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2005 AMC 10B Problem 2

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2005 AMC 10B Problem 2AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 1 of 5 · about 1 min · position P1-10

A positive number $x$ has the property that $x\%$ of $x$ is $4$ . What is $x$ ?

A)
$2$
B)
$4$
C)
$10$
D)
$20$
E)
$40$

Answer

Key insight

Translate x percent of x into (x/100) times x, so x squared equals 400 and x is 20.

Solution

The phrase " $x\%$ of $x$ " means $\frac{x}{100} \cdot x = \frac{x^2}{100}$ . Setting this equal to $4$ :
$\frac{x^2}{100} = 4 \quad\Longrightarrow\quad x^2 = 400.$
Since $x$ is positive, $x = 20$ .

Check: $20\%$ of $20$ is $0.2 \cdot 20 = 4$ .

The answer is $\boxed{\textbf{(D)}\ 20}$ .

Why this works

A percent is a hidden factor of $\frac{1}{100}$ . Once the sentence is translated literally into algebra, only a one-line equation remains. Reading "of" as multiplication and "percent" as division by $100$ turns nearly every percent word problem into arithmetic.

Alternative approach

Test the choices: $10\%$ of $10$ is $1$ , too small; $20\%$ of $20$ is $4$ , which matches. Only (D) works.

The trap

Reading the condition as x times x equals 4 (forgetting the percent) and answering 2.

Common mistakes

Reading the condition as x times x equals 4 (forgetting the percent) and answering 2.
Taking $x\%$ of $x$ to mean $x + x\%$ or $x \cdot x\%$ with the percent sign dropped, which produces other choices.

Techniques

Set up the equation/formula and compute; no special trick needed