Question 12375
<pre><font face = "courier new"><b>
A, B, and C can finish a job in 6 days. If B and C work
together, the job will take 9 days; if A and C work together,
the job will take 8 days. How many days will it take each man
to do the job alone? 
A. A in 9 days, B in 1 day, and C in 8 days 
B. A in 25 days, B in 12 2/3 day, and C in 28 days 
C. A in 20 1/2 days, B in 16 day, and C in 25 days 
D. A in 18 days, B in 24 day, and C in 14 2/5 days 
I don't have a clue where to begin with this problem. I would
appreciate any help--thanks
Let x = the number of days it would take A to do 1 job alone.
Let y = the number of days it would take B to do 1 job alone.
Let z = the number of days it would take C to do 1 job alone.
Make this chart:
           Number of jobs | Rate(jobs/days) |  Time(days)
A alone                   |                 |    
B alone                   |                 |    
C alone                   |                 |    
A,B,& C                   |                 |     
B & C only                |                 |
A & C only                |                 |
In each case 1 job was done, so fill in 1 for each of the
numbers of jobs
           Number of jobs | Rate(jobs/days) |  Time(days)
A alone          1        |                 |    
B alone          1        |                 |    
C alone          1        |                 |    
A,B,& C          1        |                 |     
B & C only       1        |                 |
A & C only       1        |                 |
Now fill in the times (numbers of days.
           Number of jobs | Rate(jobs/day)  |  Time(days)
A alone          1        |       1/x       |      x  
B alone          1        |       1/y       |      y 
C alone          1        |       1/z       |      z
A,B,& C          1        |       1/6       |      6
B & C only       1        |       1/9       |      9
A & C only       1        |       1/8       |      8
Now fill in the rates using rate = (jobs done)/(time)
The rate for A,B,& C equals the sum of their individual
rates, which must equal 1/6, so 
1/x + 1/y + 1/z = 1/6
The rate for A & C equals the sum of their individual rates,
which must equal 1/8, so 
1/x + 1/z = 1/8
The rate for B & C equals the sum of their individual rates,
which must equal 1/9, so 
1/y + 1/z = 1/9
So we have this system of equations:
1/x + 1/y + 1/z = 1/6
1/x       + 1/z = 1/8
      1/y + 1/z = 1/9
Don't clear of fractions, but solve for 1/x, 1/y, and 1/z.
Can you do this system?  If not, post again and ask how to solve it.
You'll get 1/x = 1/18, 1/y = 1/24, and 1/z = 5/72, which
means x = 18 days, y = 24 days, z = 72/5 or 14 2/5 days, and the
correct choice is D.
Edwin</b></font>