jmc

Question

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 27. What is the number of possible values for r?

Accepted Answer

The nearest fractions to 27 with numerator 1 are 13, 14; and with numerator 2 are 26, 28 = 13, 14 anyway. For 27 to be the best approximation for r, the decimal must be closer to 27 .28571 than to 13 .33333 or 14 .25. Thus r can range between 14 + 272 .267857 and 13 + 272 .309523. At r = .2679, .3095, it becomes closer to the other fractions, so .2679 r .3095 and the number of values of r is 3095 - 2679 + 1 = 417.