Question

Had then been one men less, then the number of days required to do a piece of work would have been one more. If the number of mandays required to complete the work is 56, how many workers were there?
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Answer 1

Step-by-step explanation:

Let the number of men be x and number of days taken by them to finish the work be y. Number of man-days = xy

So xy = 72 …..(1)

If one man less then number of days will be one more.

So no.of men =(x-1) and no.of days =y+1

No.of man-days = (x-1)(y+1) = xy

=> xy + x - y -1 =xy => x - y = 1 …..(2)

Combining (1) and (2) we have to find two numbers whose product is 72 and their difference is 1 .

=> x =9 and y =8

(OR) from (2) x = y+1. Using this in 1 we get (y+1)y = 72 => y^2+y-72 = 0

=> (y+9)(y-8)=0 => y =-9 or y = 8

But y is positive . So y = 8, x = y+1 =8+1=9

So no.of men worked initially = 9

According to the given problem,

Let W denotes the whole given work.

It is mentioned that,

(i) If there is one man less to complete the work W then the number of days required to complete the work W would be one more.

(ii) The initial number of man-days required to complete the work is 72.

(iii) Let N & D denote respectively the initial (number of men) & (number of days) required to complete the work W.

From (ii) & (iii) we get the following relation,

N*D = 72 ….. (1)

According to (i) we get the following relation,

(N -1)*(D + 1) = 72

or (N -1)*(72/N + 1) = 72 [from (1)]

or (N -1)*(72 + N) = 72*N

or N^2 +71*N - 72 = 72*N

or N^2 - N - 72 = 0 or (N - 9)*(N + 8) = 0 or N = 9

[N cannot be negative]

Therefore we determine that

number of men who worked initially = 9 [Ans]

Let there were intially x men were there, hence work take 72x men days. Now there are x-1 men and work to be complete in 72+1=73 so 72x=(x-1)73 or 72x=73x-73 so x=73. So initially there were 73 men worked.

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Answer 2

Step-by-step explanation:

• Let the number of workers be p and number of days be q
• So man days will be p x q = 56 ---------1
• Number of man = (p – 1)
• Number of days = (q + 1)
• So number of man days will be (p – 1) (q + 1)
• So pq = (p – 1)(q + 1)
• Simplifying  pq = pq – q + p - 1
•             So p – q = 1
•                   P – 56/p = 1 (from eqn 1)
•             So p^2 – p – 56 = 0
• Solving this quadratic equation we get
•           So p^2 – 8p + 7p – 56 = 0
•            So p(p – 8) + 7(p – 8) = 0
•          So (p – 8) (p + 7) = 0
•            Or the values p = 8, p = -7
• Since it cannot be negative, the number of workers is 8

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