SOLUTION: Betty and Karen have been hired to paint the houses in a development. Working together the women can paint a house in two-thirds the time that it takes Karen working alone. Betty t

Algebra ->  Rate-of-work-word-problems -> SOLUTION: Betty and Karen have been hired to paint the houses in a development. Working together the women can paint a house in two-thirds the time that it takes Karen working alone. Betty t      Log On


   

Question 87376This question is from textbook
: Betty and Karen have been hired to paint the houses in a development. Working together the women can paint a house in two-thirds the time that it takes Karen working alone. Betty takes 6 hours to paint a house alone. How long does it take Karen to paint a house working alone?
 This question is from textbook

Found 2 solutions by checkley75, Edwin McCravy:
Answer by checkley75(3666) About Me  (Show Source):
You can put this solution on YOUR website!
2x/3=6x/x+6 now we cross multiply
2x^2+12x=18x
2x^2+12x-18x=0
2x^2-6x=0
2x(x-3)
x-3=0
x=3 hours for Karen to paint the house alone.
proof
2*3/3=6*3/3+6
6/3=18/9
2=2

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Betty and Karen have been hired to paint the houses in a development. Working
together the women can paint a house in two-thirds the time that it takes Karen
working alone. Betty takes 6 hours to paint a house alone. How long does it
take Karen to paint a house working alone?
> 
Make this chart: 

                  no. of houses       rate in 
                     painted       houses per hour      time
Betty alone                                                 
Karen alone                                                
both together                                                  

I'll rewrite the problem emphasizing whenever the number 1 (one) is implied:

Working together the women can paint ONE house in two-thirds the time that it
takes Karen working alone to paint ONE house. Betty takes 6 hours to paint ONE
house alone. How long does it take Karen to paint ONE house working alone?

Therefore we know in all three cases we are talking about painting just ONE
house, not all the houses in the development, or even TWO of them.  Just one.
So fill in the number of houses in all three cases, which is 1 house.

                  no. of houses       rate in 
                     painted       houses per hour      time
Betty alone             1                                   
Karen alone             1                                 
both together           1                                      

The question asks:
>>...How long does it take Karen to paint ONE house working alone?...<<

So let the answer to that question be t. So fill in t as Karen's time
working alone.


                  no. of houses       rate in 
                     painted       houses per hour      time
Betty alone             1                                  
Karen alone             1                                t
both together           1                                       

>>..Betty takes 6 hours to paint ONE house alone..<<

So fill in Betty's time as 6

                  no. of houses       rate in 
                     painted       houses per hour      time
Betty alone             1                                6
Karen alone             1                                t
both together           1                                     

>>...Working together the women can paint ONE house in two-thirds
the time that it takes Karen working alone to paint ONE house...<<

So fill in (2/3)t for the time when both work together.

                  no. of houses       rate in 
                     painted       houses per hour      time
Betty alone             1                                6
Karen alone             1                                t
both together           1                             (2/3)t

Now we use RATE = (number of houses painted)/time
to fill in the three rates:

                  no. of houses       rate in 
                     painted       houses per hour      time
Betty alone             1               1/6              6
Karen alone             1               1/t              t
both together           1            1/[(2/3)t]       (2/3)t

We need to simplify that entry 1/[(2/3)t] 

                               1÷[(2t)/3]

                               1×[3/(2t)]
                          
                                 3/(2t)

                  no. of houses       rate in 
                     painted       houses per hour      time
Betty alone             1               1/6              6
Karen alone             1               1/t              t
both together           1             3/(2t)          (2/3)t

Now that we have the table all filled, to get the equation
we use this:

Betty's rate working alone + 

       Karen's rate working alone = 

                         their combined rate working together 

The combined rate is the rate at which the house is being 
painted when they both work on it together.

So the equation is

                             1/6 + 1/t = 3/(2t)

Clear of fractions by multiplying thru by 6t

                                t + 6 = 9 
                                    t = 3

So it takes Karen 3 hours to paint ONE house by herself.

Edwin