Inflate Rate

• Ginni takes 120 minutes to inflate the castle, so \begin{equation} R_G =  \frac{1}{120}  \end{equation}
• Sheryl takes 150 minutes to inflate the castle, so \begin{equation} R_S = \frac{1}{150}  \end{equation}

The castle is also inflated if Ginni and Sheryl work together for 45 minutes, and if Elon helps the last 13 minutes, making it a total of 58 minutes. How long would we expect Ginni and Sheryl to take if they had done it together?

• If you sum up Ginni's and Sheryl's inflation rates, you get \begin{equation} R_{G + S} = \frac{1}{120} + \frac{1}{150} = \frac{3}{200} \end{equation}
• In other words, we would expect it to take 66.67 minutes to be fully inflated if only Ginni and Sheryl had worked together.

After 45 minutes, the percentage of the castle that Ginni and Sheryl already inflated is equal to: \begin{equation} 45 * \frac{3}{200} = \frac{27}{40} \end{equation} Let's denote Elon's rate as: \begin{equation} R_E = \frac{1}{x} \end{equation} The combined rate for the last 13 minutes is equal to: \begin{equation} R_{G + S + E} = \frac{1}{120} + \frac{1}{150} + \frac{1}{x} \end{equation} Since they completed the remaining $\frac{13}{40}$ of the castle in 13 minutes, their combined rate is equal to: \begin{equation} R_{G + S + E} = \frac{\frac{13}{40}}{13} = \frac{13}{520} \end{equation} Now we have that \begin{equation} R_{G + S + E} = \frac{1}{120} + \frac{1}{150} + \frac{1}{x} = \frac{13}{520} \end{equation} \begin{equation} \frac{1}{x} = \frac{13}{520} - \frac{3}{200} = \frac{1}{100} \end{equation} So, it would take Elon 100 minutes to inflate the castle by his own.