Beginning Algebra, 11th Edition

Solving a Problem about Work Rates

“If Joe can paint a house in 3 hr and Sam can paint the same house in 5 hr, how long

does it take for them to do it together?” (Source: The movie Little Big League.)

Step 1 Readthe problem again. We are looking for time working together.

Step 2 Assign a variable.Let the number of hours it takes Joe and Sam to

paint the house, working together.

Certainly, xwill be less than 3, since Joe alone can complete the job in

3 hr. We begin by making a table. Based on the preceding discussion, Joe’s

rate alone is job per hour, and Sam’s rate is job per hour.^1315

x=

EXAMPLE 3

470 CHAPTER 7 Rational Expressions and Applications

(continued)

From Little Big League

Fractional Part of

Time Working the Job Done When

Rate Together Working Together

Joe x

Sam x^1

5

(^1) x
5
1
3 x
1
(^3) Sum is 1
whole job.
Step 3 Write an equation.
Fractional part Fractional part 1 whole job.
done by Joe done by Sam

= 1

1

5

+ x

1

3

x

+ =

⎧⎪⎪⎪⎨⎪⎪⎪⎩ ⎧⎪⎪⎨⎪⎪⎩ ⎧⎪⎪⎨⎪⎪⎩ Together, Joe and Sam

complete 1 whole job. Add

their individual fractional

parts and set the sum equal

to 1.

Step 4 Solve. Multiply by the LCD, 15.

Distributive property

Combine like terms.

Divide by 8.

Step 5 State the answer.Working together, Joe and Sam can paint the house in

hr, or hr.

Step 6 Checkto be sure the answer is correct. NOW TRY

NOTE An alternative approach in work problems is to consider the part of the job

that can be done in 1 hr. For instance, in Example 3Joe can do the entire job in 3 hr

and Sam can do it in 5 hr. Thus, their work rates, as we saw in Example 3,are and

respectively. Since it takes them xhours to complete the job working together, in 1 hr

they can paint of the house.^1 x

1

5 ,

1

3

(^158178)
x=

15

8

8 x= 15

5 x+ 3 x= 15

15 a

1

3

xb + 15 a

1

5

xb = 15112

15 a

1

3

x+

1

5

xb = 15112

NOW TRY
EXERCISE 3
Sarah can proofread a manu-
script in 10 hr, while Joyce
can proofread the same manu-
script in 12 hr. How long will
it take them to proofread the
manuscript if they work
together?

NOW TRY ANSWER

3.^6011 hr, or 5 115 hr

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