 QUESTION

# 150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on the second day , 4 more workers dropped out on the third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed.

Hint: If you look at the given data, it gives you an idea that it is in a sequence. Observe the difference in the series and apply appropriate formulas related to that series, you’ll get the answer.

As we know,

Total numbers of workers day = 150

Except for the first day,everyday 4 workers left the job.

So,second day number of workers = 146

Third day number of workers = 142

And this continued till n number of days

As we know the formula,

Sum of n terms in AP, $S = n/2[2a + (n - 1)*d]$

So,total work = $n/2[2a + (n - 1)*d]$

Here, a = 150, d = - 4

Putting these values in the above equation,we get

Total work = $n/2[2*150 + (n - 1)( - 4)]$

= $(152n - 2{n^2})$   (Eq 1)

If all the 150 workers are working for n days

Then,total work = 150(n - 8)   (Eq 2)

Comparing both (1) and (2), we get,

$152n - 2{n^2} = 150(n - 8)$

$\Rightarrow$  $152n - 2{n^2} = 150n - 1200$

$\Rightarrow$  $152n - 150n - 2{n^2} = - 1200$

$\Rightarrow$  $2{n^2} - 2n - 1200 = 0$

$\Rightarrow$  ${n^2} - n - 600 = 0$

$\Rightarrow$  $(n - 25)(n + 24) = 0$

$\Rightarrow$  n = 25,n = - 24

As we know,the number of days cannot be negative.

So,number of days = 25