Difference between revisions of "2011 AMC 10B Problems/Problem 15"

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== Solution ==
 
== Solution ==
 
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Simplify each operation and see which ones hold true.
Just simplify each operation and see which ones hold true.
 
  
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
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\text{III.} \qquad x @ (y @ z) &= (x @ y) @ (x @ z)\\
 
\text{III.} \qquad x @ (y @ z) &= (x @ y) @ (x @ z)\\
 
x @ \frac{y+z}{2} &= \frac{x+y}{2} @ \frac{x+z}{2}\\
 
x @ \frac{y+z}{2} &= \frac{x+y}{2} @ \frac{x+z}{2}\\
\frac{2x+y+z}{2} &= \frac{2x+y+z}{2}
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\frac{2x+y+z}{4} &= \frac{2x+y+z}{4}
 
\end{align*}</cmath>
 
\end{align*}</cmath>
  

Revision as of 11:06, 13 July 2021

Problem

Let $@$ denote the "averaged with" operation: $a @ b = \frac{a+b}{2}$. Which of the following distributive laws hold for all numbers $x, y,$ and $z$? \[\text{I. x @ (y + z) = (x @ y) + (x @ z)}\] \[\text{II. x + (y @ z) = (x + y) @ (x + z)}\] \[\text{III. x @ (y @ z) = (x @ y) @ (x @ z)}\]

$\textbf{(A)}\ \text{I only} \qquad\textbf{(B)}\ \text{II only} \qquad\textbf{(C)}\ \text{III only} \qquad\textbf{(D)}\ \text{I and III only} \qquad\textbf{(E)}\ \text{II and III only}$

Solution

Simplify each operation and see which ones hold true.

\begin{align*} \text{I.} \qquad x @ (y + z) &= (x @ y) + (x @ z)\\ \frac{x+y+z}{2} &= \frac{x+y}{2} + \frac{x+z}{2}\\ \frac{x+y+z}{2} &\not= \frac{2x+y+z}{2} \end{align*}

\begin{align*} \text{II.} \qquad x + (y @ z) &= (x + y) @ (x + z)\\ x+ \frac{y+z}{2} &= \frac{2x+y+z}{2}\\ \frac{2x+y+z}{2} &= \frac{2x+y+z}{2} \end{align*}

\begin{align*} \text{III.} \qquad x @ (y @ z) &= (x @ y) @ (x @ z)\\ x @ \frac{y+z}{2} &= \frac{x+y}{2} @ \frac{x+z}{2}\\ \frac{2x+y+z}{4} &= \frac{2x+y+z}{4} \end{align*}

$\boxed{\textbf{(E)} \text{II and III only}}$

See Also

2011 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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

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