Adding in rows

Congratulations to Soh Yong Sheng from Raffles Institution, Singapore for this excellent solution.

We have $0 < a < b$ which means $1/a > 1/b$ and so $3 + 1/a > 3 + 1/b$. Flipping over the fraction, we will get 1/(3 + 1/a) < 1/(3 + 1/b) and the inequality remains the same way round when 2 is added. Flipping over again for the last time we get $$\frac{1}{2+\frac{1}{3+\frac{1}{a}}}$$ is greater than $$\frac{1}{2+\frac{1}{3+\frac{1}{b}}}$$

The second part is a further expansion of the first, and in the process of repeating the above we know that it involves just one more flipping over of the fraction, thus $$\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{a}}}}$$ is less than the same thing with $b$ in place of $a$ as the inequality would be reversed again.

Lastly the continued fractions are expanded all the way down to $100 + 1/a$ and $100 + 1/b$. Observe the above process, we can tell that if the last or biggest number is odd then the continued fraction with a in it is bigger. If the last or biggest number is even then the continued fraction with $b$ in it is bigger. Each successive continued fraction involves one more 'flipping over' and reverses the inequality one more time. The following continued fraction is smaller than the same thing with $b$ in place of $a$: $$\frac{1}{{\displaystyle 2+\frac{1}{{\displaystyle 3+\frac{1}{{\displaystyle 4+\cdots +\frac{1}{{\displaystyle 99+\frac{1}{{\displaystyle 100+\frac{1}{{\displaystyle a}}}}}}}}}}}}$$