Ginni needs 120 minutes to inflate a massive bouncy castle by herself, while Sheryl would puff away for 150 minutes on her own. They both start inflating a castle together. After 45 minutes of teamwork, Elon jumps in and starts to help inflate as well. With the three of them going at it, the bouncy castle is fully inflated 13 minutes after Elon started helping. How many minutes would it take Elon to inflate a castle on his own?

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.

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