2021 AMC 10A Problem 8

The student's answer came out too small because he used a smaller number. So look at exactly what he left out.

1. The correct number is $1.ababab\ldots$ and he used $1.ab$. The difference is the tail he dropped: $0.00ababab\ldots$
For example, if $ab$ were $12$: $1.121212\ldots - 1.12 = 0.001212\ldots$
2. Let $N$ be the two-digit number $\underline{a}\,\underline{b}$. A known fact about repeating decimals: $0.ababab\ldots = \frac{N}{99}$. (For example, $0.121212\ldots = \frac{12}{99}$.)
3. The dropped tail $0.00abab\ldots$ is that same decimal moved two places to the right, which divides it by $100$. So the tail is $\frac{N}{9900}$.
4. Multiplying by $66$, his answer was short by
$$66 \times \frac{N}{9900} = \frac{N}{150}.$$
5. This shortfall is $0.5$, so $\frac{N}{150} = \frac{1}{2}$ and $N = 75$.
6. Check: $66 \times 1.\overline{75} = 66 \times \left(1 + \frac{75}{99}\right) = 66 + 50 = 116$, while $66 \times 1.75 = 115.5$. The difference is $0.5$.

The answer is $\boxed{\textbf{(E)}\ 75}$.

Subtracting the wrong product from the right one leaves only the piece of the number that was ignored, so you never need $a$ and $b$ separately. A two-digit repeating block is $\frac{N}{99}$, and every place it is shifted to the right divides it by another $10$.

Test the choices using the shortfall $\frac{N}{150}$ from step 4: $N = 45$ gives $0.3$, $N = 60$ gives $0.4$, and $N = 75$ gives $0.5$. Only $75$ works.

Let $N = \underline{a}\,\underline{b}$ be the two-digit number. The correct factor is $1.\overline{ab}$ and the student used $1.ab$; their difference is the part he dropped:
$$1.\overline{ab} - 1.ab = 0.00\overline{ab} = \frac{1}{100}\cdot 0.\overline{ab} = \frac{N}{9900}.$$

Multiplying by $66$ changes the product by $66 \cdot \frac{N}{9900} = \frac{N}{150}$, and this must equal $0.5$:
$$\frac{N}{150} = \frac{1}{2} \quad\Longrightarrow\quad N = 75.$$

The answer is $\boxed{\textbf{(E)}\ 75}$.