SOLUTION: 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 how many women are required to complete a task in a day?

Algebra ->  Rate-of-work-word-problems -> SOLUTION: 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 how many women are required to complete a task in a day?      Log On


   

Question 854704: 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 how many women are required to complete a task in a day?
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
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%2Fx_days of 1%2Fx jobs per day

Then 2 men's combined working rate is 2 times 1%2Fx or 2%2Fx 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%2Fy_days or 1%2Fy jobs per day.

Then 3 women's combined working rate is 3 times 1%2Fy or 3%2Fy 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%2F8_days of 1%2F8 jobs per day.

The first equation comes from: 
  
%28matrix%284%2C1%2C%0D%0A%0D%0A2%2C+%22men%27s%22%2C+work%2Crate%29%29%22%22%2B%22%22%28matrix%284%2C1%2C%0D%0A%0D%0A3%2C+%22women%27s%22%2C+work%2Crate%29%29%22%22=%22%22%28matrix%284%2C1%2C%0D%0A%0D%0ATheir%2C+combined%2C+work%2Crate%29%29

2%2Fx%22%22%2B%22%223%2Fy%22%22=%22%221%2F8   

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

Exactly the same way, the second equation is

4%2Fx%22%22%2B%22%225%2Fy%22%22=%22%221%2F12

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

system%282%2Fx%2B3%2Fy=1%2F8%2C4%2Fx%2B5%2Fy=1%2F12%29

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%28-4%2Fx-6%2Fy=-2%2F8%2C4%2Fx%2B5%2Fy=1%2F12%29

Add the two equations term by term:

-1%2Fy=-2%2F8%2B1%2F12

-1%2Fy=-1%2F4%2B1%2F12

-1%2Fy=-3%2F12%2B1%2F12

-1%2Fy=-2%2F12

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