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2001 AMC 10 Problem 7

2001 AMC 10 Problem 7AMC 10 Step by Step

Name: Date: September 13, 2026

Fractions & Decimals · difficulty 2 of 5 · about 2 min · position P1-10

When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number?

A)
$0.0002$
B)
$0.002$
C)
$0.02$
D)
$0.2$
E)
$2$

Answer

Key insight

Moving the decimal point four places right multiplies by 10^4, so 10^4 x = 4/x gives x^2 = 4 * 10^(-4) and x = 2 * 10^(-2).

Solution

Let the number be $x > 0$ . Shifting the decimal point four places to the right produces $10^4 x$ . The condition says
$10^4 x = \frac{4}{x}.$
Multiply both sides by $x$ : $10^4 x^2 = 4$ , so $x^2 = \dfrac{4}{10^4}$ . Taking the positive square root,
$x = \frac{2}{10^2} = 0.02.$

Check: $0.02$ shifted becomes $200$ , and $4 \cdot \frac{1}{0.02} = 4 \cdot 50 = 200$ .

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

Why this works

"Move the decimal point $k$ places right" is just multiplication by $10^k$ ; once that is written down, the sentence becomes a one-variable equation. Keep the power of $10$ as a power (not as a string of zeros) so that the square root is $10^{2}$ rather than a miscounted decimal.

Alternative approach

Test the choices. For (C): $0.02 \to 200$ and $\frac{4}{0.02} = 200$ , equal. For (D): $0.2 \to 2000$ but $\frac{4}{0.2} = 20$ . The shifted number grows while the reciprocal shrinks, so only one choice can balance.

The trap

Taking the square root of 4/10000 as 2/100 with the wrong number of zeros, landing on 0.002 or 0.2.

Common mistakes

Taking the square root of 4/10000 as 2/100 with the wrong number of zeros, landing on 0.002 or 0.2.
Setting up $10^4 x = 4x$ (four times the number instead of four times its reciprocal), which has no positive solution.

Techniques

Use the answer choices (mod checks, size, form) to eliminate or select · Set up the equation/formula and compute; no special trick needed