Question 958982
three men A B C working together can do a job in 6 hrs less time than A alone, in 1 hr less time than B alone and in one half the time needed by C when working alone. then A and B together can do a job in.
<pre>
Let x = the number of hours it takes A to do 1 job working alone.
Let y = the number of hours it takes B to do 1 job working alone.
Let z = the number of hours it takes C to do 1 job working alone.
Let t = the number of hours it takes A,B, and C to do 1 job working together.

Then
</pre>
three men A B C working together can do a job in 6 hrs less time than A alone,
<pre>
      t = x - 6
(1)   x = t + 6
</pre>
in 1 hr less time than B alone 
<pre>
      t = y - 1
(2)   y = t + 1
</pre>
and in one half the time needed by C when working alone. 
<pre>
      t = {{{z/2}}} 
     2t = z
(3)   z = 2t      

A's working rate = {{{ matrix(1,2,1,job)/ matrix(1,2,x,hr) }}} {{{""=""}}}{{{ matrix(1,2,1/x,job/hr) }}} 

B's working rate = {{{ matrix(1,2,1,job)/ matrix(1,2,y,hr) }}} {{{""=""}}}{{{ matrix(1,2,1/y,job/hr) }}}

C's working rate = {{{ matrix(1,2,1,job)/ matrix(1,2,z,hr) }}} {{{""=""}}}{{{ matrix(1,2,1/z,job/hr) }}}

The combined working rate of all three working together = {{{ matrix(1,2,1,job)/ matrix(1,2,t,hr) }}} {{{""=""}}}{{{ matrix(1,2,1/t,job/hr) }}}

{{{(matrix(5,1,"A's",working,rate,in,job/hr))}}}{{{""+""}}}{{{(matrix(5,1,"B's",working,rate,in,job/hr))}}} {{{""+""}}} {{{(matrix(5,1,"C's",working,rate,in,job/hr))}}} {{{""=""}}} {{{(matrix(6,1,Their,combined,working,rate,in,job/hr))}}}

(4)    {{{1/x}}}{{{""+""}}}{{{1/y}}}{{{""+""}}}{{{1/z}}} {{{""=""}}} {{{1/t}}}

Substituting from (1), (2), and (3) into (4)

      {{{1/(t+6)}}}{{{""+""}}}{{{1/(t+1)}}}{{{""+""}}}{{{1/(2t)}}} {{{""=""}}} {{{1/t}}}

Multiply through by the LCD, and get t = {{{2/3}}} hour, after dicarding
the negative value for t.

Substituting in (1), (2), and (3) we get that

A can do 1 job in 20/3 hours.
B can do 1 job in 5/3 hours.
C can do 1 job in 4/3 hours.

But that's not what we are asked to find. What we are asked to find is
given by this sentence:
</pre>
A and B together can do a job in ____ hours.
<pre>
A's working rate = {{{ matrix(1,2,1,job)/ matrix(1,2,20/3,hr) }}} {{{""=""}}}{{{ matrix(1,2,1/(20/3),job/hr) }}} {{{""=""}}} {{{ matrix(1,2,3/20,job/hr) }}}
 
B's working rate = {{{ matrix(1,2,1,job)/ matrix(1,2,5/3,hr) }}} {{{""=""}}}{{{ matrix(1,2,1/(5/3),job/hr) }}} {{{""=""}}} {{{ matrix(1,2,3/5,job/hr) }}} 

Suppose it takes them h hours to complete the job.  Then

The combined working rate of A&B working together = {{{ matrix(1,2,1,job)/ matrix(1,2,h,hr) }}} {{{""=""}}}{{{ matrix(1,2,1/h,job/hr) }}}


{{{(matrix(5,1,"A's",working,rate,in,job/hr))}}}{{{""+""}}}{{{(matrix(5,1,"B's",working,rate,in,job/hr))}}} {{{""=""}}} {{{(matrix(6,1,"A&B's",combined,working,rate,in,job/hr))}}}

{{{3/20}}}{{{""+""}}}{{{3/5}}} {{{""=""}}} {{{1/h}}}

Multiply through by an LCD, solve and get h = 4/3 hr or 1 hour and 20 minutes.

Edwin</pre>