2007 AMC 10A Problem 2

P1-10~1 minPrint

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

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

Show solution without answering

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.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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)