Find $x$ for $\left(\frac1{1\times101} + \frac1{2\times102} + \dots +\frac1{10\times110}\right)x = \frac1{1\times11} + \frac1{2\times12}...$ $$\left(\frac1{1\times101} + \frac1{2\times102} + \dots +\frac1{10\times110}\right)x = \frac1{1\times11} + \frac1{2\times12} + \dots +\frac1{100\times110}$$
Find x
My younger sister in grade 5 had this question in a test. But I, a college student, still can't solve this. What a shame :<

One can group the $100$ summands on the right side by the ending digit of the denominators. For example, taking the numbers ending in 2:
$$\frac{1}{2\times 12}+\frac{1}{12 \times 22}+ ... +\frac{1}{92 \times 102}=\\
= \frac{1}{10}\left(\frac{1}{2} -\frac{1}{12}\right) +\frac{1}{10}\left(\frac{1}{12} -\frac{1}{22}\right) +\cdots  +\frac{1}{10}\left(\frac{1}{92}- \frac{1}{ 102}\right) =\\ =\frac{1}{10}\left(\frac{1}{2}-\frac{1}{102}\right)=\\ =10\left(\frac{1}{2 \times 102}\right)\\
$$
Therefore, $x=10$
This can be formalized, of course. But I don't think that this (even without the formalization) is appropiate for a 10 years old...
