Find the value of $\frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90}$ Find the value of $p+q$, where $p$ and $q$ are two positive integers such that $p$ and $q$ have no common factor larger than $1$ and 
$$\frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} = \frac{p}{q}.$$
By using Wolfram Alpha, I obtain the sum is $\frac{3}{20}$. But I have no idea on how to obtain the sum.
Any hint would be appreaciated. 
[The question is taken from SMO 2015 Junior section]

We  know that 
$$
S_n=\frac{1}{1 \cdot 2}+ \frac{1}{2 \cdot 3}+ \cdots+ \frac{1}{(n-1)n}=1-\frac{1}{n}
$$
Then yours sum is exactly $$S_{10}-S_4=(1-\frac{1}{10})-(1-\frac{1}{4})=\frac{1}{4}-\frac{1}{10}=\frac{5-2}{20}=\frac{3}{20}.$$
