Question

Q. 150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on 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.

Answer 1

150 workers---1st day
146 workers--- 2nd day
142(3rd)---138(4th)---134(5th)---...'x' days.
The total work was finished in 8 days more than the usual time.So, actually the work would have completed in (x-8) days, with no men dropping.

Total work = Men*Days = 150*(x-8) = 150x - 1200 ....(i)

Now, since 4 men drops each day 150,146,142....... x terms.
Its an A.P, with 'n' = x, a1=150, d=-4
Sum of an A.P = n/2 * [a1 + an] = n/2[ 2a1 + (n-1)d]
Sum = x/2 [ 2*150 + (x-1)*-4]               
          = x/2[300 - 4x +4]        
          = x[150-2x+2]= x[152 - 2x]
This sum is the total work done where 4 men dropped in each day.
Tot work = x[152 - 2x].....(ii)

Total work is same, hence equate (i) & (ii)
150x - 1200 = x[152 - 2x]
150x - 1200= 152x - 2x^2
2x^2 -2x - 1200 = 0
x^2 - x - 600 = 0
x^2 -25x + 24x - 600 = 0
x(x-25) + 24(x-25) = 0(x+24)*(x-25) = 0
x = -24, 25
Since days can't be negative, x = 25 days
So, the work was now completed in 25 days.[If no men had dropped, the work would have been completed in: 25 - 8 = 17 days.]

Final answer to this question: Work was completed in 25 days, with 4 men dropping from 2nd day.
Hope it helps.[Assumption to be made: every worker had the same efficiency]

Answer 2

Question

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• 150 workers were engaged to finish a job in a Certain number of days. 4 workers dropped number of day, 4 more workers drop on third day so on. It took 8 more days to finish the work find the number pf days in which the work was completed.

$\sf \large \:\underline{ \purple{Given} }:-$

:−

• 150 workers were engaged to finish a job in a Certain number of days.

• 4 workers dropped number of day.

• 4 more workers drop on third day so on.

$\sf \large \:\underline{ \purple{To \: Find }}:- </p><p>$

:−

Find the number pf days in which the work was completed.

$\sf \large \:\underline{ \purple{Solution }}:-$

:−

$\boxed{ \sf \: sum \: of \: n \: terms \: = \frac{n}{2} ( \:2a + (n - 1) \times d) } </p><p>$

∴ Total number of workers who would have worked all n days = 150 (n – 8)

$\begin{gathered}\sf \to \: n (152 - 2n) = 150 (n - 8) \\ \\ \sf \to \: 152n - 2n^2 = 150n - 1200 \\ \\ \sf \to \: 2n^2 - 2n - 1200 = 0 \\ \\ \sf \to \: n^2 - n - 600 = 0 \\ \\ \sf \to \: n^2 - 25n + 24n - 600 = 0 \\ \\ \sf \to n(n - 25) + 24 (n + 25) = 0 \\ \\ \sf \to \: (n - 25) (n + 24) = 0 \\ \\ \sf \to \: n - 25 = 0 \: or \: n + 24 = 0 \\ \\ \sf \to \orange{ n = 25 \: or \: n = \: - 24}\: \huge \dag\\\end{gathered} </p><p>$

$\sf \underline{ \red{n = 25 } \: (Number \: of \: days \: cannot \: be \: negative)} \\ \\ </p><p></p><p> </p><p> </p><p></p><p>\text{ \green{ \underline{{Thus, the work is completed in 25 days.}}}} </p><p></p><p>$