AMC 10 · 2008 · #2

Grade 6 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
fraction-arithmeticfraction-multiplication identify-subproblemsconvert-to-algebra ↑ Prerequisites: fraction-arithmetic
📏 Medium solution 💡 1 insight
Problem
Two fractions are added together. Find the reciprocal of that sum.

Pick an answer.

(A)
$\frac{6}{7}$
(B)
$\frac{7}{6}$
(C)
$\frac{5}{3}$
(D)
3
(E)
$\frac{7}{2}$
How to solve
Strategy Identify Subproblems

The phrase 'reciprocal of a sum' hides two separate jobs. First collapse 1/2+2/3 into a single fraction, because you cannot flip a sum term by term. Then invert that single fraction. Naming the sum s and the answer r keeps the two jobs from blurring, and the defining equation r × s = 1 gives a test that turns the final flip from a memorised rule into something checkable.

1STEP 1

Write down what reciprocal means

A reciprocal is whatever multiplies the sum to one.

s=1/2+2/3, r × s=1
2STEP 2

Rewrite both fractions as sixths

A common denominator lets the fractions add.

1/2=(1 × 3)/(2 × 3)=3/6, 2/3=(2 × 2)/(3 × 2)=4/6
3STEP 3

Add to get the single fraction s

Adding gives the single fraction 7/6.

s=3/6+4/6=7/6
4STEP 4

Propose the flipped fraction

Flipping it proposes 6/7.

s=7/6 ⟶ r ?= 6/7
5STEP 5

Check the product is 1

The product really is one, so the answer is 6/7, choice (C).

7/6×6/7=(7 × 6)/(6 × 7)=42/42=1
Answer
6/7
Size check first: 2/3 is more than 1/2, so s ＞ 1/2+1/2=1, and s ＜ 1/2+1=3/2. A number bigger than 1 has a reciprocal smaller than 1, so r must sit between 2/3 and 1, and 6/7≈ 0.857 does. The other four choices are all at least 7/6, so they are all too big. Worth being clear about what that size check does and does not do: it only rules the other options out, and it works here only because exactly one listed value is below 1. It never shows 6/7 is the reciprocal of anything. The product check 7/6×6/7=1 is the part that actually confirms the answer, and it would still confirm it with no answer choices printed at all.
💡Key takeaway

You cannot flip a sum piece by piece, so add the fractions into one fraction first, flip that, then multiply back to make sure you get 1.

  • Write down what reciprocal means
  • Rewrite both fractions as sixths
  • Add to get the single fraction s
  • Propose the flipped fraction
  • Check the product is 1

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