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2015 AMC 10B Problem 7

2015 AMC 10B Problem 7AMC 10 Step by Step

Name: Date: October 28, 2026

Functions · difficulty 1 of 5 · about 2 min · position P1-10

Consider the operation "minus the reciprocal of," defined by $a\diamond b=a-\frac{1}{b}$ . What is $((1\diamond2)\diamond3)-(1\diamond(2\diamond3))$ ?

A) $-\dfrac{7}{30}$
B) $-\dfrac{1}{6}$
C) $0$
D) $\dfrac{1}{6}$
E) $\dfrac{7}{30}$

Answer

Key insight

Work from the inside parentheses out. The left side is 1/2 - 1/3 = 1/6, the right side is 1 - 3/5 = 2/5, and 1/6 - 2/5 = -7/30.

Solution

The rule is: $a \diamond b$ means "take $a$ , then subtract $1$ over $b$ ". Work from the innermost parentheses outward.

Left part, $(1 \diamond 2) \diamond 3$ .

$1 \diamond 2 = 1 - \dfrac12 = \dfrac12$ .
$\dfrac12 \diamond 3 = \dfrac12 - \dfrac13 = \dfrac36 - \dfrac26 = \dfrac16$ .

Right part, $1 \diamond (2 \diamond 3)$ .

$2 \diamond 3 = 2 - \dfrac13 = \dfrac53$ .
$1 \diamond \dfrac53 = 1 - \dfrac{1}{5/3}$ . One over $\dfrac53$ is $\dfrac35$ (flip the fraction), so this is $1 - \dfrac35 = \dfrac25$ .

Subtract.

$\dfrac16 - \dfrac25 = \dfrac{5}{30} - \dfrac{12}{30} = -\dfrac{7}{30}$ .

The answer is $\boxed{\textbf{(A)}\ -\frac{7}{30}}$ .

Why this works

A made-up operation is just a formula, so plug in carefully one layer at a time. The two groupings give different results, which is the point of the problem: for this operation the parentheses matter. Keep every step as an exact fraction, and remember that one over a fraction is that fraction flipped.

The trap

Working out 1 diamond (5/3) as 1 - 5/3, forgetting to flip 5/3 into its reciprocal 3/5.

Concise solution

Work from the innermost parentheses outward.

Left expression: $1 \diamond 2 = 1 - \tfrac12 = \tfrac12$ , then
$\tfrac12 \diamond 3 = \tfrac12 - \tfrac13 = \tfrac16.$

Right expression: $2 \diamond 3 = 2 - \tfrac13 = \tfrac53$ , then
$1 \diamond \tfrac53 = 1 - \frac{1}{5/3} = 1 - \tfrac35 = \tfrac25.$

Difference:
$\tfrac16 - \tfrac25 = \tfrac{5}{30} - \tfrac{12}{30} = -\tfrac{7}{30}.$

The answer is $\boxed{\textbf{(A)}\ -\frac{7}{30}}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 16: Functions
Introduction to Number Theory, Chapter 11: Decimals and Fractions
Introduction to Algebra, Chapter 17: Graphing Functions (§17.2)
Prealgebra, Chapter 6: Decimals (§6.3, §6.4)

Techniques

Set up the equation/formula and compute; no special trick needed