AMC 10 · 2005 · #2
Grade 8 arithmeticA positive number x has the property that x% of x is 4. What is x?
Pick an answer.
AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A positive number $x$ satisfies the condition that $x\%$ of $x$ equals $4$. Find $x$.
Givens: $x$ is a positive number; Taking $x\%$ of that same number $x$ gives exactly $4$; Answer choices: (A) $2$, (B) $4$, (C) $10$, (D) $20$, (E) $40$
Unknowns: The value of the positive number $x$
Understand
Restated: A positive number $x$ satisfies the condition that $x\%$ of $x$ equals $4$. Find $x$.
Givens: $x$ is a positive number; Taking $x\%$ of that same number $x$ gives exactly $4$; Answer choices: (A) $2$, (B) $4$, (C) $10$, (D) $20$, (E) $40$
Plan
Primary tool: #13 Convert to Algebra
Secondary: #8 Analyze the Units, #3 Eliminate Possibilities
The sentence "$x\%$ of $x$ is $4$" is a word statement hiding one equation, so Tool #13 (Convert to Algebra) turns it into $\dfrac{x}{100}\cdot x = 4$ and solves for $x$. Tool #8 (Analyze the Units) is the key unlock: "percent" means "per hundred," so $x\%=\dfrac{x}{100}$, 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 $4$.
Execute — Answer: D
6.RP.A.3 Step 1 Rewrite the percent
- "Percent" means "per hundred," so $x\%$ is $\dfrac{x}{100}$.
- The phrase "$x\%$ of $x$" then means $\dfrac{x}{100}$ multiplied by $x$.
- Notice the unknown $x$ shows up twice: once inside the percent and once as the amount being taken.
💡 A percent is just a fraction out of $100$, and "of" means multiply.
6.EE.B.6 Step 2 Write the equation
- Set that product equal to the given value $4$.
- Multiplying $\dfrac{x}{100}$ by $x$ combines the two copies of $x$ into $x^2$, so the statement becomes a single clean equation in $x$.
💡 Turning the words into one equation lets the unknown be solved for directly.
6.EE.B.7 Step 3 Clear the denominator
- Undo the division by $100$ by multiplying both sides by $100$.
- The left side becomes plain $x^2$, and the right side becomes $4\times100=400$.
💡 Whatever is done to one side of an equation must be done to the other to keep it balanced.
8.EE.A.2 Step 4 Take the positive square root
- Find the number whose square is $400$.
- Since $20^2=400$, the square root gives $x=20$.
- The equation $x^2=400$ also has the solution $x=-20$, but the problem says $x$ is positive, so that one is thrown out.
- Checking against the list, $20$ is choice (D); a quick test confirms $20\%$ of $20 = 0.2\times20 = 4$.
💡 Squaring is undone by a square root, and "positive number" picks the positive root.
6.RP.A.3 "Percent" means "per hundred," so $x\%$ is $\dfrac{x}{100}$. The phrase "$x\%$ o 6.EE.B.6 Set that product equal to the given value $4$. Multiplying $\dfrac{x}{100}$ by $ 6.EE.B.7 Undo the division by $100$ by multiplying both sides by $100$. The left side bec 8.EE.A.2 Find the number whose square is $400$. Since $20^2=400$, the square root gives $ Review
Reasonableness: Test the answer straight against the words: $20\%$ of $20$ is $0.2\times20=4$, 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.
Alternative: Skip the algebra and test the five choices directly (Tool #3). Compute $x\%$ of $x$ for each: $2\to0.04$, $4\to0.16$, $10\to1$, $20\to4$, $40\to16$. Only $x=20$ hits $4$, giving (D) — the same answer with no square root needed.
CCSS standards used (min grade 8)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Reading $x\%$ as $\dfrac{x}{100}$ and interpreting "$x\%$ of $x$" as $\dfrac{x}{100}\cdot x$.)6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Writing the condition as the equation $\dfrac{x^2}{100}=4$ in the single unknown $x$.)6.EE.B.7Solve real-world problems by writing and solving equations of the form px = q (Multiplying both sides by $100$ to reduce the equation to $x^2=400$.)8.EE.A.2Use square root and cube root symbols to represent solutions (Solving $x^2=400$ by taking the positive square root to get $x=20$.)
⭐ "Percent" means "out of $100$" and "of" means multiply, so "$x\%$ of $x$" is $\dfrac{x^2}{100}$ — set that equal to $4$ and the positive square root gives $x=20$.
⭐ "Percent" means "out of $100$" and "of" means multiply, so "$x\%$ of $x$" is $\dfrac{x^2}{100}$ — set that equal to $4$ and the positive square root gives $x=20$.
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