# Assuming no gain or loss in efficiency, how long should it take each person to complete the job alone in the following case:Three painters, Beth, Bill, and Edie working together can paint a home in...

Assuming no gain or loss in efficiency, how long should it take each person to complete the job alone in the following case:

Three painters, Beth, Bill, and Edie working together can paint a home in 8 hours. Bill and Edie together have painted a similar house in 16 hours. One day all three work together for 6 hours, after which Edie left. Beth and Bill required 3 more hours to finish.

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Let the time it takes Beth to complete the job alone be X, the time taken by Bill be Y and the time taken by Edie be Z.

If the time taken by any of them to paint the house is T, the rate at which the house is painted is R = 1/T

The three working together can paint a home in 8 hours.

=> 1/X + 1/Y + 1/Z = 1/8 ...(1)

Bill and Edie together have painted a similar house in 16 hours.

=> 1/Y + 1/Z = 1/16 ...(2)

If all the three work together for 6 hours they finish 6*(1/8) of the job. Then Edie leaves and Beth and Bill require 3 more hours to complete the job. 3*(1/X + 1/Y) = 1 - (6/8) = 1/4

=> 1/X + 1/Y = 1/12 ...(3)

From (1) and (2)

1/X = 1/8 - 1/16 = 1/16

From (1) and (3)

1/Z = 1/8 - 1/12 = 1/24

Substituting these in (1) gives 1/Y = 1/8 - 1/16 - 1/24 = 1/48

**Working alone Beth finishes the job in 16 hours, Bill finishes it in 48 hours and Eddie takes 24 hours.**