We know that John can paint the house in 28 hours so we can use the following equation to determine the John’s work rate.

$$\left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{ of John}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{ of John}}\end{array} \right) = \left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{ of John}}\end{array} \right)\left( {28} \right) = 1\hspace{0.25in} \Rightarrow \hspace{0.25in}{\mbox{Work Rate of John = }}\frac{1}{{28}}$$

Similarly, if we let $t$ be the amount of time it takes Dave to paint the house by himself we have the following relationship between the time and work rate of Dave.

$$\left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{ of Dave}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{ of Dave}}\end{array} \right) = \left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{ of Dave}}\end{array} \right)\left( t \right) = 1 \hspace{0.25in} \Rightarrow \hspace{0.25in}{\mbox{Work Rate of Dave = }}\frac{1}{t}$$ Show Step 3

We can now plug in the information from the second step as well as the fact that it takes John and Dave 17 hours to paint the house by themselves into the word equation from the first step to get,

$$\begin{align*}\left( {\frac{1}{{28}}} \right)\left( {17} \right) + \left( {\frac{1}{t}} \right)\left( {17} \right) & = 1\\ \frac{{17}}{{28}} + \frac{{17}}{t} & = 1\end{align*}$$ Show Step 4

Now we can solve this for $t$.

$$\begin{align*}\frac{{17}}{t} & = 1 - \frac{{17}}{{28}}\\ \frac{{17}}{t} & = \frac{{11}}{{28}}\\ 17 & = \frac{{11}}{{28}}t\\ \frac{{476}}{{11}} & = t\hspace{0.25in} \Rightarrow \hspace{0.25in}t = 43.2727\end{align*}$$

So, it will take Dave approximately 43.2727 hours to paint the house by himself.