9. Investment funds A, B and C claim that they can earn profits of $200 \%, 300 \%$ and $500 \%$ respectively in one year. Tommy has $\$ 90000$ and plans to invest in these funds. However, he knows that only one of these funds can achieve its claim while the other two will close down. He has thought of an investment plan which can guarantee a profit of at least $\$ n$ in one year. Find the greatest possible value of $n$.
(1 mark)
甲、乙、丙三個投資基金分別聲稱可在一年內賺取 $200 \%$ 、 $300 \%$ 和 $500 \%$ 的利潤。湯美有 90000 元，他打算用這筆錢投資在這些基金上。可是他知道，只有一個基金可以兌現承諾, 其餘兩個則會倒閉。他想到一個投資計劃, 可以確保一年後獲得最少 $n$ 元的淨利潤。求 $n$ 的最大可能值。
(1 分)
9. 30000
9. Suppose Tommy invests $\$ a, \$ b$ and $\$ c$ on funds $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ respectively. Clearly, as long as a positive net profit can be guaranteed, we must take $a+b+c=90000$. Then we want to maximise
$$
n=\min \{3 a, 4 b, 6 c\}-90000
$$
subject to the above conditions.
For maximality, we must have $3 a=4 b=6 c$. This is because when $3 a, 4 b$ and $6 c$ are not all equal, say, if $3 a<4 b$ and $3 a \leq 6 c$, then Tommy can shift some investments from fund $\mathrm{B}$ to fund $\mathrm{A}$ (or some to fund $\mathrm{C}$ as well, if necessary). In this way the value of $3 a$ increases while the values of $4 b$ and $6 c$ will remain greater than or equal to $3 a$ when the shift is sufficiently small. Therefore $n$ increases, i.e. the value of $n$ cannot be maximum when $3 a, 4 b$ and $6 c$ are not all equal.
Hence, in order to attain maximality, we want $a: b: c=\frac{1}{3}: \frac{1}{4}: \frac{1}{6}=4: 3: 2$. In this case, the maximum value of $n$ is
$$
3 a-90000=3 \times 90000 \times \frac{4}{4+3+2}-90000=30000 .
$$