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