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


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.

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


1 / 20
= x


Peter can do  
1 / 25
  of the job in 1 day


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

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, they can do

1 / 5
 +    
1 / 20
 of the job in 1 minute


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.

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


1 / 20
= x


Peter can do  
1 / 25
  of the job in 1 day


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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