Question

Jane, Paul and Peter can finish painting the fence in 2 hours. If Jane does the job alone she can finish it in 5 hours. If Paul does the job alone he can finish it in 6 hours. How long will it take for Peter to finish the job alone?

Answer 1

Answer: 7.5 hours (or 7 hours 30 minutes)

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Explanation:

Jane does the job alone and she can finish it in 5 hours. Her rate is 1/5 of a job per hour. By "job", I mean painting the entire fence. Notice that multiplying 1/5 by the number of hours she works will yield the value 1 to indicate one full job is done.

Through similar reasoning, Paul's rate is 1/6 of a job per hour.

Let x be the time, in hours, it takes Peter to get the job done if he worked alone. His rate is 1/x of a job per hour.

Combining the three individual rates gives

1/5 + 1/6 + 1/x = (6x)/(30x) + (5x)/(30x) + (30)/(30x)

1/5 + 1/6 + 1/x = (6x+5x+30)/(30x)

1/5 + 1/6 + 1/x = (11x+30)/(30x)

The expression (11x+30)/(30x) is the total rate if the three people worked together. This is assuming neither worker slows another person down.

Set this equal to 1/2 as this is the combined rate (based on the fact everyone teaming up gets the job done in 2 hours). Then solve for x

(11x+30)/(30x) = 1/2

2(11x+30) = 30x*1 .... cross multiply

22x+60 = 30x

60 = 30x-22x

60 = 8x

8x = 60

x = 60/8

x = 7.5

It takes Peter 7.5 hours, or 7 hours 30 minutes, to get the job done if he worked alone.

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Here's another equation to solve though its fairly the same idea as above

1/5 + 1/6 + 1/x = 1/2

30x*(1/5 + 1/6 + 1/x) = 30x*(1/2) ... multiply both sides by LCD

30x(1/5) + 30x(1/6) + 30x(1/x) = 30x(1/2)

6x + 5x + 30 = 15x

11x + 30 = 15x

30 = 15x-11x

30 = 4x

4x = 30

x = 30/4

x = 7.5

We get the same answer

Answer 2

Answer:  7 . 5 hrs

Step-by-step explanation:

It takes Jane 5 hours to finish the fence so she can get  $\dfrac{1}{5}$ of the job done in 1 hour.

It takes Paul 6 hours to finish the fence so he can get  $\dfrac{1}{6}$ of the job done in 1 hour.

It takes Peter x hours to finish the fence so he can get  $\dfrac{1}{x}$ of the job done in 1 hour.

Together, it takes them 2 hours to finish the fence so they can get  $\dfrac{1}{2}$ of the job done in 1 hour.

Jane + Paul + Peter = Together

$\dfrac{1}{5}\quad +\quad \dfrac{1}{6}\quad +\quad \dfrac{1}{x}\quad =\quad \dfrac{1}{2}$

Multiply everything by 30x to eliminate the denominator:

$\dfrac{1}{5}(30x) + \dfrac{1}{6}(30x) +\dfrac{1}{x}(30x) =\dfrac{1}{2}(30x)$

Simplify and solve for x:

6x + 5x + 30 = 15x

      11x + 30 = 15x

              30 = 4x

              $\dfrac{30}{4}=x$

              7.5 = x