Ch 11: Rationals G) Work Word Problems Objective: To solve word problems involving people working together to complete a task.

Ch 11: RationalsG) Work Word Problems

Objective:

To solve word problems involving people working together to complete a task.

Jeff Hal

I can paint a room in 4 hours

That’s fast!It takes me

5 hrs

How long will it take to paint a room if we work together?

Demonstration

Jeff

1 hr 1 hr 1 hr 1 hr

I can get ¼ of the room

painted each hour that I work

I can paint a room in 4 hours

1 room4 hours

Work RateJeff =

Hal

I can get ⅕ of the room

painted each hour that I work

1 hr1 hr1 hr1 hr1 hr

It takes me 5 hrs to paint a room

1 room5 hours

Work RateHal =

Jeff Hal

How long will it take to paint a room if we work together?

I can get ¼ of the room

painted each hour that I work

I can get ⅕ of the room

painted each hour that I work

Work RateJeff

€

1

4

+

€

1

5

Work RateHal

= Work RateTogether

+ =

€

5

5

⎛

⎝ ⎜

⎞

⎠ ⎟

€

4

4

⎛

⎝ ⎜

⎞

⎠ ⎟

205 + 4

=

€

9

20per hr =

€

20

9hrs

1. Calculate the rate of work for each person.

2. Multiply the “work rate” by the total amount of “time” to determine the amount of “Work” each person contributes to the task.

3. Add the “Work” for each person and set that value equal to 1 task.

Use the table below to set up the equation.

Rules

Work Rate Time Work

×

×

=

=

Person 1+

=Person 2

1 task

Jeff can paint a room in 4 hours. It takes Hal 5 hours to paint the same room. How long will it take them if they work together?

Work Rate Time Work Done

Jeff

Hal

1415

x

x

14

x

15

x

1 room

Example 1

× =

× =

+15

x = 114

x

€

5

5

⎛

⎝ ⎜

⎞

⎠ ⎟

€

4

4

⎛

⎝ ⎜

⎞

⎠ ⎟

€

20

20

⎛

⎝ ⎜

⎞

⎠ ⎟

5 4 20x x+ =9 20x =

x =209

=229

hrs.

One water hose can fill a pool in 10 hours. A different hose only takes 6 hours. How long would it take if both hoses are used?

Work Rate Time Work Done

Hose 1

Hose 2

€

1

10

€

1

6

x

x

€

1

10x

€

1

6x

1 pool

Example 2

× =

× =

+

€

1

6x = 1

€

1

10x

€

3

3

⎛

⎝ ⎜

⎞

⎠ ⎟

€

5

5

⎛

⎝ ⎜

⎞

⎠ ⎟

€

30

30

⎛

⎝ ⎜

⎞

⎠ ⎟

3 5 30x x+ =8 30x = x =

308

=334

hrs.

Julie can complete a wedding cake in 8 hours. Marty can put one together in 10 hours. If Julie and Marty work together for 4 hours, how long will it take Julie to finish the job alone?

Work Rate Time Work Done

Hose 1

Hose 2

€

1

8

€

1

10

€

4 + x

8

€

4

101 cake

Example 3

× =

× =

4

4

+ x

+

€

4

10= 1

€

4 + x

8

€

5

5

⎛

⎝ ⎜

⎞

⎠ ⎟

€

4

4

⎛

⎝ ⎜

⎞

⎠ ⎟

€

40

40

⎛

⎝ ⎜

⎞

⎠ ⎟

€

20 + 5x +16 = 40

€

5x = 4

€

x =4

5

€

≈48min

Working alone, Matt can clean an attic in 11 hours. One day his friend Kim helped him and it only took 4.95 hours. How long would it take Kim to do it alone?

Example 4

Work Rate Time Work Done

Matt

Kim

€

1

11

€

1

x

€

4.95

11

€

4.95

x1 attic

× =

× =

4.95

4.95

+

€

4.95

x= 1

€

4.95

11

€

x

x

⎛

⎝ ⎜

⎞

⎠ ⎟

€

11

11

⎛

⎝ ⎜

⎞

⎠ ⎟

€

11x

11x

⎛

⎝ ⎜

⎞

⎠ ⎟

€

4.95x + 54.45 =11x

€

54.45 = 6.05x

€

x =54.45

6.05

€

=9hrs

Classwork

1) It takes Wilbur 9 hours to mop a warehouse. Bill can mop the same warehouse in 10 hours. How long will it take them if they work together?

Work Rate Time Work

×

×

=

=

_______

_______

€

x =90

19≈ 4.7hrs

2) Working alone, Ryan can pick 40 bushels of apples in 10 hours. Darryl can pick the same amount in 15 hours. How long will it take them if they work together?

Work Rate Time Work

×

×

=

=

_______

_______

€

x =60

10= 6hrs

3) It takes Ming 14 hours to tar a roof. Willie can tar the same roof in 8 hours. If they work together, how long will it take them?

Work Rate Time Work

×

×

=

=

_______

_______

€

x =56

11≈ 5.1hrs

4) Working alone, Scott can dig a 10’ x 10’ hole in 8 hours. Mark can dig the same hole in 9 hours. How long will it take if they work together?

Work Rate Time Work

×

×

=

=

_______

_______

€

x =72

17≈ 4.2hrs

5) Beth can oil the lanes in a bowling alley in 10 hours. One day her friend Shawna helped her and it only took 4.44 hours. How long would it take Shawna to do it alone?

Work Rate Time Work

×

×

=

=

_______

_______

€

x =44.4

5.56= 8hrs

6) Joe can tile the kitchen floor in 7 hours. His wife decided to help him and they got the job done in 3.93 hours. How long would it have taken his wife to do it by herself?

Work Rate Time Work

×

×

=

=

_______

_______

€

x =27.51

3.07= 9hrs