Question 854704
<pre>
If 2 men and 3 women can complete a task in 8 days; and 4 men and 5 women can complete a task in 12 days then 
Suppose 1 man can complete a task alone in x days.  Then 1 man's working rate is 1 job per x days or {{{1_job/x_days}}} of {{{1/x}}} jobs per day

Then 2 men's combined working rate is 2 times {{{1/x}}} or {{{2/x}}} jobs per how many women are required to complete a task in a day?
day.

Suppose 1 woman can complete a task alone in y days.  Then 1 woman's working rate is 1 job per y days or {{{1_job/y_days}}} or {{{1/y}}} jobs per day.

Then 3 women's combined working rate is 3 times {{{1/y}}} or {{{3/y}}} jobs per day.

>>2 men and 3 women can complete a task in 8 days<<

Their combined work rate is 1 job per 8 days or {{{1_job/8_days}}} of {{{1/8}}} jobs per day.

The first equation comes from: 
  
{{{(matrix(4,1,

2, "men's", work,rate))}}}{{{""+""}}}{{{(matrix(4,1,

3, "women's", work,rate))}}}{{{""=""}}}{{{(matrix(4,1,

Their, combined, work,rate))}}}

{{{2/x}}}{{{""+""}}}{{{3/y}}}{{{""=""}}}{{{1/8}}}   

>>4 men and 5 women can complete a task in 12 days<<

Exactly the same way, the second equation is

{{{4/x}}}{{{""+""}}}{{{5/y}}}{{{""=""}}}{{{1/12}}}

So we have the system of two equations in two unknowns:

{{{system(2/x+3/y=1/8,4/x+5/y=1/12)}}}

To solve that DO NOT CLEAR OF FRACTINS! Instead use
elimination just as they are.  To make the first terms
cancel multiply the first equation by -2

{{{system(-4/x-6/y=-2/8,4/x+5/y=1/12)}}}

Add the two equations term by term:

{{{-1/y=-2/8+1/12}}}

{{{-1/y=-1/4+1/12}}}

{{{-1/y=-3/12+1/12}}}

{{{-1/y=-2/12}}}

Cross multiply:

{{{-2y = -12}}}

{{{y = 6}}}

So 1 woman can complete 1 task in 6 days.

>>how many women are required to complete a task in a day?<<

So 6 women can complete 1 task in 1 day.

Edwin</pre>