AMC 10 · 2005 · #2

Grade 8 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

percentagequadratic-equations convert-to-algebra ↑ Prerequisites: percentagequadratic-equations

📏 Medium solution 💡 2 insights

Problem

A positive number x satisfies the condition that x% of x equals 4. Find x.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Convert to Algebra

The sentence "x% of x is 4" is a word statement hiding one equation, so Tool #13 (Convert to Algebra) turns it into x/100 · x = 4 and solves for x. Tool #8 (Analyze the Units) is the key unlock: "percent" means "per hundred," so x%=x/100, which stops the common mistake of reading x% as just x. Tool #3 (Eliminate Possibilities) is a fast backstop — because the choices are given, the found value can be checked against them, and only 20 makes x% of x equal 4.

1STEP 1

Rewrite the percent

"Percent" means per hundred, so x% is $\frac{x}{100}$ , and "of" means multiply — so "x% of x" is $\frac{x}{100}\cdot x$ .

x% of x = x/100 · x

2STEP 2

Write the equation

Set that product equal to 4; the two copies of x combine into $x^2$ , giving one clean equation $\frac{x^2}{100} = 4$ .

x/100 · x = 4 ⟹ x²/100 = 4

3STEP 3

Clear the denominator

Multiply both sides by 100 to undo the division: the left side becomes plain $x^2$ and the right becomes 400.

x²/100 = 4 ⟹ x² = 400

4STEP 4

Take the positive square root

$<span class="hl-ask">20</span>^2 = 400$ and the negative root is rejected since x is positive, so x is 20, choice (D) — indeed 20% of 20 is 4.

x² = 400 ⟹ x = √(400) = 20 → (D)

Answer

Test the answer straight against the words: 20% of 20 is 0.2×20=4, which matches exactly. The other choices fail — for example 40% of 40 = 16 (too big) and 10% of 10 = 1 (too small) — and they even bracket the answer, since the percent-of-itself value grows as x grows. A tempting trap is to read "x% of x" as just x=4 and pick (B); that ignores the "percent" and the "of," both of which shrink the result.

💡Key takeaway

"Percent" means "out of 100" and "of" means multiply, so "x% of x" is x²/100 — set that equal to 4 and the positive square root gives x=20.

Rewrite the percent
Write the equation
Clear the denominator
Take the positive square root

Rewrite the percent

Write the equation

Clear the denominator

Take the positive square root

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