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 or and whose denominator is an integer. The closest such fraction to is . What is the number of possible values for
Solution:
Let be the set of real numbers that can be written as fractions whose numerators are or and whose denominators are integers. Because is the greatest element of that is less than , and is the least element of that is greater than , we must find the number of values for that are closer to than to or . Because can be expressed as a four-place decimal, the inequality
implies that . Thus there are possible values for .
Query. The length of the interval is . How many four-place decimals might there be in an arbitrary interval of length ?
The problems on this page are the property of the MAA's American Mathematics Competitions