is: a. nonactive? b. athletic?
There are approximately 8,000 to 10,000 professional online book dealers, selling books with an average price range of $10ā$19 per book. Source: www.bookologist.com
272 Division and Proportions in Algebra
- Kenji can wash k dishes per minute. His sister Suna is twice as fast. a. How many dishes can Suna wash per minute? b. How many minutes does Suna spend per dish?
- Suppose Chip is baking cookies for a school bake sale. His oven bakes 36 cookies every 10 minutes. He wants to bake d dozen cookies for the sale. How long will it take him to bake all of the cookies?
- In 2006, Gordy Savela ran 300 miles across northern Minnesota in 12 days. a. What distance did Gordy average each day? b. If a marathon is 26.2 miles, how many marathons did Gordy run during his journey?
- People sometimes go for a walk after a big meal to burn off calories. To lose 1 pound, a person must burn 3,500 calories. Suppose a person burns about 300 calories per hour by walking. a. If a person walks for 2.5 hours, how many pounds will the person lose? b. If a person walks for 2.5 hours, how many ounces will the person lose? c. If a person walks for h hours, how many ounces will the person lose? d. If a person walks for m minutes, how many ounces will the person lose?
REVIEW
- Over the past five days, Mykia has run 3.6 miles one day, 2.9 miles each of two days, 4.2 miles one day, and 1.9 miles one day. Calculate the average number of miles run per day. Write your answer as a rate. (Lesson 5-3)
- For what value(s) of n is 19 ______ 19 +- 22 nn undefined? (Lesson 5-3)
- A box is __^13 as long, 1 __ 3 as wide, and __ 31 as high as a crate. How many of these boxes will fit in the crate? (Lesson 5-1)
In 20ā22, use the formula H = ________M^ +^ 2 F +^^5. It predicts the adult height H
of a boy based on his motherās height M and his fatherās height F,
where all measurements are given in inches. The formula
H = M________^ +^ 2 F -^^5 applies to the adult height of girls. (Lesson 4-7)
- Solve each formula for M.
Chapter 5
Bake sales have long been one of the most popular ways of raising funds for schools, social clubs, and other organizations.
Multiplying and Dividing Rates 273
- Bookerās fatherās height is 73 inches, and his motherās height is 68 inches. How tall does the formula predict Booker will be when he is an adult?
- Predict your adult height using this formula.
Multiple Choice In 23ā25, decide whether each equation has A no solution. B one solution. C more than one solution. (Lesson 4-5)
__ 5 x + 5 = 5
__ y 4 +^4 =^
__ y 4
- __ 3 z + 3 = (^3) ( __ z 3 + (^3) )
- Solve 8(ā3 d - 4) ā¤ 13(2 - d ) - (7 d - 2). (Lessons 4-4, 2-1)
EXPLORATION
- a. Here is a problem found in many puzzle books. If a hen and a half can lay an egg and a half in a day and a half, how long will it take 24 hens to lay 24 eggs? Explain how you got your answer. b. Generalize Part a. If a hen and a half can lay an egg and a half in a day and a half, how long will it take h hens to lay e eggs?
Lesson 5-
QY ANSWERS a. __p 16 c b. __p 12 c
