AMC 10 · 2014 · #2
Grade 8 arithmeticPick an answer.
AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The fraction is really three small jobs stacked together, so Tool #7 (Identify Subproblems) tackles them one at a time: first add the two equal powers on top, then add the two equal reciprocals on the bottom, then divide. Tool #3 (Eliminate Possibilities) is a quick guard — the bottom is a tiny number (1/4), so dividing by it must make the answer much bigger than the top value 16, which rules out (A) 16 immediately.
Add the two equal powers on top
Two copies of 2³ = 8 double to 16, which is the same as 2 · 2³ = 2⁴.
Two copies of a power add up to double that power, the same as bumping the exponent up by one.
Two copies of a power add up to double that power, the same as bumping the exponent up by one.
▸ Why?
A shared power can be lifted out of the sum at once, leaving a plain count of copies.
▸ Why?
An exponent counts how many times a factor is used, so doubling adds exactly one more use.
Add the two equal reciprocals on the bottom
A negative exponent flips it, so 2⁻³ = 1/8 and the bottom is 1/8 + 1/8 = 1/4.
A negative exponent flips the power to the bottom of a fraction, turning 2⁻³ into 1/8.
8.EE.A.1Identify SubproblemsDivide the top by the bottom
Dividing by 1/4 means multiplying by 4, so 16 · 4 = 64 — choice (E), far above 16, so (A) is out.
Dividing by a quarter counts how many quarters fit inside — four times as many as the number itself.
6.NS.A.1Eliminate PossibilitiesBreak a messy fraction into top, bottom, and divide — and remember a negative exponent flips the power into a fraction's bottom.
- Add the two equal powers on top
- Add the two equal reciprocals on the bottom
- Divide the top by the bottom