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Difference between revisions of "1997 AIME Problems/Problem 5"

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== Problem ==
 
== Problem ==
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The number <math>r</math> can be expressed as a four-place decimal <math>0.abcd,</math> where <math>a, b, c,</math> and <math>d</math> represent digits, any of which could be zero.  It is desired to approximate <math>r</math> by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to <math>r</math> is <math>\frac 27.</math>  What is the number of possible values for <math>r</math>?
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The number <math>r</math> can be expressed as a four-place [[decimal]] <math>0.abcd,</math> where <math>a, b, c,</math> and <math>d</math> represent [[digit]]s, any of which could be zero.  It is desired to approximate <math>r</math> by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to <math>r</math> is <math>\frac 27.</math>  What is the number of possible values for <math>r</math>?
  
 
== Solution ==
 
== Solution ==
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{{solution}}
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The nearest fractions to <math>\frac 27</math> with numerator <math>1</math> are <math>\frac 13, \frac 14</math>; and with numerator <math>2</math> are <math>\frac 26, \frac 28 = \frac 13, \frac 14</math> anyway. For <math>\frac 27</math> to be the best approximation for <math>r</math>, the decimal must be closer to <math>\frac 27 \approx .28571</math> than to <math>\frac 13 \approx .33333</math> or <math>\frac 14 \approx .25</math>.
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Thus <math>r</math> can range between <math>\frac{\frac 14 + \frac{2}{7}}{2} \approx .267857</math> and <math>\frac{\frac 13 + \frac{2}{7}}{2} \approx .309523</math>. At <math>r = .2678, .3096</math>, it becomes closer to the other fractions, so <math>.2679 \le r \le .3095</math> and the number of values of <math>r</math> is <math>3095 - 2679 + 1 = \boxed{417}</math>.
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=1997|num-b=4|num-a=6}}
 
{{AIME box|year=1997|num-b=4|num-a=6}}
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[[Category:Intermediate Number Theory Problems]]
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{{MAA Notice}}

Latest revision as of 10:46, 4 June 2021

Problem

The number $r$ can be expressed as a four-place decimal $0.abcd,$ where $a, b, c,$ and $d$ represent digits, any of which could be zero. It is desired to approximate $r$ by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to $r$ is $\frac 27.$ What is the number of possible values for $r$?

Solution

The nearest fractions to $\frac 27$ with numerator $1$ are $\frac 13, \frac 14$; and with numerator $2$ are $\frac 26, \frac 28 = \frac 13, \frac 14$ anyway. For $\frac 27$ to be the best approximation for $r$, the decimal must be closer to $\frac 27 \approx .28571$ than to $\frac 13 \approx .33333$ or $\frac 14 \approx .25$.

Thus $r$ can range between $\frac{\frac 14 + \frac{2}{7}}{2} \approx .267857$ and $\frac{\frac 13 + \frac{2}{7}}{2} \approx .309523$. At $r = .2678, .3096$, it becomes closer to the other fractions, so $.2679 \le r \le .3095$ and the number of values of $r$ is $3095 - 2679 + 1 = \boxed{417}$.

See also

1997 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
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All AIME Problems and Solutions

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