Question

John and Susan can clean their house in 5 hours. Working alone, Susan can clean the house one hour faster than John can clean the house. How long does it take each person to clean the house alone?

Answer 1

Let J be the time (in hours) John takes to clean the house alone, and S be the time (in hours) Susan takes to clean the house alone since they can clean the house in 5 hours together, and Susan can clean the house one hour faster than John, then we can set the following system of equations:

$\begin{gathered} \frac{1}{S}\cdot5+\frac{1}{J}\cdot5=1, \\ S=J-1. \end{gathered}$

Substituting the second equation in the first one we get:

$\frac{5}{J-1}+\frac{5}{J}=1.$

Multiplying the above equation by (J-1)J we get:

$(\frac{5}{J-1}+\frac{5}{J})\times((J-1)J)=1\times(J-1)J\text{.}$

Simplifying the above equation we get:

$\begin{gathered} \frac{5}{J-1}\times(J-1)J+\frac{5}{J}\times(J-1)J=(J-1)J, \\ 5J+5(J-1)=J^2-J, \\ 10J-5=J^2-J\text{.} \end{gathered}$

Solving the above equation for J we get:

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