Percentage Problem Trying to help a student with this, but I cannot see how I am doing it wrong. Here is the problem and the answer from the book.
Q. Tina invested $30,000$ in a stock. In the first year, the
stock increased in value by $10\%$. In the second year,
the stock decreased in value by $20\%$. What percent-
age gain is required in the third year for Tina’s stock
to return to its original value? (Round to the nearest
tenth of a percent.)
A. $13.6\%$
My work:
The first year as $30000*10\%=3000$,  $30000+3000=33000$
Second year as $33000*-20\%=-6600$, $33000-6600=26400$
Third year as $26400/30000=0.88,$ $1-0.88=.12=12\%$
By steps, please, what am I doing wrong?
 A: Your steps were fine until you calculated the third-year percentage. It looks like you calculated the percentage growth relative to the original amount ($\$30,000$) rather than relative to the amount after the second year ($\$26,400$).
You can think of this like an algebraic equation:
$$\begin{align*}
26 \ 400 \times  \left(1 + \frac{x}{100} \right) &= 30 \ 000\\[5pt]
1 + \frac{x}{100} &= \frac{30 \ 000}{26 \ 400}\\[5pt]
\frac{x}{100} &= \frac{3 \ 600}{26 \ 400} \\[5pt]
x &= \frac{3 \ 600}{264} \approx 13.6 \\[5pt]
\end{align*}$$
In short, the amount Tina needs to make up - $\$3,600$ - is $12\%$ of $\$30,000$ as you have calculated, but that is not the value the question is looking for. The question is looking for the amount that Tina needs to gain based on what she has after the second year. $\$3,600$ is $13.6\%$ of $\$26,400$, which is why it is the answer.
A: The mistake appears to be in the last percantage calculation. You set
$$100\%=30'000$$
$$88\%=26'400,$$
so you got $12\%$. The order is incorrect. You should have set
$$100\%=26'400$$
$$113.\dot{6}\dot{3}\%=30'000,$$
So the difference is rounded $13.6\%$.
A: Those $30,000$ don't matter at all.  After the first period Tina has $1.1$-times as before; after the second period only $80$% of that amount, that is in total $1.1\cdot0.8=0.88$-times of the original amount: a loss of $12$ percent.
To come out even the change rate $r$ in the third period must satisfy $0.88\cdot r=1$, that is $r=1/0.88\approx 1.136$.
A: You have $\frac{26400}{30000}$ but you should have $\frac{30000}{26400}$.
