43/197: [0;4,1,1,2,1,1,3]

17/77: [0;4,1,1,8]

Then make a new continued fraction keeping matching terms and where the terms differ then increment the smaller one by 1 and stop.

Then the continued fraction for a/b becomes [0;4,1,1,3].  This is the continued fraction for 7/32.

What if you are like me and barely know of continued fractions? There is a fairly effective algorithm that can find the answer to the question with not too much work.

Start with r1 < a/b < r2 (all values assumed to be positive)

Is there at least one integer in the interval from r1 to r2?  

If so set a/b equal to the smallest such integer and solve for a and b.

If not then take the reciprocals of everything (r1, r2, and a/b).  Find the largest integer less than the new interval. 

Then make a new r1, r2, and a/b fraction by subtracting that integer and head back to the start.

For this problem we start with 43/197 < a/b < 17/77

There is no integer in the interval.  

Then 1/r2=77/17 and 1/r1=197/43 and 1/(a/b)=b/a.  

4 is the largest integer less than 77/17.  

Then 9/17 < a/(b-4a) < 25/43

There is an integer in the interval.  The smallest is 1.

Then (23a-5b)/(2b-9a)=1 reduces to 32a=7b, which makes a/b=7/32.

Note, if we take some other integers in this interval, like 2, we get a different but not ideal answer.  (23a-5b)/(2b-9a)=2 reduces to 41a=9b, implying a/b=9/41, which is precisely the second-best answer that Charlie's program found.