Financial Economics: Optimal Risky Portfolio question - involves Sharpe ratio

I think I've managed to solve this one, but there's a part or two I'm unsure of due to some of the question's wording. I'll first list the question.

Quote:

Your parents find out that you are taking financial economics and come to you for financial advice. A family friend has asked them to buy equity in a gold mine that she is developing in northern Ontario. The expected return of the gold mine is 9% and its standard deviation is 40%. Currently your parents divide their savings between a risk-free asset (Government of Canada bonds) and a large, well-diversified portfolio of stocks. The expected return of the risk-free asset is 3%, and the expected return of the portfolio of risky stocks is 9% with a standard deviation of 25%. The gold mine’s returns are independent of your parents’ stock portfolio. They are considering investing 5% of their money in the gold mine. Would you advise them to undertake this investment?

Here's my answer:

, , , , ,

(this is something I'm unsure of, but I'm assuming the covariance is zero because the question says the gold mine's returns are independent of the stock portfolio)

I first compute the Sharpe ratio of the portfolio:

Then, for the Sharpe ratio of the new portfolio, we need the mean and variance of said new portfolio.

This is less than

I now compute the Sharpe ratio of the new portfolio:

Since the Sharpe ratio of the new portfolio is higher than that of the original portfolio, it would be advisable to undertake the investment.

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I'm not sure if I made a mistake somewhere, but if I did, could you show/correct it?