2001 AMC 10 Problem 7

1. Moving the decimal point one place to the right multiplies by $10$ (for example $0.3 \to 3$). Four places multiplies by $10 \times 10 \times 10 \times 10 = 10000$.
2. Call the original number $x$. After the move it is $10000x$.
3. "Four times the reciprocal" is $4 \cdot \frac{1}{x} = \frac{4}{x}$. So
$$10000x = \frac{4}{x}.$$
4. Multiply both sides by $x$ to clear the fraction: $10000x^2 = 4$.
5. Divide by $10000$: $x^2 = \dfrac{4}{10000}$.
6. Take the square root of the top and bottom separately. $\sqrt{4} = 2$ and $\sqrt{10000} = 100$. The number is positive, so
$$x = \frac{2}{100} = 0.02.$$
7. Check: moving the point in $0.02$ four places gives $200$. The reciprocal of $0.02$ is $50$, and $4 \times 50 = 200$. They match.

The answer is $\boxed{\textbf{(C)}\ 0.02}$.

"Move the decimal point $k$ places to the right" always means "multiply by $10$ written $k$ times." Once you write that down, the sentence is a simple equation. Keeping $10000$ as a whole number (rather than counting zeros in a decimal) makes the square root easy: $\sqrt{10000} = 100$.

Test the choices. For each one, compare the shifted number with $4$ divided by the number.

Only $0.02$ makes the two columns equal.