AMC 10 · 2016 · #1

Grade 8 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

exponentsfraction-arithmeticorder-of-operations identify-subproblems ↑ Prerequisites: exponents

📏 Short solution 💡 1 insight

Problem

Evaluate the expression 2a⁻¹+(a⁻¹)/(2)/a when a=

\frac{1}{2}

Pick an answer.

(A)

(B)

(C)

$\frac{5}{2}$

(D)

(E)

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

This is one compound calculation stacked on top of another, so Tool #7 (Identify Subproblems) is the natural fit: first figure out what a⁻¹ is, then build the numerator, then divide by a. Each piece is a small, safe step. Tool #4 (Introduce a Variable) keeps the meaning of a⁻¹ front and center — a negative exponent is just $\frac{1}{a}$ — so the symbol does not become a trap.

1STEP 1

A negative exponent flips the base into its reciprocal, so a⁻¹ = $\frac{1}{a}$ , and the reciprocal of $\frac{1}{<span class="hl-ask">2</span>}$ is 2.

a⁻¹=

\frac{1}{a}

=1/1/2=2

2STEP 2

Substitute a⁻¹ = 2 into the top 2a⁻¹ + $\frac{a⁻¹}{2}$ : that is 2·2 + $\frac{2}{2}$ = 4 + 1 = 5.

2a⁻¹+

\frac{a⁻¹}{2}

=2(2)+

\frac{2}{2}

=4+1=5

3STEP 3

Divide the top by a: 5 ÷ ( $\frac{1}{2}$ ) = 5·2 = 10, which is choice (D).

\frac{5}{a}

=5/1/2=5· 2=10 → (D)

Answer

The denominator a=

\frac{1}{2}

is small, so dividing by it should make the value much bigger than the numerator 5. Getting 10 — twice the numerator — fits that expectation, and the choices 1, 2,

\frac{5}{2}

are all too small to come from dividing by a half. So (D) is the sensible size.

💡Key takeaway

Turn the negative exponent into a reciprocal first, then it is just plug-in-and-compute that you already know.

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