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Difference between revisions of "1987 AIME Problems/Problem 8"

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What is the largest positive integer <math>n</math> for which there is a unique integer <math>k</math> such that <math>\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}</math>?
 
What is the largest positive integer <math>n</math> for which there is a unique integer <math>k</math> such that <math>\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}</math>?
 
== Solution ==
 
== Solution ==
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 +
'''Solution 1'''
 
Multiplying out all of the [[denominator]]s, we get:
 
Multiplying out all of the [[denominator]]s, we get:
  
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Since <math>91n - 104k < n + k</math>, <math>k > \frac{6}{7}n</math>. Also, <math>0 < 91n - 104k</math>, so <math>k < \frac{7n}{8}</math>. Thus, <math>48n < 56k < 49n</math>. <math>k</math> is unique if it is within a maximum [[range]] of <math>112</math>, so <math>n = 112</math>.
 
Since <math>91n - 104k < n + k</math>, <math>k > \frac{6}{7}n</math>. Also, <math>0 < 91n - 104k</math>, so <math>k < \frac{7n}{8}</math>. Thus, <math>48n < 56k < 49n</math>. <math>k</math> is unique if it is within a maximum [[range]] of <math>112</math>, so <math>n = 112</math>.
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 +
'''Solution 2'''
 +
Flip all of the fractions for
 +
 +
<math>\frac{15}{8} > \frac{k + n}{n} > \frac{13}{7}</math>
 +
 +
<math>105n > 56 (k + n) > 104n</math>
 +
 +
<math>49n > 56k > 48n</math>
 +
 +
Continue as in Solution 1.
  
 
== See also ==
 
== See also ==

Revision as of 15:25, 15 March 2011

Problem

What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}$?

Solution

Solution 1 Multiplying out all of the denominators, we get:

\begin{align*}104(n+k) &< 195n< 105(n+k)\\ 0 &< 91n - 104k < n + k\end{align*}

Since $91n - 104k < n + k$, $k > \frac{6}{7}n$. Also, $0 < 91n - 104k$, so $k < \frac{7n}{8}$. Thus, $48n < 56k < 49n$. $k$ is unique if it is within a maximum range of $112$, so $n = 112$.

Solution 2 Flip all of the fractions for

$\frac{15}{8} > \frac{k + n}{n} > \frac{13}{7}$

$105n > 56 (k + n) > 104n$

$49n > 56k > 48n$

Continue as in Solution 1.

See also

1987 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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