Question

A few workers finished their work in 14 days. If there were 4 more workers and each of them would work 1 hour longer, this work would have been done in 7 days. How many workers were there and for how many hours a day did they work?

Answer 1

My attempt that didn't succeed:
d - work rate, A -work, n - number of hours worked each day, x - number of workers

d = A/(14nx)
7(x+4)(n+1)*A/(14nx) = A | dividing by A/(14nx)
(x+4)(n+1) = 2nx
nx + x + 4n + 4 = 2nx
x + 4n + 4 - nx = 0 | That's where I stumbled across a wall

Answer 2

It is very confusing. I will try to solve it and then see if I can find your error, if there is one.

Answer 3

Thank you very much!

Answer 4

Sorry, still struggling. I'm sure the answer is obvious but I am not seeing it right off.

But I like a challenging question in the morning, instead of the usual, "HELP I need to find the slope of a line".

Answer 5

I think that the answer involves some kind of factoring, since there's only one equation to be made, but I am not sure...

Answer 6

Each worker works h hours a day.
All workers work at the same rate.
In 14 days, each worker worked 14h hours.
The number of workers is n.
The total number of hours used to complete the job is
14hn
With 4 more workers working, and all workers working 1 more hour per day, then you would have
7(h + 1)(n + 4) hours to complete the job

The total number of man hours needed to complete the job is the same in either situation, so

14hn = 7(h + 1)(n + 4)

This is one equation in 2 unknowns.
We need another equation, but I'm not sure what to do now.

Also, gtg.
I'll try to come back later.

Answer 7

Okay...

Answer 8

Trying the reverse...

Answer 9

Alright, these number seem to work but are not very logical....

if I have 12 workers working 2 hours a day that equals 24 man-hours
now if I add 4 workers and now 16 workers work 3 hours equals 48 man-hours

48 men working one hour can finish a job TWICE as fast as 24 men working one hour

7 days is half of 14

Answer 10

While explaining this to my daughter, I found a few more solutions to the problem. Unless I can find another equation, I can't determine the correct answer.

This solution seems the most logical:
However, the one with 5 workers working 9 hours for 45 man-hours and then adding four workers would make 9 workers at 10 hours each for a product of 90 man-hours.

whr - w - 4hr = 4
8(3) - 8 - 12 = 4

6(5) - 6 - 4(5) = 4
5(9) - 5 - 4(9) = 4

It also works for 8 workers working 3 hours 24
12 4 48

It also works for 6 workers working 5 hours 30
10 6 60

It also works for 5 workers working 9 hours 45
9 10 90

Answer 11

I noticed that from my previous work I could have made an equality w = 4*(h+1)/(h-1). Since the day has only 24 hours and number of workers must be an integer (h+1)/(h-1) must also be an integer. Looking at all the values I got the same answers as yours ;)
The only problem that i have is that perhaps there are values of h that aren't integers, but still satisfy the '(h+1)/(h-1) must be an integer' requirement. Maybe you have any idea how to prove that there are / are not some? @retirEEd

Answer 12

There were some cases where you could have had workers with non-integers hours.

For example 7 workers working 3 and 2/3 hours, 10 workers working 10 and 1/3 hours.

I used desmos graphing to find the integer answers.... but here is an example of a non-integer hour answer.