How can I show that $\displaystyle \frac {3}{10} $ is $\displaystyle \frac {1}{7} $ of the way from $\displaystyle \frac {1}{4} $ to $\displaystyle \frac {3}{5} $?
First you need to find out the length of "the whole way" from $\displaystyle \frac{1}{4}$ to $\displaystyle \frac{3}{5}$, which you do by subtraction:
$\displaystyle \frac{3}{5} - \frac{1}{4} = \frac{12}{20} - \frac{5}{20} = \frac{7}{20}$.
So what's 1/7 of 7/20? 1/20. So now the question becomes, is $\displaystyle \frac{3}{10}$ a distance of $\displaystyle \frac{1}{20}$ from $\displaystyle \frac{1}{4}$?
$\displaystyle \frac{3}{10} - \frac{1}{4} = \frac{6}{20} - \frac{5}{20} = \frac{1}{20}$
and the answer is yes.
We figured out that $\displaystyle \frac{3}{5}$ and $\displaystyle \frac{1}{4}$ are $\displaystyle \frac{7}{20}$ apart. So a number which is $\displaystyle \frac{1}{7}$ of the way from $\displaystyle \frac{1}{4}$ to $\displaystyle \frac{3}{5}$ is going to be a distance of $\displaystyle \frac{1}{20}$ from $\displaystyle \frac{1}{4}$. Therefore, the desired answer is $\displaystyle \frac{1}{4} + \frac{1}{20} = \frac{5}{20} + \frac{1}{20} = \frac{6}{20} = \frac{3}{10}$.