Percents

[Nightmare]

A,B,C can do a piece of work in 36,54,and 72 days respectively. they started the work but A left 8 days before the completion of the work while B left 12 days before the completion. the number of days for which c worked is?

[ygdrasil24]

You may find number of approaches for such questions. I personally find below method to be shortest.
A=36
B=24
C=72.

Take LCM of these numbers to get 18X12.
So A will finish 18X12/36=6 units in 1 day, similarly B=4 units, C=3 units.
Let total work to be completed is 18X12(This is imp number to choose)
Let C worked for x days
so A worked for x-8, B for x-12, C for x
so A's Rate X Days Worked + .... = total work
6(x-8)+4(x-12)+3x=18X12, solve for x to get 24.

I know this method works well, but I have a general question , is it OK to assume two values in one linear equation, here we assumed, 18X 12 and x, had it been ratio or similar, the assumed expression might have got canceled.Please comment.

[Atekihcan]

In one day,

• A does 1/36 of the work
  B does 1/54 of the work
  C does 1/72 of the work
  
  Together all three of them does (1/36 + 1/54 + 1/72) = (1/18)*(1/2 + 1/3 + 1/4) = (1/18)*(6 + 4 + 3)/12 = 13/12*18

A and C together worked for (12 - 8) = 4 days.
In these 4 days, they completed 4*(1/36 + 1/72) = 4*(2/72 + 1/72) = 4*3/72 = 12/72 of the work

C alone worked for 8 days.
In these 8 days, C completed 8/72 of the work.

So, in last 12 days (12/72 + 8/72) = 20/72 of the work was done.
Hence, the rest of the work, i.e. (1 - 20/72) = 52/72 of the work was done by all of the together.

Now, all of the together takes 12*18/13 days to complete the job.
So, they will take (12*18/13)*(52/72) = 12 days to complete 52/72 of the work.

So, C worked for (12 + 12) = 24 days.