# Solving Combined Work Algebra Problems | 7 Examples

TR Smith is a product designer and former teacher who uses math in her work every day.

Combined work problems in algebra ask how much time it takes for two or more people to complete a job working together, given the amount of time it takes for them to complete the job individually. Luckily there is a simple formula to solve all of these problems!

Suppose there are n people and each of them can complete a certain task in X1 minutes, X2 minutes, X3 minutes, ... Xn minutes respectively. The total number of minutes T it takes to complete the task if all of them work together is given by the equation

1/X1 + 1/X2 + 1/X3 + ... + 1/Xn = 1/T

or equivalently

T = [ 1/X1 + 1/X2 + 1/X3 + ... + 1/Xn ]^(-1)

The unit of time doesn't necessarily have to be minutes; it may be seconds, hours, days, etc. The important thing is to make sure each variable measures the same unit. In these equations, the terms 1/X1, 1/X2, 1/T are rates because they contain a time unit in the denominator. With rate problems, you can always add the rates to find the combined rate.

Here are some examples to help you solve a variety of combined work or combined rate math problems in algebra.

## Example 1: Painting Hotel Rooms

All the rooms in a hotel are identical. Working together, John and Frank can paint one room in one hour and 20 minutes. Working by himself, John can paint one room in three hours. How long would it take Frank to paint four rooms painting alone?

Let T be the total number of minutes it takes for John and Frank to paint one room together, J be the number of minutes it takes for John to complete a room by himself, and F be the number of minutes for Frank to complete a room by himself. From the problem we have T = 80 and J = 180. The combined work formula gives us

1/J + 1/F = 1/T
1/180 + 1/F = 1/80
1/F = 1/80 - 1/180
1/F = 5/720 = 1/144
F = 144 minutes
F = 2 hours and 24 minutes

If Frank can paint one room in two hours and 24 minutes, then he can paint four rooms in nine hours and 36 minutes.

## Example 2: Three People Working Together

It takes Sandra three hours to stuff 200 envelopes. Tim can stuff 200 envelopes in one and a half hours. Jenny can stuff 200 envelopes in one hour. If they all work together to stuff 200 envelopes, how long will it take?

Here we have S = 3 hours, T = 1.5 hours, and J = 1 hour. The combined rate formula gives us

1/S + 1/T + 1/J = 1/T
1/3 + 1/1.5 + 1/1 = 1/T
1/3 + 2/3 + 3/3 = 1/T
2 = 1/T
T = 1/2 hours

The three of them working together can complete the job in half an hour, or 30 minutes.

## Example 3: Percentages and Optimization

Nick can mow his family's big backyard in one hour using the good lawn mower; it takes his sister Jana twice as long to cut the grass with the same machine. With the crappy lawn mower, both Nick and Jana take 20% longer to cut the grass. This afternoon Nick and Jana are going to work together to mow the lawn. Who should use the good lawn mower and who should use the crappy lawn mower so that the job gets done as quickly as possible?

Case 1: Nick uses the good lawn mower and Jana uses the crappy one. In this scenario, N = 60 minutes and J = 144 minutes. This gives us

1/N + 1/J = 1/T
1/60 + 1/144 = 1/T
17/720 = 1/T
T = 43.35 minutes

Case 2: Nick uses the crappy mower and Jana uses the good one. In this scenario N = 72 minutes and J = 120 minutes. This gives us

1/N + 1/J = 1/T
1/72 + 1/120 = 1/T
8/360 = 1/T
T = 45 minutes

By analyzing the two cases we can see that the job gets done slightly faster if Nick uses the good lawn mower and his sister uses the crappy one.

## Example 4: Four People Working Together

Keisha, Clive, Minhee, and Juan are teaching assistants for a college biology course. The course coordinator needs them to write 200 multiple-choice questions for the online quiz bank. Since Keisha and Clive are native English speakers, it takes each of them 3 hours to write to write 200 questions, working individually. Since Minhee and Juan are not native speakers, it takes them 6 hours and 5 hours respectively to write the questions, working individually. If the four of them meet to write all the questions in one sitting, how long will it take? Approximately how many questions does each write?

1/K + 1/C + 1/M + 1/J = 1/T
1/3 + 1/3 + 1/6 + 1/5 = 1/T
10/30 + 10/30 + 5/30 + 6/30 = 1/T
31/30 = 1/T
T = 30/31 hours
T = 0.9677 hours
T 58 minutes.

Therefore, it takes the group a total of 58 minutes to write all 200 questions.

As for how many questions each contributes, Keisha and Clive both contribute 10/31 of the questions, Minhee contributes 5/31 of them, and Juan 6/31. This works out to about 64.5 questions for each of Keisha and Clive, about 32 questions for Minhee, and about 39 questions for Juan. You can check that 64.5 + 64.5 + 32 + 39 = 200.

## Example 5: Tangled Rates

Mark and Nina can complete a task in three and a half hours if they work together. Nina and Paul can complete the same task in three hours if they work together. Paul and Mark can complete the same task in five and a half hours working together. How long does it take each person individually?

This problem gives us a system of three equations in three variables

1/M + 1/N = 1/3.5
1/N + 1/P = 1/3
1/P + 1/M = 1/5.5

If we subtract the third equation from the second we get

1/N - 1/M = 1/3 - 1/5.5

If we add this new equation to the first, we get

2/N = 1/3.5 + 1/3 - 1/5.5
N = 462/101 hours

Using the value of N, we can solve for M and P using the first and second equations. This gives us

M = 462/31 hours
P = 462/53 hours

Converting these improper fraction hours to minutes gives us

Nina's time ≈ 274 minutes
Mark's time ≈ 894 minutes
Paul's time ≈ 523 minutes

## Example 6: Solving a Cubic Equation

Jessica can sort a big tub of LEGOs 21 minutes faster than Luke can. Dave can sort the same tub four minutes faster than Jessica can. Working together, the three of them can sort the entire tub in 5 minutes flat. How long does it take each of them individually?

Let x be the number of minutes it takes for Luke to sort the LEGOs. Then it takes Jessica x - 21 minutes, and it takes Dave x - 25 minutes. This gives us the combined work equation

1/x + 1/(x-21) + 1/(x-25) = 1/5

We can give the left-hand side a common denominator to make it a single fraction, which gives us the simpler equation

(3x^2 - 92x + 525)/(x^3 - 46x^2 + 525x) = 1/5

Now we can make it a polynomial equation by cross-multiplying

15x^2 - 460x + 2625 = x^3 - 46x^2 + 525x
x^3 - 61x^2 + 985x - 2625 = 0

The solutions to this equation are

x = 35
x = 13 + sqrt(94) ≈ 22.695
x = 13 - sqrt(94) ≈ 3.305

The correct answer has to be at least as big as 25 to avoid negative rates, therefore the correct answer is x = 35. This means Luke can sort the LEGOs in 35 minutes, Jessica can do it in 14 minutes, and Dave can do it in 10 minutes.

## Example 7: Unit Conversion in Combined Work

Becky's pool's dimensions are 4 meters wide by 6 meters long by 2.5 meters deep. Becky can fill her pool from two different water sources. Water from a hose attached to Source A flows at a rate of 48 liters per minute. Water from a hose attached to Source B flows at a rate of 75 liters per minute. If Becky uses both water sources simultaneously, how long will it take to fill her pool to 80% of its capacity?

The first step is to figure out the amount of water needed in liters. Since there are 1000 liters in a cubic meter, Becky needs (80%)(4*6*2.5)(1000) = 48000 liters.

Next, we need to find the total amount of time it will take each source to provide 48000 liters, on its own. Source A will take 48000/48 = 1000 minutes. Source B will take 48000/75 = 640 minutes.

Working simultaneously, the total time T it will take to provide 48000 liters is given by the equation

1/1000 + 1/640 = 1/T

Solving this gives us

41/16000 = 1/T
T = 16000/41 minutes
T ≈ 390 minutes
T ≈ 6 and a half hours.

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