AMC 10 · 2013 · #1

Grade 5 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

Add the numbers in each group to form the two fractions in the expression (2+4+6)/(1+3+5) - (1+3+5)/(2+4+6), where the second fraction is the first one turned upside down, then subtract the second from the first. What single number does this expression equal?

Pick an answer.

(A)

-1

(B)

$\frac{5}{36}$

(C)

$\frac{7}{12}$

(D)

$\frac{49}{20}$

(E)

$\frac{43}{3}$

AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The expression is built from small pieces, so Tool #7 (Identify Subproblems) breaks it into three easy jobs: add each group of numbers, reduce the two fractions, then subtract. Tool #3 (Eliminate Possibilities) uses the shape of the problem as a guardrail — the first fraction is bigger than 1 and the second is smaller than 1, so their difference is a small positive number, which throws out the negative choice (A) and the large choices (D) and (E) before any careful arithmetic.

1STEP 1

Add the numbers in each group

The top group is 2+4+6=12, the bottom 1+3+5=9, so the expression becomes 12/9-9/12 — the same two numbers, swapped.

2+4+6 = 12, 1+3+5 = 9 → 12/9-9/12

2STEP 2

Reduce each fraction

Divide top and bottom of each by 3: 12/9 becomes 4/3 and 9/12 becomes 3/4, so the problem is 4/3-3/4 — two flips of each other.

12/9=4/3, 9/12=3/4 → 4/3-3/4

3STEP 3

Subtract using a common denominator

Over the common denominator 12, 4/3=16/12 and 3/4=9/12, so 16/12-9/12=7/12 — choice (C).

4/3-3/4=16/12-9/12=7/12 → (C)

Answer

7/12

The first fraction 4/3≈1.33 is a bit above 1 and the second 3/4=0.75 is a bit below 1, so their difference should be a small positive number near 0.5. The result 7/12≈0.58 fits exactly. A negative answer like (A) is impossible because the bigger fraction is in front, and the large choices (D) 2.45 and (E) ≈14.3 are far too big.

💡Key takeaway

Add the repeated groups first, then subtract the two fractions over a common denominator — a bigger-than-one fraction minus a smaller-than-one fraction leaves a small positive number.

Add the numbers in each group
Reduce each fraction
Subtract using a common denominator

Add the numbers in each group

Reduce each fraction

Subtract using a common denominator

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