AMC 10 · 2009 · #2

Grade 6 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
fraction-arithmeticorder-of-operations identify-subproblems ↑ Prerequisites: fraction-arithmetic
📏 Short solution 💡 1 insight
Problem
A compound fraction has numerator 1/3-1/4 and denominator 1/2-1/3, and the long fraction bar between them means divide the top value by the bottom value. Find the single number this compound fraction equals.

Pick an answer.

(A)
$\frac 14$
(B)
$\frac 13$
(C)
$\frac 12$
(D)
$\frac 23$
(E)
$\frac 34$

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The expression is really three tasks stacked together: simplify the top, simplify the bottom, then divide one by the other. Tool #7 (Identify Subproblems) handles them one at a time instead of all at once, so no common-denominator step gets skipped. Tool #3 (Eliminate Possibilities) is a quick sanity guard: the top (1/12) is much smaller than the bottom (1/6) but not tiny, so the ratio should be a modest fraction near one-half — which lets us reject the extreme-looking choices at a glance.

1STEP 1

Simplify the top

Take the numerator alone. Over the least common denominator 12, 1/3=4/12 and 1/4=3/12, so 4/12-3/12=1/12.

1/3-1/4=4/12-3/12=1/12
2STEP 2

Simplify the bottom

Same move on the denominator: 2 and 3 share the denominator 6, so 1/2=3/6 and 1/3=2/6, giving 3/6-2/6=1/6.

1/2-1/3=3/6-2/6=1/6
3STEP 3

Divide top by bottom

The bar now says 1/12÷1/6, and dividing by a fraction means multiplying by its flip: 1/12×6/1=6/12=1/2, choice (C).

1/12/1/6=1/12×6/1=6/12=1/2 → (C)
Answer
1/2
A quick size check confirms it: the top 1/12 is exactly half of the bottom 1/6 (since 1/6=2/12), so their ratio has to be 1/2. Because top and bottom are close in size and the top is the smaller one, the answer had to be a fraction a bit under 1, not something tiny like 1/4 nor near 3/4 — and 1/2 fits.
💡Key takeaway

Simplify the top and the bottom into single fractions first, then a big fraction bar just means divide — and dividing by a fraction is flip-and-multiply.

  • Simplify the top
  • Simplify the bottom
  • Divide top by bottom

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