AMC 10 · 2016 · #1

Grade 8 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
exponentsfraction-arithmeticorder-of-operations identify-subproblems ↑ Prerequisites: exponents
📏 Short solution 💡 1 insight
Problem
An expression mixes a reciprocal with a division by the same value. Find its value.

Pick an answer.

(A)
1
(B)
2
(C)
$\frac{5}{2}$
(D)
10
(E)
20
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 1/a — so the symbol does not become a trap.

1STEP 1

Compute the reciprocal first

The reciprocal comes out to 2.

a⁻¹=1/a=1/1/2=2
2STEP 2

Simplify the top

The top then simplifies to 5.

2a⁻¹+(a⁻¹)/2=2(2)+2/2=4+1=5
3STEP 3

Divide by the small fraction

Dividing by the small fraction gives 10, choice (D).

5/a=5/1/2=5 · 2=10 → (D)
Answer
10
The denominator a=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, 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.

More like this

Same archetype — closest grade level first.