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2007 AMC 10A Problem 2

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2007 AMC 10A Problem 2AMC 10 Step by Step

Name: Date: October 28, 2026

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

Define $a@b = ab - b^{2}$ and $a\#b = a + b - ab^{2}$ . What is $\frac {6@2}{6\#2}$ ?

A) $- \frac {1}{2}$
B) $- \frac {1}{4}$
C) $\frac {1}{8}$
D) $\frac {1}{4}$
E) $\frac {1}{2}$

Answer

Key insight

The strange symbols are just recipes. Put 6 in place of a and 2 in place of b in each recipe, work out the two numbers, then divide.

Solution

Each symbol is a rule for combining two numbers. Here $a = 6$ and $b = 2$ . In both rules it is $b$ that gets squared, so $b^2 = 4$ .

1. Top: $6@2$ . The rule is $ab - b^2$ :
$6 \cdot 2 - 2^2 = 12 - 4 = 8.$
2. Bottom: $6\#2$ . The rule is $a + b - ab^2$ . Here $ab^2$ means $6 \times 4 = 24$ :
$6 + 2 - 24 = 8 - 24 = -16.$
3. Divide.
$\frac{8}{-16} = -\frac{1}{2}.$

The answer is $\boxed{\textbf{(A)}\ -\frac{1}{2}}$ .

Why this works

A made-up symbol like $@$ is only a shortcut for a formula. The whole problem is careful substitution. Writing the numbers into the formula before doing any arithmetic helps you square the right letter and keep track of the minus sign.

The trap

Squaring the wrong letter (a instead of b), or losing the minus sign so the answer comes out as positive one half.

Concise solution

Substitute $a = 6$ , $b = 2$ into each rule, paying attention to which letter is squared.

Numerator: $6@2 = 6\cdot 2 - 2^2 = 12 - 4 = 8$ .

Denominator: $6\#2 = 6 + 2 - 6\cdot 2^2 = 8 - 24 = -16$ .

Quotient: $\dfrac{8}{-16} = -\dfrac{1}{2}$ .

The answer is $\boxed{\textbf{(A)}\ -\frac{1}{2}}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 16: Functions
Introduction to Algebra, Chapter 17: Graphing Functions (§17.2)
Intermediate Algebra, Chapter 19: Functional Equations (§19.1, §19.2)

Techniques

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