Four identical glasses are shown below. One glass is empty, and the other 3 glasses are $\dfrac{1}{4}$ full, $\dfrac{1}{2}$ full, $\dfrac{4}{5}$ full of water, respectively. If the water were redistributed equally among the 4 glasses, what fractional part of each glass would be filled?

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The total amount of water is the sum of each fractional value:

$$\frac{1}{4}+\frac{1}{2}+\frac{4}{5}$$ $$=\frac{5}{20}+\frac{10}{20}+\frac{16}{20}$$ $$= \frac{31}{20}$$

Dividing this among four glasses:

$$=\frac{31}{20} \div 4$$ $$= \frac{31}{20}\cdot \frac{1}{4}$$ $$= \boxed{\frac{31}{80}}$$