Claire, age 16, told us that the answer was 9 (which is right), but she didn't tell us how she worked it out.

Sally showed us her workings (she told us she used the hint!).

$\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\dots +\frac{1}{\sqrt{99}+\sqrt{100}}=\frac{1-\sqrt{2}}{1-2}+\frac{\sqrt{2}-\sqrt{3}}{2-3}+\frac{\sqrt{3}-\sqrt{4}}{3-4}+\dots +\frac{\sqrt{99}-\sqrt{100}}{99-100}$ (multiplying top and bottom of each fraction).
So the sum is $-(1-\sqrt{2}+\sqrt{2}-\sqrt{3}+\sqrt{3}-\sqrt{4}+\dots +\sqrt{99}-\sqrt{100})=-(1-10)=9$.