AMC 10 · 2011 · #1

Grade 5 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
order-of-operationsfraction-arithmetic identify-subproblems ↑ Prerequisites: fraction-arithmetic
📏 Short solution 💡 1 insight
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Problem
Two fractions built from the same two sums are reciprocals of each other. Find their difference.

Pick an answer.

(A)
-1
(B)
$\frac{5}{36}$
(C)
$\frac{7}{12}$
(D)
$\frac{147}{60}$
(E)
$\frac{43}{3}$
How to solve
Strategy Identify Subproblems

The expression stacks several small jobs — two sums, two fractions, one subtraction — so Tool #7 (Identify Subproblems) does them one at a time: add the numbers, build and simplify each fraction, then subtract. Tool #3 (Eliminate Possibilities) gives a fast sanity check: the bigger fraction (4/3) comes first and the smaller (3/4) is subtracted, so the result is positive but small — that alone kills the negative (A) and the large answers (D) and (E).

1STEP 1

Add the evens and the odds

The two groups add to 12 and 9.

2+4+6 = 12, 1+3+5 = 9
2STEP 2

Build and simplify the two fractions

Both fractions reduce nicely.

12/9 = 4/3, 9/12 = 3/4
3STEP 3

Subtract using a common denominator

Subtracting gives 7/12, choice (C).

4/3 - 3/4 = 16/12 - 9/12 = 7/12 → (C)
Answer
7/12
The expression is x - tfrac1x with x=4/3, which is a bit bigger than 1. Subtracting a number from its slightly-smaller reciprocal gives a small positive result, so an answer between 0 and 1 is exactly right — 7/12≈ 0.58 fits. This immediately rules out the negative (A), the greater-than-2 (D), and the huge (E); and 5/36≈ 0.14 (B) is far too small.
💡Key takeaway

Add each little pile into one number first, simplify the fractions, then match denominators before subtracting.

  • Add the evens and the odds
  • Build and simplify the two fractions
  • Subtract using a common denominator

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