Optimize Work and Time Equations: Formula for Calculating Completion Time"

[GlassBones]

Homework Statement

A does 80% of a work in 20 days. He then calls in B and they together finish the remaining work in 3 days. How long B alone would take to do the whole work?

Relevant Equations

None

I came up with this formula.

A takes 20 days to have 80% completion. Simple algebra gets that A takes 25 days to complete finish by themselves. Now I'm trying to come up with a formula.

$1/.8(1/25) = 1/20$ with 80% completion. Now I'm working with the remaining 20%. $1/.2(1/25+1/b)$, where b is the total amount of days B takes to finish 1 work. The sum should be 1/23, ie:

$1/.8(1/25)+1/.2(1/25+1/b) = 1/23$

So I should isolate b to find the number of days B takes to finish alone. In doing so I get a negative number...

What's wrong with my reasoning?

 

[phinds]

How much work does A do per day as a function of the total work to be done?

 

[GlassBones]

...as a function of the total work to be done

I know A can complete work alone in 25 days. So 1 day A will do 1/25. Is this what your asking? Sorry just confused

 

[RPinPA]

I know A can complete work alone in 25 days. So 1 day A will do 1/25. Is this what your asking? Sorry just confused

Right. That's usually what you want to do in these problems, is figure out the fraction per day that each person does.

So A working alone does 1/25 of the job in a day.

Now, given that the team of A + B does 0.20 of a job in 3 days, how much of a job does the team of A + B do in one day?

Do you see what knowing that tells you about B's work output?

 

[RPinPA]

$1/.8(1/25)+1/.2(1/25+1/b) = 1/23$

So I should isolate b to find the number of days B takes to finish alone. In doing so I get a negative number...

What's wrong with my reasoning?

This does look like it may be doing some of the correct calculations but I can't quite understand your equations.

What does 1/.8 represent? Why are you dividing by the fraction of the job done?

If your first term is equal to 1/20, the reciprocal of the number of days A works, then your second term will be 1/3, the reciprocal of the number of days A + B work together to complete the 0.20 of the job.

But then your equation says (1/20) + (1/3) = (1/23) which is obviously not true.

I think your individual terms on the left are OK, but all you have to do is use the fact that the second term is 1/3. They don't add up to 1/23. And it would be even easier if you worked with the number of days, rather than its reciprocal.

 

[GlassBones]

But then your equation says (1/20) + (1/3) = (1/23) which is obviously not true.

...yes there's the issue with my reasoning. I can't believe I made this mistake. Thanks everything makes sense

 

[SammyS]

How much work does A do per day as a function of the total work to be done?

I know A can complete work alone in 25 days. So 1 day A will do 1/25. Is this what your asking? Sorry just confused

An alternate way to answer @phinds 's question is:

A can do 4% of a whole work in one day.

Then the question can be answered as follows.

The remaining 20% of a whole work takes A&B 3 days to complete of which A does 3 × 4% = 12%.

So in 3 days B did 8% of a whole work.

Let x be the number of days for B alone to do (100% of) a whole work. Then we have the following proportion.

$\displaystyle \frac{8\%}{100\%}=\frac{3}{x}$

 

[WWGD]

You can also compute what percent of the work A has done by the 3rd day and take it from there. B will do the remainder and you can compare the rates.