# Working together word problems

We will solve "working together word problems" with proportion. It may easier to tackle than algebra. Problem #1:

Maria can clean all bathrooms in her house in 5 minutes. Her husband can do the same in 20 minutes. How long will it take them if they work together?

One important observation is that an entire task or job can be represented with the number 1.

If it takes Maria 5 minutes to do the entire job, then she can do
1 / 5
of the job in 1 minute

By the same fashion, if it takes the husband 20 minutes to do the entire job, then he can do
1 / 20
of the job in 1 minute.

Together, the amount of the job they can do in 1 minute is

 1 / 5 +     1 / 20
Together, the amount of the job they can do in 1 minute is

 1 / 5 +     1 / 20

 1 / 5 +   1 / 20 =   4 / 20 +   1 / 20 =   5 / 20

Together, they can do
5 / 20
of the job in 1 minute

Now set up a proportion using the following logic.

If
5 / 20
of the job can be done in 1 minute, then the entire job or 1 can be done in x minutes

5 / 20
_____________     1 minute

1       _____________       x minutes

Cross multiply

5 / 20
× x = 1 × 1

5 / 20
× x = 1

Multiply both sides of the equation by
20 / 5

The left side gives the expression below:

 5 / 20 ×   20 / 5 × x =   100 / 100 = 1x = x

Therefore, x = 1 ×
20 / 5
= 4

Together, they can clean all bathrooms in 4 minutes.

Problem #2

Say it is a big house project and Maria could finish the project in 5 days and the husband could do it in 20 days.

How many days will it take husband and wife to finish the work?

Problem #1 and problem #2 are similar working together word problems. The only difference is that instead of using hours, problem #2 is using days.

Therefore, it would take them 4 days to finish.

 1 / 5 +   1 / 20 =   4 / 20 +   1 / 20 =   5 / 20

Together, they can do
5 / 20
of the job in 1 minute

Now set up a proportion using the following logic.

If
5 / 20
of the job can be done in 1 minute, then the entire job or 1 can be done in x minutes

5 / 20
_____________     1 minute

1       _____________       x minutes

Cross multiply

5 / 20
× x = 1 × 1

5 / 20
× x = 1

 5 / 20 ×   20 / 5 =   100 / 100 = 1

Therefore, x =
20 / 5
= 4

Together, they can clean all bathrooms in 4 minutes.

Problem #2

Say it is a big house project and Maria could finish the project in 5 days and the husband could do it in 20 days.

How many days will it take husband and wife to finish the work?

It is the exact same problem as problem #1 except that instead of using hours, we are using days.

Therefore, it would take them 4 days to finish.

## Some working together word problems can be complicated. Problem #3 is a little challenging!

Problem #3:

It can take Peter 25 days to build a house garage all by himself. If he works with his friend, together they can build the house garage in 20 days.

How fast can the friend work alone?

Set up a proportion to find out how much of the work can be done by both of them in 1 day.

let x represent the amount of work they can do in 1 day. Notice again how we use 1 to represent the entire job

1       _____________       20 days

x       _____________       1 day

Cross multiply

1 × 1 = x × 20

1 = 20x

Divide both sides by 20

 1 / 20 =   20 / 20 x

 1 / 20 =   20 / 20 x

1 / 20
= x

The amount of the job Peter can do in 1 day is

1 / 25

The amount of the job the friend can do in 1 day is

 1 / 20 -   1 / 25
 1 / 20 -   1 / 25

The amount of the job the friend can do in 1 day is

 5 / 100 -   4 / 100 or    1 / 100

The amount of the job the friend can do in 1 day is

 5 / 100 -   4 / 100 or    1 / 100

This means that it will take the friend 100 days to finish the house garage by himself.

Want more working together word problems?

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