Express 99 2/3% as a fraction? No calculator

You can almost eyeball this, which is a useful technique for dealing with multiple choice quizzes against the clock.

If 100% = 1, then you're looking for a fraction slightly less than one. That lets you rule out all but two of the answers. How much less? A third of a percent. One percent is 1/100. Divide that by three to get 1/300. That points you at 299/300 being the answer.

So you want to find $$\frac{99+\frac{2}{3}}{100}.$$

The numerator is equal to $$\frac{3(99)}{3}+\frac{2}{3} = \frac{297}{3}+\frac{2}{3} = \frac{299}{3}.$$

Dividing by $100$ you get $$\frac{\frac{299}{3}}{100} = \frac{299}{3(100)}=\frac{299}{300}.$$

The reason you are getting the wrong answer is because you are rounding. The fraction $\frac{2}{3}$ is not equal to $0.66$ but rather is equal to $0.6666\cdots$ (often written $0.\overline{6}$).

$99\frac{2}{3}\%$ is $\frac{1}{3}\%$ away from $100\%$ or $1$. $\frac{1}{3}\%$ is literally $\frac{1}{300}$. The answer is therefore $\frac{299}{300}$.