Difference between revisions of "2016 AMC 10A Problems/Problem 23"

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==Solution==
 
==Solution==
  
We see that <math>a \diamond a = 1</math>, and think of division. Testing, we see that the first condition <math>a \diamond (b \diamond c) = (a \diamond b) \cdot c</math> is satisfied, because <math>\frac{a}{\frac{b}{c}} = \frac{a}{b} \cdot c</math>. Therefore, division is the operation <math>\diamond</math>. Solving the equation, <math>\frac{2016}{\frac{6}{x}} = \frac{2016}{6} \cdot x = 336x = 100</math>, so <math>x=\frac{100}{336} = \frac{25}{84}</math>, so the answer is <math>25 + 84 = \boxed{109}</math> (A)
+
We see that <math>a \diamond a = 1</math>, and think of division. Testing, we see that the first condition <math>a \diamond (b \diamond c) = (a \diamond b) \cdot c</math> is satisfied, because <math>\frac{a}{\frac{b}{c}} = \frac{a}{b} \cdot c</math>. Therefore, division is the operation <math>\diamond</math>. Solving the equation, <math>\frac{2016}{\frac{6}{x}} = \frac{2016}{6} \cdot x = 336x = 100</math>, so <math>x=\frac{100}{336} = \frac{25}{84}</math>, so the answer is <math>25 + 84 = \boxed{(A)109}</math>
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2016|ab=A|num-b=22|num-a=24}}
 
{{AMC10 box|year=2016|ab=A|num-b=22|num-a=24}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 22:32, 3 February 2016

Problem

A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q?$

$\textbf{(A) }109\qquad\textbf{(B) }201\qquad\textbf{(C) }301\qquad\textbf{(D) }3049\qquad\textbf{(E) }33,601$

Solution

We see that $a \diamond a = 1$, and think of division. Testing, we see that the first condition $a \diamond (b \diamond c) = (a \diamond b) \cdot c$ is satisfied, because $\frac{a}{\frac{b}{c}} = \frac{a}{b} \cdot c$. Therefore, division is the operation $\diamond$. Solving the equation, $\frac{2016}{\frac{6}{x}} = \frac{2016}{6} \cdot x = 336x = 100$, so $x=\frac{100}{336} = \frac{25}{84}$, so the answer is $25 + 84 = \boxed{(A)109}$

See Also

2016 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 22
Followed by
Problem 24
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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