If $\frac ab$ rounded to the nearest trillionth is $0.008012018027$, where $a$ and $b$ are positive integers, what is the smallest possible value of $a+b$?
I don't see any strategies here for solving this problem, any help? Thanks in advance!
[Math] the smallest possible value of $a+b$
algebra-precalculusfractions
Best Answer
The continued fraction representations of the limits of the interval are $$ 0.0080120180265 = [0; 124, 1, 4, 2, 1, 463872, 1, 1, 12, 1, 1, 41] \\ 0.0080120180275 = [0; 124, 1, 4, 3, 545777, 2, 13, 1, 1, 1, 1, 2] $$
The simplest continued fraction (and therefore also the simplest ordinary fraction!) in that interval is $$ [0;124,1,4,3] = \frac{16}{1997} = 0.00801201802704056084\ldots $$ and the sum of its numerator and denominator is $2013$.
(I used Wolfram Alpha to expand the continued fractions fully. For a pencil-and-paper solution one only needs to carry out the expansion until they start differing, which requires only a handful of long divisions with remainder.)