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

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

Revision as of 18:11, 3 February 2016

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

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 $25 + 84 = \boxed{109}$ (A)

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