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Standards

15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.

When two people team up to complete a job, each person completes a fraction of the whole job. You can use this idea to write and solve a rational equation to find out how long it will take to complete a job.

Let h be the number of hours Jessica and her roommate need to clean the apartment. Jessica cleans the apartment in 5 hours, so she cleans of the apartment per hour. The roommate cleans the apartment in 4 hours, so she cleans of the apartment per hour.

Additional Example 1: Application

Jessica can clean her apartment in 5 hours. Her roommate can clean the apartment in 4 hours. How long will it take to clean the apartment if they work together?

Jessica’s part

Whole Apartment

=

+

+

=

1

+

=

1

Additional Example 1 Continued

The table shows the part of the apartment that each person cleans in h hours.

Solve this equation for h.

Working together Jessica and her roommate can clean the apartment in hours or about 2 hours 13 minutes.

Additional Example 1 Continued

Multiply both sides by the LCD, 20.

Distribute 20 on the left side.

5h + 4h = 20

Combine like terms.

9h = 20

Divide by 9 on both sides.

Jessica cleans of her apartment per hour, so in hours, she cleans of the apartment. Her roommate cleans of the apartment per hour, so in hours, she cleans of the apartment. Together, they clean apartment.

Additional Example 1 Continued

Check

Let m be the number of minutes Cindy and Sara need to mow the same lawn. Cindy mows the lawn in 50 minutes, so she mows of the lawn per minute. Sara mows the lawn in 40 minutes, so she mows of the lawn per minute.

Check It Out! Example 1

Cindy mows a lawn in 50 minutes. It takes Sara 40 minutes to mow the same lawn. How long will it take them to mow the lawn if they work together?

Sara’s part

Whole lawn

=

+

+

=

1

Check It Out! Example 1 Continued

The table shows the part of the lawn that each person mows in m hours.

Solve this equation for m.

Working together Cindy and Sara can mow the lawn in minutes or about 22 minutes and 13 seconds.

Check It Out! Example 1 Continued

Multiply both sides by the LCD, 200.

Distribute 200 on the left side.

4m + 5m = 200

9m = 200

Combine like terms.

Divide by 9 on both sides.

Cindy mows of the lawn per minute, so in minutes, she mows of the lawn. Sara mows of the lawn per minute, so in minutes, she mows of the lawn. Together, they mow lawn.

Check It Out! Example 1 Continued

Check

Total (oz)

Original Solution

10

20

New Solution

10 + p

20 + p

Additional Example 2: Application

Omar has 20 oz of a snack mix that is half peanuts and half raisins. How many ounces of peanuts should he add to make a mix that is 70% peanuts?

Let p be the number of ounces of peanuts that Omar should add.

The table shows the amount of peanuts and the total amount of the mixture.

The new mixture is 70% peanuts, so Solve the equation for p.

Omar should add oz of peanuts to the mixture.

Additional Example 2 Continued

10 + p = 0.7(20 + p)

Multiply both sides by 20 + p.

Distribute 0.7 on the right side.

10 + p = 14 + 0.7p

Subtract 10 from both sides and 0.7p from both sides.

0.3p = 4

Divide both sides by 0.3.

p = 13.33…

The new solution is 80% alcohol, so

Solve the equation for a.

Check It Out! Example 2

A chemist has 500 milliliters of a solution that is half alcohol. Suppose the chemist wants a solution that is 80% alcohol. How many milliliters of alcohol should he add in this case?

250 + a = 0.8(500 + a)

Multiply both sides by 500 + a.

250 + a = 400 + 0.8a

Distribute 0.8 on the right side.

Subtract 250 from both sides and 0.8a from both sides.

0.2a = 150

a = 750

Divide both sides by 0.2.

Rate (mi/h)

Distance (mi)

Amber

16

t

Dave

16

t + 2

Additional Example 3: Application

Amber runs along a 16-mile trail while Dave walks. Amber runs 4 mi/h faster than Dave walks. It takes Dave 2 hours longer than Amber to cover the 16 miles. How long does it take Amber to complete the trip?

Let t be the time it takes Amber to run the 16 miles.

Additional Example 3 Continued

Amber is 4 mi/h faster than Dave so,

Multiply both sides by the LCD.

Distribute t(t + 2) on the right side.

16t = (t + 2)16 + t(t + 2)(–4)

Simplify.

Distribute and multiply.

16t = 16t + 32 – 4t2– 8t

Subtract 16t from both sides and rearrange terms.

0 = –4t2– 8t + 32

Additional Example 3 Continued

0 = 4t2 + 8t –32

Multiply by –1.

0 = t2 + 2t –8

Divide both sides by 4.

0 = (t + 4)(t– 2)

Factor.

t = –4 or 2

Time must be nonnegative, so –4 is an extraneous solution.

Amber makes the trip in 2 hours.

Time (h)

Rate (mi/h)

Ryan

300

t + 1

Maya

300

t

Check It Out! Example 3

Ryan drives 10 mi/h slower than Maya, and it takes Ryan 1 hour longer to travel 300 miles. How long does it take Maya to make the trip?

Let t be the time it takes Maya to drive the 300 miles.

Check It Out! Example 3 Continued

Maya is 10 mi/h faster than Ryan so,

Multiply both sides by the LCD.

Distribute t(t + 1) on the left side.

300t + t(t + 1)10 = (t + 1)(300)

Simplify.

Distribute and multiply.

300t +10t2 + 10t = 300t + 300

Check It Out! Example 3 Continued

10t2 + 10t –300 = 0

Subtract 300t from both sides and 300 from both sides.

t2 + t –30 = 0

Divide both sides by 10.

Factor.

(t + 6)(t– 5) = 0

t = –6, 5

Time must be nonegative, so –6 is an extraneous solution.

Maya makes the trip in 5 hours.

Lesson Quiz

1. Pipe A can fill a storage tank in 40 minutes. Pipe B can fill the tank in 80 minutes. How long does it take to fill the tank using both pipes at the same time?

2. An 8 oz smoothie is made up of 50% strawberries and 50% yogurt. How many ounces of strawberries should be added to make a smoothie that is 60% strawberries?

2 oz

3. Carlos bikes 3 mi/h faster than Tim. It takes Tim 1 hour longer than Carlos to bike 36 miles. How long does it take Carlos to bike 36 miles?

3 h

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