Difference between revisions of "1997 AIME Problems/Problem 5"
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== Problem == | == Problem == | ||
| + | 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 == | ||
| + | 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}} | |
| + | |||
| + | [[Category:Intermediate Number Theory Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 10:46, 4 June 2021
Problem
The number 
can be expressed as a four-place decimal 
where 
and 
represent digits, any of which could be zero. It is desired to approximate by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to is 
What is the number of possible values for ?
Solution
The nearest fractions to 
with numerator 
are 
; and with numerator 
are 
anyway. For to be the best approximation for , the decimal must be closer to 
than to 
or 
.
Thus can range between 
and 
. At 
, it becomes closer to the other fractions, so 
and the number of values of is 
.
See also
| 1997 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
