SOLUTION: A contractor undertook to do a certain piece of work in 9 days. He employed certain number of men, but 6 of them being absent from the very first day, the rest could finish the wor

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Question 1206760: A contractor undertook to do a certain piece of work in 9 days. He employed certain number of men, but 6 of them being absent from the very first day, the rest could finish the work in 15 days. The number of men originally employed were:
Found 5 solutions by ikleyn, Theo, greenestamps, josgarithmetic, Edwin McCravy:
Answer by ikleyn(52908)   (Show Source): You can put this solution on YOUR website!
.
A contractor undertook to do a certain piece of work in 9 days. He employed certain number of men,
but 6 of them being absent from the very first day, the rest could finish the work in 15 days.
The number of men originally employed were:
~~~~~~~~~~~~~~~~~~~~~ 

x originally employed men;

So, the entire job was 9x men-days, as planned originally.


But it was done by (x-6) men in 15 days; hence

    15*(x-6) = 9x.


From it,  

    15x - 90 = 9x,

    15x - 9x = 90

       6x    = 90

        x    = 90/6 = 15.


ANSWER.  15 men were employed initially;  but only 15-6 = 9 of them really worked.

Solved.



Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
the work was originally expected to be done in 9 days.

the formula i use in problems such as this is:

p * r * t = q

p is the number of people
r is the average rate that each person works at.
t is the time
q is the quantity.

q is usually 1, i.e. 1 job.
the formula becomes p * r * t = 1

when the original number of people are working, the formula becomes p * r * 9 = 1

then 6 are missing, the formula becomes (p-6) * r * 15 = 1

these 2 equations need to be solved simultaneously.

start with:
p * r * 9 = 1
(p-6) * r * 15 = 1

divide both sides of the first equation by 9 and divide both sides of the second equation by 15 to get:

p * r = 1/9
(p - 6) * r = 1/15

simplify the second equation and leave the first eqution as is to get:
p * r = 1/9
p * r - 6 * r = 1/15

subtract the second eqution from the first to get:

6 * r = 1/9 - 1/15.

place the right side of the equation under the common denominator of 45 to get:

6 * r = 5/45 - 3/45 = 2/45

solve for r to get:

r = 2/45 / 6 = 2/270 which reduces to 1/135.

now that you have r, you can solve for p.

p * r * 9 = 1
when r = 1/135, this becomes p * 1/135 * 9 = 1
solve for p to get p = 1 / 9 * 135 = 15.

the original number of people assigned to the project was 15.

(p - 6) * r * 15 = 1 becomes:
(p - 6) * 1/135 * 15 = 1
solve for (p - 6) to get (p - 6) = 1/15 * 135 = 9.
solve for p to get p = 15.

the value of p checks out ok.
the original number of people assigned to the project is 15.
after 6 are taken away, the project was then finished with 9.
the average rate that each person worked is 1/135.

15 * 1/135 * 9 = 1 is confirmed to be correct.
9 * 1/135 * 15 = 1 is also confirmed to be correct.


Answer by greenestamps(13215)   (Show Source): You can put this solution on YOUR website!


The formal algebraic solutions from the other tutors are fine.

See if this common sense quick mental solution method makes sense to you.

With the missing 6 men, the work took 15/9 times as long (15 days instead of 9).
That means the number of men actually working is 9/15 of what it was supposed to be.
But the difference between the numerator and denominator of the fraction 9/15 is 6; and that means 15 men were scheduled to do the work but only 9 men worked.

ANSWER: 15
Answer by josgarithmetic(39630)   (Show Source): You can put this solution on YOUR website!
work rate each worker, r
n, expected how many workers

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A contractor undertook to do a certain piece of work in 9 days. He employed certain number of men,
but 6 of them being absent from the very first day, the rest could finish the work in 15 days.
--------------






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Answer by Edwin McCravy(20065)   (Show Source): You can put this solution on YOUR website!
A contractor undertook to do a certain piece of work in 9 days.
He employed certain number of men, but 6 of them being absent from the very first day, the rest could finish the work in 15 days. The number of men originally employed were:
Use the worker-time-job formula, which is:



where

W1 = the number of workers in the first situation. 
T1 = the number of time units (days in this case) in the first situation.
J1 = the number of jobs in the first situation.

W2 = the number of workers in the second situation.
T2 = the number of time units (days in this case) in the second situation.
J2 = the number of jobs in the second situation.

W1 =  n             W2 = n-6     
T1 =  9             T2 = 15 
J1 =  1             J2 = 1













number of men originally employed was: 15

Edwin
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