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
|
Together, they can do
1
5
|
+
1
20
of the job in 1 minute
|
|
1
5
|
+
1
20
|
=
4
20
|
+
1
20
|
=
5
20
|
Now set up a proportion using the following logic.
1 _____________ x minutes
Cross multiply
|
5
20
|
×
20
5
|
=
100
100
= 1
|
Together, they can clean all bathrooms in 4 minutes.
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.
Problem #3: A complicated working together word problem
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
|
|
The friend can do
1
20
|
-
1
25
of the job in 1 day
|
|
The friend can do
5
100
|
-
4
100
of the job in 1 day
|
or
1
100
|
This means that it will take the friend 100 days to finish the house garage by himself
Together, they can do
|
1
5
|
+
1
20
of the job in 1 minute
|
|
1
5
|
+
1
20
|
=
4
20
|
+
1
20
|
=
5
20
|
Now set up a proportion using the following logic.
1 _____________ x minutes
Cross multiply
|
5
20
|
×
20
5
|
=
100
100
= 1
|
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.
Problem #3: A complicated working together word problem
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
|
|
The friend can do
1
20
|
-
1
25
of the job in 1 day
|
The friend can do
|
5
100
|
-
4
100
of the job in 1 day
|
or
1
100
|
This means that it will take the friend 100 days to finish the house garage by himself
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