I have a question about a relatively simple probability exercise involving a sum of components. It goes as follows:
A cargo ship has a maximum payload of $50,000$ kilograms. The ship is loaded with boxes of rice, that each have a mean weight of $50$ kilograms and a standard deviation of $5$ kilograms.
How many boxes can be loaded onto the ship, so that the odds of the maximum payload being exceeded is smaller than or equal to $0.001$?
I thought that if each box $X$ was distributed as
$$X \sim \mathcal{N} ( 50,5^2)$$
the sum $S$ of the boxes would be distributed as
$$S\sim \mathcal{N} ( n\cdot 50, n\cdot 5^2)$$
so that the solution to the question would be $$P(S \leq 50 000) = 0.999.$$
$$P\left(Z \leq\frac{50 000 - n⋅50}{5 \cdot \sqrt{n}} \right) = 0.999$$
$$\frac{50 000 - n\cdot 50}{5 \cdot \sqrt{n}}= 2.97$$
If I solve it this way, I get two complex solutions. According to the solutions manual, the answer should be $990$. Can someone tell me what I did wrong?