# Difference between revisions of "2013 AMC 12A Problems/Problem 16"

Epicwisdom (talk | contribs) (Created page with "Let pile <math>A</math> have <math>A</math> rocks, and so on. The mean weight of <math>A</math> and <math>C</math> together is <math>44</math>, so the total weight of <math>A</m...") |
Epicwisdom (talk | contribs) m (Clarification) |
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Note that <math>\frac{B}{B + C} < 1</math>, and the maximal value of this factor occurs when <math>C = 1</math> | Note that <math>\frac{B}{B + C} < 1</math>, and the maximal value of this factor occurs when <math>C = 1</math> | ||

− | Also note that <math>\frac{46}{3}</math> must cancel to give an integer value, and the only fraction that satisfies these conditions is <math>\frac{45}{46}</math> | + | Also note that <math>\frac{46}{3}</math> must cancel to give an integer value, and the only fraction that satisfies both these conditions is <math>\frac{45}{46}</math> |

Plugging in, we get | Plugging in, we get | ||

<math>44 + \frac{46}{3} * \frac{45}{46} = 44 + 15 = 59</math> | <math>44 + \frac{46}{3} * \frac{45}{46} = 44 + 15 = 59</math> |

## Revision as of 05:06, 7 February 2013

Let pile have rocks, and so on.

The mean weight of and together is , so the total weight of and is

To get the total weight of and , we need to add the total weight of and subtract the total weight of

And then dividing by the number of rocks and together, to get the mean of and ,

Simplifying,

Now, to get rid of the in the numerator, we use two definitions of the total weight of and

Substituting back in,

Note that , and the maximal value of this factor occurs when

Also note that must cancel to give an integer value, and the only fraction that satisfies both these conditions is

Plugging in, we get