Work Word Problems

1. Doug Upp can dig a ditch in 4 hours. Mark Upp can dig the same ditch in 3 hours. How long (in hours) would it take them to dig it together?

Solution

Let x = numbers of hours to dig the ditch together.

	Total Time in hours	Fractional part of job done in 1 hour
Doug	4	1/4
Mark	3	1/3
Together	x	1/x

If Doug takes 4 hours to dig the ditch, he can dig one-fourth of it in 1 hour. Mark can dig one-third of it in 1 hour. If it takes them x hours to dig it together, they can dig 1/x part of it in 1 hour together. The total of the fractional part each can dig or one-third plus one-forth equals the fractional part they can dig together in 1 hour or 1/x.

1/3 + 1/4 = 1/x

Multiply by the LCD, 12x, to clear fractions.

4x + 3x = 12

7x = 12

x = 1 5/7

So, it will take them 1 5/7 hours to dig the ditch together.

2. If Butch Err can type a paper in 5 hours and together he and Janet can type it in 2 hours, how long would it take Janet type the same paper alone?

Solution

Let x = numbers of hours for Jim to type alone.

	Total Time in hours	Fractional part of job done in 1 hour
Butch Err	5	1/5
Janet	x	1/x
Together	2	1/2

1/5 + 1/x = 1/2

Multiply by the LCD, 10x, to clear fractions.

2x + 10 = 5x

-3x = -10

x = 3 1/3

So, it would take 3 1/3 hours for Janet to type it alone.

3. Two women paint a barn. Barbara can paint it alone in 5 days, Sara in 8 days. They start to paint it together, but after 2 days Sara gets bored and Barbara finishes alone. How long does it take her to finish?

Solution

All work problems need diagrams. In the first column, show the total time for each person to do the job. Then in the next column show the fractional part of the job each person can do in one unit of time.

Let x = numbers of days for Barbara to finish barn.

 

	Total Time in days	Fractional part of job done in 1 day
Barbara	5	1/5
Sara	8	1/8

 

Next, for the third column multiply the time each person actually works by the column marked “Fractional part of job done in 1 day.” The resulting fraction in the third column is the part of the job each person does in the time worked.

 

	Total time in days	Fractional part of job done in 1 day	Fractional part of job does in 2 days	Fractional part Barbara does in x days
Barbara	5	1/5	2/5	x/5
Sara	8	1/8	2/8

Barbara does 2/5, Sara 2/8, and Barbara x/5 of the job. Add each part and you get  one complete job, equal to 1.

 

2/5 + 2/8 + x/5  = 1

Multiply by the LCD, 40, to clear fractions.

16 + 10 + 8x = 40

8x + 26 = 40

8x = 14

x = 1 3/4

So, it will take Barbara 1 3/4 hours to finish alone.

4. A swimming pool can be filled by an inlet pipe in 10 hours and emptied by an outlet pipe in 12 hours. One day the pool is empty and the owner opens the inlet to fill the pool. But he forgets to close the outlet. With both pipes open, how long will it take to fill the pool?

Solution

This time we have two pipes whose flows work against each other. The filling takes place faster than the emptying. So in any given time, the part of the job done is equal to the amount filled minus the amount emptied.

Let x = number of hours to fill pool with both pipes open.

	Total time in hours	Fractional part filled or emptied in 1 hour	Fractional part each pipe does in x  hours
Inlet pipe	10	1/10	x/10
Outlet pipe	12	1/12	x/12

Amount filled minus amount emptied equals whole job.

Equation

x/10 – x/12 = 1

Multiply by the LCD, 60, to clear fractions.

6x – 5x = 60

Answer

x = 60

 

5. Manny, Moe, and Jack work at Pepboys. If each worked alone, it would take Manny 10 hours, Moe 8 hours, and Jack 12 hours alone. One day Manny came to work early and he had worked for 2 hours when Moe and Jack arrived and all three finished the job. How long did they take to finish?

Solution

Let x = number of hours they took to finish.

	Total time in hours.	Fractional part of job done in 1 hour.	Fractional part Manny works in 2 hours.	Fractional part each works in x hours.
Manny	10	1/10	2/10	x/10
Moe	8	1/8		x/8
Jack	12	1/12		x/12

The fraction parts of the job each does added together equal the total job or 1.

Equation

2/10 + x/10 + x/8 + x/12 + 1

Multiply by the LCD, 120.

24 + 12x + 15x + 10x = 120

37x + 96

Answer

x = 2 22/37

 

6. Tim, Dick, and Harry decided to fence a vacant lot adjoining their properties. If it would take Tom 4 days to build the fence, Dick 3 days, and Harry 6 days, how long would it take them working together?

Solution

Let x = number of days to build fence together

	Total time in days	Fractional part of job in 1 day
Tom	4	1/4
Dick	3	1/3
Harry	6	1/6
Together	x	1/x

The fractional parts each person does in 1 day added together equal the fractional part done together in 1 day.

Equation

1/4 + 1/3 + 1/6 = 1/x

3x + 4x + 2x = 12

9x = 12

Answer

x = 1 1/3

 

 

7. John and Charles are linemen. John can string 10 miles of line in 3 days, while together they can string it in 1 day. How long would it take Charles alone to string the same line?

Solution

Let x = number of days for Charles to string line alone.

	Total time in days	Fractional part of job done in 1 day.
John	3	1/3
Charles	x	1/x
Together	1	1/1

Equation

1/3 + 1/x = 1/1

x + 3 = 3x

-2x = -3

Answer

x = 1 1/2 days

 

8. A firm advertises for workers to address envelopes. Patty says she will work 1OO hours. Herb will work for 80 hours. If each can address 10,000 envelopes in the time they work, how long would it take them to address 10,000 envelopes if they work together?

Solution

Let x = number of hours to address envelopes together.

 

	Total time in hours.	Fractional part of job done in 1 hour.
Patty	100	1/100
Herb	80	1/80
Together	x	1/x

Equation

1/100 + 1/80 = 1/x

Multiply by the LCD, 400x.

4x + 5x = 4009x = 400

Answer

x = 44 4/9 hours.

 

 

9. Jimmy D Lock cataloged a group of incoming library books in 8 hours. Eva Lu Shun can catalog the same number of books in 4 hours. Phil T Hans can catalog them in 6 hours. How long would it take if they all worked together?

Solution

Let x = number of hours for all three working together to catalog books.

	Total time in hours	Fractional part of job in 1 hour
Jimmy D Lock	8	1/8
Eva Lu Shun	4	1/4
Phil T Hans	6	1/6
Together	x	1/x

The total of the fractional parts of the job done by each in 1 hour equals the fractional part done by all of them working together.

Equation

1/8 + 1/4 + 1/6 = 1/x

Multiply by the LCD, 24x, to clear fractions.

3x + 6x + 4x = 24

13x = 24

Answer

x = 1 11/13

 

 

10. One machine can complete a job in 10 minutes. If the same job is done by this machine and an older machine working together, the job can be completed in 6 minutes. How long would it take the older machine to do the job alone?

Solution

Let x = number of minutes for older machine to do job.

	Total Time in Minutes	Fractional part of job done in 1 minute
First Machine	10	1/10
Older Machine	x	1/x
Together	6	1/6

Equation

1/10 + 1/x = 1/6

Multiply by the LCD, 30x, to clear fractions.

3x + 30 = 5x

-2x = -30

Answer

x = 15

 

 

11. A pond is being drained by a pump. After 3 hours, the pond is half empty. A second pump is put into operation, and together the two pumps finish emptying the pond in half an hour. How long would it take the second pump to drain the pond if it had to do the same job alone?

Solution

If the first pump drains half the pond in 3 hours, it could drain the pond completely in 6 hours. If the two pumps grain the last half of the pond in half an hour, together they could drain the whole pond in 1 hour.

Let x = number of hours it would take second pump to empty pond.

	Total time in hours	Fractional part of job done in 1 hour
First Pump	6	1/6
Second Pump	x	1/x
Together	1	1/1

Equation

1/6 + 1/x = 1/1

Multiply by the LCD, 6x, to clear fractions.

x + 6 = 6x

-5x = -6

Answer

x = 1 1/5

 

 

12. Eb and Flo are bricklayers. Eb can lay bricks for a fireplace and chimney in 5 days. With Flo’s help he can build it in 2 days. How long would it take Flo to build it alone?

Solution

Let x = number of days it would take father alone.

	Total time in days	Fractional part of job done in 1 day
Eb	5	1/5
Flo	x	1/x
Together	2	1/2

The fractional part Jim can do plus the fractional part his father can do equal the fractional part they can do together.

Equation

1/5 + 1/x = 1/2

Multiply by the LCD, 10x, to clear fractions

2x + 10 = 5x

-3x = -10

Answer

x = 3 1/3

 

13. Lotta Spences can paint a car in 8 hours. Ernest Worker can paint the same car in 6 hours. They start to paint the car together. After 2 hours, Lotta leaves for lunch and Ernest finishes painting the car alone. How long does it take Ernest to finish?

Solution

Let x = number of hours it would take Lotta to finish.

	Total time to do job in hours.	Fractional part of job done in 1 hour.	Fractional part each does in 2 hours.	Fractional part of job done in x hours.
Lotta	8	1/8	2(1/8)	—
Ernest	6	1/6	2(1/6)	x(1/6)

 

Lotta and Ernest each work for 2 hours. In that time each did two times the fractional part he could do in 1 hour or 2(1/8) and 2(1/6). Lotta left and only Ernest worked for x hours, so he did x(1/6) of the job. When the job is completed, all the fractional parts of the job done add up to 1.

Equation

2(1/8) + 2(1/6) + x (1/6) = 1

2/8 + 2/6 + x/6 = 1

Reduce fractions

1/4 + 1/3 + x/6 = 1

Multiply by the LCD, 12.

3 + 4 + 2x = 12

2x + 7 = 12

2x = 5

x = 2 1/2

Answer

x = 2 1/2 hours

 

 

14. Bill, Bob, and Barry are hired to paint signs. In 8 hours Bill can paint I sign, Bob can paint 2 signs, and Barry can paint 1 1/3 signs. They all come to work the first day, but Barry doesn’t like the job and quits after 3 hours. Bob works half an hour longer than Barry and quits. How long will it take Bill to finish the two signs they were supposed to paint?

Solution

Let x = number of hours it would take Bill to finish two sign.

The total job in this problem is painting two signs. So the total time for each person is the time it takes him to paint two signs. Bill can paint one sign in 8 hours or two signs in 16 hours. Bob can paint two signs in 8 hours. Barry can paint 1 1/3 signs in 8 hours.

8 ÷ 1 1/3 = 8 × 3/4 = 6 hours to paint one sign.  Barry can paint two signs in 12 hours.

	Time to paint two signs in hours	Fractional part of job done in 1 hour	Fractional part each does in 3 hours.	Fractional part each does in 1/2 hours.	Fractional part each does in x hours.
Bill	16	1/16	3(1/16)	(1/2)(1/16)	x(1/16)
Bob	8	1/8	3(1/8)	(1/2)(1/8)
Barry	12	1/12	3(1/12)

Equation

3/16 + 3/8 + 3/12 + 1/32 + 1/16 + x/16 = 1

Multiply by the LCD, 96

18 + 36 + 24 + 3 + 6 + 6x = 96

90 + 6x = 96

6x = 9

x = 1 1/2

Answer

x = 1 1/2

 

 

15. Art Tillery has to grind the valves on his Mercedes. He estimates it will take him 8 hours. His friends Moe Tell and Fran Tick have done the same job in 10 and 12 hours, respectively. Art had worked for 2 hours when Moe came along and helped him for 2 hours. Moe worked another half an hour alone and then Fran came along and together they finished the job. How long did it take them to finish?

Solution

Let x = number of hours Moe and Fran took to finish the job.

Fractional part each person does in:

 

Fractional part each person does in:

	Total time in hours	Fractional part of job done in 1 hour.	2 hours	2 hours	1/2 hour	x hours
Art Tillery	8	1/8	2(1/8)	2(1/8)
Moe Tell	10	1/10		2(1/10)	1/2(1/10)	x(1/10)
Fran Tick	12	1/12				x(1/12)

Equation

The total of all fractional parts of the job equals 1.

2(1/8) + 2(1/8) + 2(1/10) + 1/2(1/10) + x(1/10) + x(1/12 = 1

1/4 + 1/4 + 1/5 + 1/20 + x/10 + x/12 = 1

Multiply by the LCD, 60, to clear fractions.

15 + 15 + 12 + 3 + 6x + 5x = 60

11x = 15

Answer

x =1 4/11 hours

 

 

16. Mr. Caleb hires Jay and Debbie to prune his vineyard. Jay can prune the vineyard in 50 hours, and Debbie can prune it in 40 hours. How many hours will it take them working together?

Solution

Let x = number of vines Jay and Debbie can prune together in 1 hour.

	Total time to do job in hours.	Fractional part of job done in 1 hour.
Jay	50	1/50
Debbie	40	1/40
Together	x	1/x

Equation

The fractional parts of the job each can do in 1 hour equals the fractional part they can do together.

1/50 + 1/40 = 1/x

Multiply by the LCD (200x) to clear fractions.

4x + 5x = 200

9x = 200

Answer

x = 22 2/9 hours

 

 

17. A high school hiking club held a car wash to raise money for equipment. Rudy, Cheryl, Tom, and Pat volunteered to help. Rudy could wash a car in 10 minutes, Cheryl in 12,Tom in 8, and Pat in 15. They all started on the first car, but after 2 minutes another car came in and Pat and Tom went to work on it. One minute later Rudy quit to take care of another customer. How long did it take Cheryl to finish the first car alone?

Solution

Let x = number of minutes Cheryl took to finish first car

	Total time in minutes	Fractional part of job done in 1 minute.	2 minutes	1 minutes	x minutes
Ruby	10	1/10	2(1/10)	1/10
Cheryl	12	1/12	2(1/12)	1/12	x(1/12)
Tom	8	1/8	2(1/8)
Pat	15	1/15	2(1/15)

Equation

The sum of the fractional parts of the job done equals 1.

2(1/10) + 2(1/12) + 2(1/8) + 2(1/15) + 1/10 + 1/12 + x/12 = 1

1/5 + 1/6 + 1/4 + 1/15 + 1/10 + 1/12 + x/12 = 1

Multiply by the LCD, 60, to clear fractions.

12 + 10 + 15 + 8 + 6 + 5 + 5x = 60

5x = 4

Answer

x = 4/5 hour

 

 

18. Harry can paint a room in 3 hours, and Kerry can paint it in 4 hours. How long will it take if they work together?

Solution

Let x = numbers of hours to paint a room together.

	Total Time in hours	Fractional part of job done in 1 hour
Harry	3	1/3
Kerry	4	1/4
Together	x	1/x

If Harry takes 3 hours to paint a room, he can dig one-third of it in 1 hour. Kerry can paint one-forth of it in 1 hour. If it takes them x hours to paint it together, they can paint 1/x part of it in 1 hour together. The total of the fractional part each can paint or one-third plus one-forth equals the fractional part they can paint together in 1 hour or 1/x.

1/3 + 1/4 = 1/x

Multiply by the LCD, 12x, to clear fractions.

4x + 3x = 12

7x = 12

x = 1 5/7

So, it will take them 1 5/7 hours to paint the ditch together.

 

 

19. Matthew can build a block wall in 3 days. Andy can build the wait in 5 days. How long will it take if they work together?

Solution

Let x = numbers of hours to build a block wall together.

	Total Time in days	Fractional part of job done in 1 day
Matthew	3	1/3
Andy	5	1/5
Together	x	1/x

If Matthew takes 3 days to build a block wall, he can build one-third of it in 1 day. Andy can build one-fifth of it in 1 day. If it takes them x days to build it together, they can dig 1/x part of it in 1 day together. The total of the fractional part each can build or one-third plus one-forth equals the fractional part they can build together in 1 day or 1/x.

1/3 + 1/5 = 1/x

Multiply by the LCD, 15x, to clear fractions.

5x + 3x = 15

8x = 15

x = 1 8/15

So, it will take them 1 8/15 days to dig the ditch together.

 

 

20. Pump A can fill a tank in 8 hours. Pump B can fill the tank in 6 hours. How long will it take to fill the tank using both pumps?

Solution

Let x = numbers of hours to fill the tank together.

	Total Time in hours	Fractional part of job done in 1 hour
Pump A	8	1/8
Pump B	6	1/6
Together	x	1/x

If Pump A takes 8 hours to fill a tank, it can fill one-eighth of it in 1 hour. Pump B can fill one-sixth of it in 1 hour. If it takes them x hours to fill it together, they can fill 1/x part of it in 1 hour together. The total of the fractional part each can fill or one-eighth plus one-sixth equals the fractional part they can dig together in 1 hour or 1/x.

1/8 + 1/6 = 1/x

Multiply by the LCD, 24x, to clear fractions.

3x + 4x = 24

7x = 24

x = 3 3/7

So, it will take them 3 3/7 hours to fill the tank together.

 

 

21. To do a job alone, it would take Jennifer 5 hours, Bob 8 hours, and George 10 hours. How long would it take if they all work together? .

Solution

Let x = number of days to do a job together

	Total time in days	Fractional part of job in 1 day
Jennifer	5	1/5
Bob	8	1/8
George	10	1/10
Together	x	1/x

The fractional parts each person does in 1 day added together equal the fractional part done together in 1 day.

Equation

1/5 + 1/8 + 1/10 = 1/x

Multiply through by 40x

8x + 5x + 4x = 40

17x = 40

Answer

x = 2 6/40 or 2 3/20

So, it will take them 2 3/20 hours to do the job together.

 

 

22. Susan and Mary working together can rake a lawn in 2 hours. Susan can do the job alone in 3 hours. How long would it take Mary lo rake the lawn alone?

Solution

Let x = numbers of hours to rake a lawn together.

	Total Time in hours	Fractional part of job done in 1 hour
Susan	2	1/2
Mary	3	1/3
Together	x	1/x

If Susan takes 4 hours to rake the lawn, he can rake one-half of it in 1 hour. Mary can rake one-third of it in 1 hour. If it takes them x hours to rake it together, they can rake 1/x part of it in 1 hour together. The total of the fractional part each can rake or one-half plus one-third equals the fractional part they can rake together in 1 hour or 1/x.

1/2 + 1/3 = 1/x

Multiply by the LCD, 6x, to clear fractions.

3x + 2x = 6

5x = 6

x = 5/6

So, it will take them 5/6 hours to rake the lawn together.

 

 

 

23. Pipe A can empty a pool in 8 hours. It Pipe B is also used, the pool can be emptied in 3 hours. How long would it take Pipe B, by itself, to empty the pool?

Solution

Let x = numbers of hours for Pipe B to fill the pool by itself.

	Total Time in hours	Fractional part of job done in 1 hour
Pipe A	8	1/8
Pipe B	x	1/x
Together	3	1/3

If Pipe A takes 8 hours to empty a pool, he can empty one-eight of it in 1 hour. Pipe B can empty one-x of it in 1 hour. If it takes them 3 hours to empty it together, they can empty 1/3 part of it in 1 hour together. The total of the fractional part each can dig or one-eighth plus one-x equals the fractional part they can dig together in 1 hour or 1/3.

1/8 + 1/x = 1/3

Multiply by the LCD, 24x, to clear fractions.

3x + 24 = 8x

5x = 24

x = 4 4/5

So, it will take Pipe B 4 4/5 hours to empty the pool by itself.

 

24. Noah can build an ark in 40 days. Together, Noah and his wife can build the ark in 24 days. How long would it take Noah’s wife working alone?

Solution

Let x = numbers of days for Noah’s wife to build the ark alone.

	Total Time in days	Fractional part of job done in 1 day
Noah	40	1/40
Wife	x	1/x
Together	24	1/24

If Noah takes 40 days to build an ark, he can build one-fortieth of it in 1 day. His wife can build one-x of it in 1 day. If it takes them x days to build it together, they can build 1/24 part of it in 1 day together. The total of the fractional part each can build or one-fortieth plus one-x equals the fractional part they can build together in 1 day or 1/24.

1/40 + 1/x = 1/24

Multiply by the LCD, 120x, to clear fractions.

3x + 120 = 5x

2x = 120

x = 60

So, it will take Noah’s wife 60 days to build the ark alone.