150 men were hired to finish a construction. 150 men worked the first day,146 the second, 142 the third day and so on. If the work was completed with 7 day delay, in how many days would the work have been completed if 150 men worked every day
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Let, 150 men would have completed the work in 'n' days working all days.
Now no. of men & no. of days are inversely proportional. So, we get,
150n = constant ----------->(1)
But, from 2nd day, 4 men left and so on and it took 8 days more than the normal to finish the work, ie, (n + 8) days. Consequently,
150 + 146 + 142 +.......to (n+8) terms = constant
Left hand side is an A.P, with 1st term =150, common difference = -4
So, using formula,
(n+8)*1/2*[2*150+(n+8-1)(-4)]= constant ------->(2)
From equations,
(n+8)[300+(n+7)(-4)]=300n
Solving the quadratic equation, we get,
n = 17, or -32(neglected)
Hence, work is completed in,
n+8
= 17 + 8
= 25 days
Pls, verify if it's correct. I ain't sure.
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A2A
Let x days be required to finish the work by 150 workers.
Therefore, total man days is 150x.
In 1 day, 1 person can finish 1/(150x) amount of work.
First day, 150 people work,
Total work done is 150/(150x)
Second day, 4 drop off, hence 146 work,
Total work done is 150/(150x) + 146/(150x)
This continues for (x+8) days.
Therefore,
150/(150x) + 146/(150x) + .... (x+8) terms = 1
(1/150x)*{150 + 146 + .... (x+8) terms} = 1
The numbers form an AP with n = (x+8) terms, a = 150 & d = -4
Sum of terms of an AP is given by
(n/2)*{2a + (n-1)d}
Here,
(1/150x) * [{(x+8)/2} * {2*150 + (x+8-1)*(-4)}] = 1
(1/150x) * [{(x+8)/2} * 2{150 + (x+7)*(-2)}] = 1
(1/150x) * [(x+8) * {150 + (-2x-14)}] = 1
[(x+8) * {150 -2x-14}] = 150x
[(x+8) * {136 -2x}] = 150x
136x -2x^2 + 1088 -16x -150x = 0
-2x^2 +1088 -30x = 0
x^2 + 15x - 544 = 0
(x+32)(x-17) = 0
x = -32 or x = 17
As x is no. of days, it cannot be negative.
Therefore, x = 17.
The no. of days to finish the work is x + 8 = 25 days.
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If I understood your question correctly, then 4 workers dropped from second day.
Let total days be 'N'.
Total work done = 150*N (man days)
Work done on first day = 150*1 = 150 (man days)
Work done by using 146 workers on second day = 146 man days
and so on.. for (N+8) days.
So 150*N = 150 + 146 + 142 + .....(N+8) terms........... + (150 - 4(N+8-1)) ..{A . P.}
150*N = 150(N+8) - {0 + 4 + 8 + 12 + ....... + 4(N+7)}
The bold region is in A. P. which has sum = (N+8)(4(N+7))/2
So
150*8 = 2(N+7)*(N+8) => N = 17, -32.
Actual expected days is 17 days but it is done in 25 days at dropping 4 on each day.
Note: Sorry for wrong answer earlier.
the answer of your question is 25 days