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Working together the women can paint a house in two-thirds the lime that it takes Karen working alone. Betty takes 6h to paint a house alone. How long does it take Karen to paint a house working alone?
let karen take x hours to do the job alone.
so she does 1/x of the job in 1 hour.
Betty takes 6 hours working alone
so she does 1/6 of the job in 1hour.
Both together they do 1/6 + 1/x of the job in 1 hour
= (x+6)/6x of the job in 1 hour.
both together they take 2/3 *x = 2x/3 hours
or they do 3/2x of the job in 1 hour.
(x+6)/6x = 3/2x
divide by 2x
Karen takes 3 hours to do the job alone.