Question 887869
If it takes 2 days for 5 workers to finish painting a wall at a rate of 6 hours
per day, then how long will it take 7 workers to finish painting a wall at a
rate of 7 hours per day? 
<pre>
First of all 2 days at 6 hours per day is 12 hours.

So first of all we'll pretend the problem just speaks of hours like this:
</pre>
It takes 12 hours for 5 workers to finish painting a wall. How many hours
will it take 7 workers to finish painting a wall?
<pre>
The least common multiple of 5 workers and 7 workers is 35 workers.

35 workers is 7 times 5, so 35 workers can finish painting 7 walls in 12 hours.

7 workers is only {{{1/5}}} as many as 35, so it will take 7 workers 5 times as long
or 60 hours to paint 7 walls.  So to paint only 1 wall will take only {{{1/7}}}th as
long as it takes to paint 7 walls.  {{{1/7}}}th of 60 hours is or {{{60/7}}} or {{{8&4/7}}} hours.  
So they'll work 1 7-hour day and will finish after {{{1&4/7}}} hours on the
second day.

------------------------

You can also use the worker-time-job formula, which is:

{{{(W[1]T[1])/J[1]}}}{{{""=""}}}{{{(W[2]T[2])/J[2]}}}

where

W<sub>1</sub> = the number of workers in the first situation.
T<sub>1</sub> = the number of time units (hours in this case) in the first situation.
J<sub>1</sub> = the number of jobs in the first situation.

W<sub>2</sub> = the number of workers in the second situation.
T<sub>2</sub> = the number of time units (hours in this case) in the second situation.
J<sub>2</sub> = the number of jobs in the second situation.

W<sub>1</sub> =  5             W<sub>2</sub> = 7     
T<sub>1</sub> = 12             T<sub>2</sub> = the unknown quantity 
J<sub>1</sub> =  1             J<sub>2</sub> = 1

{{{(W[1]T[1])/J[1]}}}{{{""=""}}}{{{(W[2]T[2])/J[2]}}}

{{{(5*12)/1}}}{{{""=""}}}{{{(7*T[2])/1}}}

{{{60}}}{{{""=""}}}{{{7*T[2]}}}

{{{60/7}}}{{{""=""}}}{{{T[2]}}}

{{{8&4/7}}} hours.  So they'll work 1 7-hour day and will finish after
{{{1&4/7}}} hours on the second day.

Edwin</pre>