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

Latest revision as of 15:59, 16 December 2020

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 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

$\begin{array}{ccccc}\frac{15}{8} &>& \frac{k + n}{n} &>& \frac{13}{7}\\ 105n &>& 56 (k + n)& >& 104n\\ 49n &>& 56k& >& 48n\end{array}$

Continue as in Solution 1.

Solution 3

Flip the fractions and subtract one from all sides to yield $\frac{7}{8}>\frac{k}{n}>\frac{6}{7}.$ Multiply both sides by $56n$ to get $49n>56k>48n.$ This is equivalent to find the largest value of $n$ such that there is only one multiple of 56 within the open interval between $48n$ and $49n$ . If $n=112,$ then $98>k>96$ and $k=97$ is the unique value. For $n\geq 113,$ there is at least $(49\cdot 113-48\cdot 113)-1=112$ possible numbers for $k$ and there is one $k$ every 56 numbers. Hence, there must be at least two values of $k$ that work. So, the largest value of $n$ is $\boxed{112}$ .

Solution 4

Notice that in order for $k$ to be unique, $\frac{n}{n + k+ 1} \le \frac{8}{15}$ and $\frac{n}{n+ k-1} \ge \frac{7}{13}$ must be true. Solving these inequalities for $k$ yields $\frac{7}{6}(k-1) \le n \le \frac{8}{7}(k+1)$ .

Thus, we want to find $k$ such that $\frac{7}{6}(k-1)\le \frac{8}{7}(k+1)$ . Solving this inequality yields $k \le 97$ , and plugging this into $\frac{n}{n+k} < \frac{7}{13}$ in the original equation yields $n \le 112$ so the answer is $\boxed{112}$ .

@@ Line 12: / Line 12: @@
 Flip all of the fractions for
-<cmath>\begin{align*}\frac{15}{8} > \frac{k + n}{n} > \frac{13}{7}\\
+<cmath>\begin{array}{ccccc}\frac{15}{8} &>& \frac{k + n}{n} &>& \frac{13}{7}\\
-n > 56 (k + n) > 104n\\
+n &>& 56 (k + n)& >& 104n\\
-n > 56k > 48n\end{align*}</cmath>
+n &>& 56k& >& 48n\end{array}</cmath>
 Continue as in Solution 1.
+==Solution 3==
+Flip the fractions and subtract one from all sides to yield <cmath>\frac{7}{8}>\frac{k}{n}>\frac{6}{7}.</cmath>  Multiply both sides by <math>56n</math> to get <cmath>49n>56k>48n.</cmath> This is equivalent to find the largest value of <math>n</math> such that there is only one multiple of 56 within the open interval between <math>48n</math> and <math>49n</math>.  If <math>n=112,</math> then <math>98>k>96</math> and <math>k=97</math> is the unique value. For <math>n\geq 113,</math> there is at least <math>(49\cdot 113-48\cdot 113)-1=112</math> possible numbers for <math>k</math> and there is one <math>k</math> every 56 numbers. Hence, there must be at least two values of <math>k</math> that work.  So, the largest value of <math>n</math> is <math>\boxed{112}</math>.
+==Solution 4==
+Notice that in order for <math>k</math> to be unique, <math>\frac{n}{n + k+ 1} \le \frac{8}{15}</math> and <math>\frac{n}{n+ k-1} \ge \frac{7}{13}</math> must be true. Solving these inequalities for <math>k</math> yields <math>\frac{7}{6}(k-1) \le n \le \frac{8}{7}(k+1)</math>.
+Thus, we want to find <math>k</math> such that <math>\frac{7}{6}(k-1)\le \frac{8}{7}(k+1)</math>. Solving this inequality yields <math>k \le 97</math>, and plugging this into <math>\frac{n}{n+k} < \frac{7}{13}</math> in the original equation yields <math>n \le 112</math> so the answer is <math>\boxed{112}</math>.
 == See also ==
@@ Line 25: / Line 34: @@
 [[Category:Intermediate Algebra Problems]]
 [[Category:Intermediate Number Theory Problems]]
+{{MAA Notice}}

1987 AIME (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
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