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# help with rates

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Bill and Peter can paint a 75 ft wall together in 7 hours. because Bill has more experience he can paint alone in two hours quicker than Peter. how long can it take Peter to paint the wall alone? round to the nearest minute.

Jan 4, 2021

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Let  the  portion of the wall that   Bill  can  paint in  one hour  be  1/x  =  his rate  where x is the  number  of  hours that Bill takes to  paint the wall by himself

Let  the portion of the wall that  Peter can paint in one hour  = 1/ ( x + 2)   = his rate  where x + 2  is the  number of hours that Peter  takes to paint the wall by himself

And

Bill's rate * 7hours  +  Peter's rate * 7 hours =   1  whole job done.....so...

1/x * 7   + 1/(x + 2) * 7  = 1

7/x  +  7/(x + 2) =  1

7 ( x + 2)    +  7x

______________   =     1

x ( x + 2)

14x  + 14  =   x ( x + 2)

14x + 14 = x^2  + 2x    rearrange as

x^2  - 12x  -  14 =   0        complete the square on x

x^2  -12x  +  36   =   14  +   36

(x - 6)^2 =    50        take the positive root

x - 6 =  sqrt (50)

x  =    sqrt (50 ) +  6

x +  2 =       sqrt (50 )  +  8   ≈  15.07  hrs =  Peter's time  =  15  +  .07 *60   ≈  15 hrs  4 min   Jan 4, 2021
edited by CPhill  Jan 5, 2021