Difference between revisions of "2007 AMC 10B Problems/Problem 2"

Revision as of 22:35, 12 January 2018

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

$\textbf{(A) } -16 \qquad\textbf{(B) } -8 \qquad\textbf{(C) } 0 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 16$

Substitute and simplify. $(3+5)5 - (5+3)3 = (3+5)2 = (8)2 = \boxed{\textbf{(E) }16}$

Note that $(a \star b) - (b \star a) = (a+b)b - (b+a)a = (a+b)(b-a) = b^2 - a^2 = 5^2 - 3^2 = \boxed{\textbf{(E) }16}$

2007 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

@@ Line 11: / Line 11: @@
 ==Solution 2==
-Note that <math>\begin{align*}
+Note that
-(a \star b) - (b \star a) &= (a+b)b - (b+a)a \\
+<math>(a \star b) - (b \star a) = (a+b)b - (b+a)a
-&= (a+b)(b-a) \\
+= (a+b)(b-a)
-&= b^2 - a^2 \\
+= b^2 - a^2
-&= 5^2 - 3^2 = \boxed{\textbf{(E) }16}
+= 5^2 - 3^2 = \boxed{\textbf{(E) }16}</math>
-\end{align*}</math>
 ==See Also==