i didn't know (immediately) the minimum number of at bats to have a batting average of .399 , although i'm sure i worked it out later, probably by trial and error. what i did know is a decent lower bound on it; from the inequalities

.5 / 1000 < 2/5 - p/q < 1.5 / 1000 .

it follows that q > (1000 / 1.5) / 5 = 133.333333... which is far too many at bats to be believable. to find the minimum number of at bats, we can use continued fractions, although it's not quite as simple as i thought it should be. we seek the "simplest" fraction (i.e. with smallest denominator) in the closed interval [.3985 , .3995] . the continued fraction expansions of .3985 and .3995 are 1/(2 + 1/(1 + 1/(1 + 1/(26 + 1/15)))) and 1/(2 + 1/(1 + 1/(1 + 1/(79 + 1/(2 + 1/2))))) which have convergents 1/2 , 1/3 , 2/5 , 53/133 , 797/2000 and 1/2 , 1/3 , 2/5 , 159/398 , 320/801 , 799/2000 respectively. this means that 1/2 is above the range, 1/3 is below the range, and 2/5 is above the range. then the convergents diverge (*groan*) but we have that 53/133 is below the lower bound. this convergent 53/133 occurs (from the previous two convergents 1/3 and 2/5 ) as (1 + 2n) / (3 + 5n) , where n is the largest such that the resulting fraction is on the opposite side of the target (.3985) as the preceding convergent. then for n + 1 , the corresponding fraction is on the same side (above) of the target as 2/5 , so this is the simplest fraction we want, (2 + 53) / (5 + 133) . mike