Work and Time calculation

Question

A is thrice as good as a workman as $B$ and therefore is able to finish a job in $60$ days less than $B$. How much time will they take to finish the same job if they work together?

My attempt:

Let's say that the amount of work done by $B$ in $1$ day = $1 \over B$

As $A$ is $3$ times better than $B$, hence the amount of work done by $A$ in $1$ day=$3 \over B$

The difference in times to complete the same work is $60$ days.

Hence, ${3 \over B} - {1 \over B} = {1 \over 60}$

Solving which gives me B as 120 days and A as 40 days. Working together, they can complete the same job in ${ 1 \over {1 \over 120} + {1 \over 40}}= 30$ days.

But the correct answer, as given in the question, is something else.

What did I do wrong?

Answer 1

I learned how to do this with a table, so let's see if I can format it all correctly here. (Sorry in advance, my LaTex friends)

We know that rate (r)*time (t)=work, and that the work is the same for all jobs.

     Rate     Time      Work
A     3r      t-60        1 

  ------------------------- 

B     r       t           1

  ------------------------- 

Both  4r      ?           1

Now, we know that their rate together is 4r because rates can be added when they combine their abilities. If we set up a system:

   rt = 1
   3r(t-60) = 1

so

   3rt-180r = rt
   2rt = 180r
   t = 90 days

Then we can fill in our chart and figure out some other stuff.

     Rate     Time      Work
A     1/30     30       1 

  ------------------------- 

B     1/90(r)  90      1

  ------------------------- 

Both  4/90     x       1

Using again our equation:

   1 = 4x/90
   90/4 = x
   x= 45/2 days

That was a lot of formatting work, so please at least admire it if/when/before editing it.

Answer 2

Let A takes $d$ days to complete the work, so B will take $3d$ days.

So, $3d-d=60\implies d=30$

So, A takes $30$ days , B takes $90$ days individually to complete the work.

So, A & B,together will do $\frac{1}{30}+\frac{1}{90}=\frac{2}{45}$ part of the whole work in one day.

So, together they will need $\frac{45}{2}$ days to complete the task.