Work Problems

Mrs. Napholtz's Math Site

Work Problems

General Information Needed to Solve Work Problems

rate X time = work
When a job is complete, the amount of work done is 1 whole job.
You may use minutes instead of hours, but use the same units throughout the problem.
The time it takes an individual to do the job alone is used for their rate.
Note: If it takes a person n hours to complete a job, they do 1/n of the job in 1 hour, so their rate is 1/n.
Examples:
- If it takes Sam 4 hours to do the job, use 1/4 as his rate.
- If it takes Mary 24 minutes to do the job, use 1/24 as her rate.
The time noted in the "time" column represents the time actually spent as the people do the job together.

Solved Problems

Pipe A can fill a tank in 4 h. Pipe B can fill it in 9 h less than the time it takes pipe C, a drain pipe to empty the tank. When all 3 pipes are open, it takes 2 h. to fill the tank. How much time is required for pipe C to empty the tank if pipes A and B are closed?

	Rate	Time	Work
Pipe A	1/4	2	1/2
Pipe B	1/(n - 9)	2	2/(n - 9)
Pipe C	1/n	2	2/n

Note: Pipes A and B do "positive" work, while Pipe C does "negative" work.
1/2 + 2/(n - 9) - 2/n = 1
Multiply by 2n(n - 9) to clear the fractions:
n(n - 9) + 4n - 4(n - 9) = 2n(n - 9)
n² - 9n + 4n - 4n + 36 = 2n² - 18n
-n² + 9n + 36 = 0
n² - 9n - 36 = 0
(n - 12)(n + 3) = 0
n = 12 is the only solution.

One computer can process a bank's statements in 30 minutes, and a second computer can process the same number of statements in 20 minutes. How long would it take to process the bank's statements using both computers together?

	Rate	Time	Work
One Computer	1/30	n	n/30
Other Computer	1/20	n	n/20

Note: The amount of work done by the two computers combined must equal 1 whole job.
n/30 + n/20 = 1 Multiply by 60.
2n + 3n = 60
5n = 60
n = 12
Answer: 12 minutes

Paul can wash and wax a car in 4 hours. Chris can do the same job in 3 hours. How long would it take them to complete the job if they work together?

	Rate	Time	Work
Paul	1/4	n	n/4
Chris	1/3	n	n/3

Note: The amount of work done by Paul and the amount of work done by Chris combined must equal 1 whole job.
n/4 + n/3 = 1
3n/12 + 4n/12 = 1
7n/12 = 1 Multiply by 12/7.
n = 12/7
Answer: 1 and 5/7 hours, or approx. 1 hours and 43 minutes.

One printing press can finish a job in 8 hours. The same job would take a second press 12 hours. How long would it take both presses together?

	Rate	Time	Work
One press	1/8	n	n/8
The other press	1/12	n	n/12

Note: The amount of work done by the presses combined must equal 1 whole job.
n/8 + n/12 = 1
3n/24 + 2n/24 = 1
5n/24 = 1 Multiply by 24/5.
n = 24/5
Answer: 4 and 4/5 hours, or 4 hours and 48 minutes.

An installer can carpet a room in 3 hours. An assistant takes 4.5 hours to do the same job. If the assistant helps for 1 hour and then is called away, how long will it take the installer to finish?

Note: 4.5 hours = 9/2 hours. 1/(9/2) is equal to 2/9 !!!!

	Rate	Time	Work
Installer	1/3	n	n/3
Assistant	2/9	1	2/9

Note: The assistant works for only 1 hour and completes 2/9 of the job.
n/3 + 2/9 = 1
3n/9 + 2/9 = 1
3n/9 = 7/9 Multiply by 9/3.
n = 7/3
By the time the job is done, the installer will have worked a total of 7/3 or 2 and 1/3 hours.
Answer: Since the assistant works for the first hour, the installer will take an additional 1 and 1/3 hour or 1 hour and 20 minutes to complete the job after the assistant leaves.

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