2007 AMC 10B Problem 2

P1-10~1 minPrint

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

Define the operation $\star$ by $a \star b = (a+b)b.$ What is $(3 \star 5) - (5 \star 3)?$

Show solution without answering

Key insight

The new symbol is just a recipe: add the two numbers, then multiply by the second one. Swapping the order changes which number you multiply by.

Solution

The rule $a \star b = (a+b)b$ says: add the two numbers, then multiply by the second number. Watch which number is second each time.

1. For $3 \star 5$ the second number is $5$ :
$3 \star 5 = (3 + 5) \times 5 = 8 \times 5 = 40.$
2. For $5 \star 3$ the second number is $3$ :
$5 \star 3 = (5 + 3) \times 3 = 8 \times 3 = 24.$
3. Subtract:
$40 - 24 = 16.$

The answer is $\boxed{\textbf{(E)}\ 16}$ .

Why this works

A made-up operation is only a formula with blanks; fill the blanks in the right order. A quick check: both results start with the same sum $8$ , so the difference is $8$ times the difference of the multipliers, $8 \times (5 - 3) = 16$ . That also tells you the sign is positive.

Concise solution

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

Apply the definition to each pair, keeping track of which number is the second argument:
$3 \star 5 = (3 + 5)\cdot 5 = 40, \qquad 5 \star 3 = (5 + 3)\cdot 3 = 24.$
Then $40 - 24 = 16$ .

The answer is $\boxed{\textbf{(E)}\ 16}$ .

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)