Question 387867
If Michael can paint a house in 8 hours and a Samantha can paint a house in 2
hours, how long will it take them together? 
<pre>
You can do it in your head without algebra, but you're probably supposed
to do it by algebra, so we''l do it both ways.  First, in your head:

In the 8 hours Michael takes to paint only one house, Samantha will have
painted 4 houses.  So in 8 hours together they will have painted 5 houses. So
it'll only take them 1/5th of 8 hours to paint just one house, and 1/5 of 8 is
8/5 hours.  That's 1 3/5 hours or 1 hour and 36 minutes.

Now we do it using algebra:

Make this chart:

             Houses painted     Time in hours   Rate in houses/hour
Michael
Samantha
Together

Let x be the time it takes them to paint 1 house working together, so
fill in 1 for the number of houses, and x for the time.

             Houses painted     Time in hours   Rate in houses/hour
Michael
Samantha
Together           1                  x


>>...Michael can paint ONE house in 8 hours...<<

So we fill in 1 for his number of houses painted and 8 for his time:

             Houses painted     Time in hours   Rate in houses/hour
Michael            1                  8               1/8
Samantha                              
Together           1                  x               1/x



>>...Samantha can paint a house in 2 hours...<<


So we fill in 1 for her number of houses painted and 2 for her time:

             Houses painted     Time in hours   Rate in houses/hour
Michael            1                  8
Samantha           1                  2 
Together           1                  x

Now we fill the rate in houses/hour by dividing houses painted by hours
in all three cases:

             Houses painted     Time in hours   Rate in houses/hour
Michael            1                  8                 1/8 
Samantha           1                  2                 1/2
Together           1                  x                 1/x 

Now we form our equation from the rates:

        Michael's rate + Samantha's rate = their rate together
              1/8      +      1/2        =        1/x

Multiply through by the LCD of 8x

                                  x + 4x = 8
                                      5x = 8
                                       x = 8/5
                                       x = 1 3/5 hours or 1 hour 36 minutes.

Edwin</pre>