MHB How many days will it take B to finish the remaining work?

[mathdad]

A can do a piece of work in 25 days and B can do the same work in 30 days. They work together for 5 days and then A leaves. B will finish the remaining work in...

a) 21 days
b) 11 days
c) 20 days
d) 19 days

Is the following equation correct?

1/25 + 1/B = 1/5

 

[MarkFL]

A can do a piece of work in 25 days and B can do the same work in 30 days. They work together for 5 days and then A leaves. B will finish the remaining work in...

a) 21 days
b) 11 days
c) 20 days
d) 19 days

Is the following equation correct?

1/25 + 1/B = 1/5

I would first look at:

$$\text{lcm}\,(25,30)=150$$

We observe that in 150 days, A can complete 6 tasks and B can complete 5 tasks, and so working together they can complete 11 tasks in 150 days. The means that working together they can complete one task in 150/11 days. So, if they work together for 5 days, what portion of the task remains to be done?

 

[mathdad]

I don't follow the logic.

 

[MarkFL]

I don't follow the logic.

In order to answer this question, we need to know how long it takes A and B working together to complete one task, because we need to know how much of the task will be remaining after 5 days. We'll call this length of time T. Instead of the method I presented, we could also write:

$$\frac{1}{25}+\frac{1}{30}=\frac{1}{T}$$

This equation arises from looking at what gets done in 1 unit of time (days in this problem). In 1 day, A can complete 1/25 of a task and B can complete 1/30 of a task, and so the LHS of the equation represents their combined effort in 1 day. Since we've defined T to be the amount of time it takes A and B to complete a task working together, their combined efforts for 1 day, must equal 1/T.

What do you get when you solve for T?

 

[mathdad]

In order to answer this question, we need to know how long it takes A and B working together to complete one task, because we need to know how much of the task will be remaining after 5 days. We'll call this length of time T. Instead of the method I presented, we could also write:

$$\frac{1}{25}+\frac{1}{30}=\frac{1}{T}$$

This equation arises from looking at what gets done in 1 unit of time (days in this problem). In 1 day, A can complete 1/25 of a task and B can complete 1/30 of a task, and so the LHS of the equation represents their combined effort in 1 day. Since we've defined T to be the amount of time it takes A and B to complete a task working together, their combined efforts for 1 day, must equal 1/T.

What do you get when you solve for T?

I got T = 150/11.

 

[MarkFL]

I got T = 150/11.

Yes, and so if A and B have worked together for 5 days, the portion of the job remaining would be:

$$\frac{\dfrac{150}{11}-5}{\dfrac{150}{11}}=?$$

 

[mathdad]

I got 19/30.

 

[MarkFL]

I got 19/30.

Good...and so if 19/30 of the job remains and it takes B 30 days to complete the entire task, how long will it take for B to complete the 19/30 of the task that remains?

 

[mathdad]

(19/30)(30) = 19 days

 

[MarkFL]

(19/30)(30) = 19 days

Yes...I concur. :D