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2007 AMC 10A Problem 2

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2007 AMC 10A Problem 2AMC 10 Step by Step

Name: Date: September 13, 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

A custom operation is just a formula; substitute a = 6 and b = 2 into each definition separately, then divide.

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

Why this works

Invented symbols like $@$ and $\#$ are functions of two inputs; the only skill tested is careful substitution. Write the definition with the values in place before simplifying so that " $b^2$ " and " $ab^2$ " are computed with the right variable.

The trap

Squaring the wrong variable (computing ab - a^2 or a + b - a^2 b) or dropping the minus sign in the final fraction.

Common mistakes

Squaring the wrong variable (computing $ab - a^2$ or $a + b - a^2 b$ ) or dropping the minus sign in the final fraction.
Reading $ab^2$ as $(ab)^2 = 144$ , which gives a denominator of $-136$ and no matching choice.

Techniques

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