Competition · AMC preparation · step 4 of 4

AMC 10 · 2011B · #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

Problem

This expression subtracts

\frac{1+3+5}{2+4+6}

from

\frac{2+4+6}{1+3+5}

, so the two fractions are reciprocals of each other. What single value does the whole expression equal?

Pick an answer.

(A)

-1

(B)

$\frac{5}{36}$

(C)

$\frac{7}{12}$

(D)

$\frac{147}{60}$

(E)

$\frac{43}{3}$

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

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

Add the piles first: the evens make 12 and the odds make 9, so every sum can be replaced by its value.

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

2STEP 2

Build and simplify the two fractions

Now it is $\frac{12}{9}-\frac{9}{12}$ ; dividing by 3 gives $\frac{4}{3}$ and $\frac{3}{4}$ , reciprocals of each other.

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

3STEP 3

Subtract using a common denominator

Over the common denominator 12, the two become $\frac{16}{12}-\frac{9}{12}$ , and 16-9 leaves $\frac{7}{12}$ , choice (C).

4/3 - 3/4 = 16/12 - 9/12 = 7/12 → (C)

Answer

7/12

The expression is

x-\frac{1}{x}

with

x=\frac{4}{3}

, which is a bit bigger than 1. Subtracting the slightly-smaller reciprocal from the number gives a small positive result, so an answer between 0 and 1 is exactly right —

\frac{7}{12}\approx 0.58

fits. This immediately rules out the negative (A), the greater-than-2 (D), and the huge (E); and

\frac{5}{36}\approx 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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Add the evens and the odds

Build and simplify the two fractions

Subtract using a common denominator

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