AMC 10 · 2005 · #2
Grade 8 arithmeticPick an answer.
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.
Rewrite the percent
Percent means per hundred, so the phrase becomes a fraction.
A percent is just a fraction out of 100, and "of" means multiply.
6.RP.A.3Analyze The UnitsWrite the equation
The unknown appears twice, giving a plain square.
Turning the words into one equation lets the unknown be solved for directly.
6.EE.B.6Convert To AlgebraClear the denominator
Clearing the denominator gives 400.
Whatever is done to one side of an equation must be done to the other to keep it balanced.
6.EE.B.7Convert To AlgebraTake the positive square root
Only the positive root is allowed, giving 20, choice (D).
Squaring is undone by a square root, and "positive number" picks the positive root.
Taking a square root undoes the squaring, and the word positive picks which of the two roots is meant.
▸ Why?
A percent is a fraction out of a hundred, so the words turn into one equation with a single unknown.
▸ Why?
A root reverses a power exactly, so applying it recovers the number that was squared.
"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