To solve this question, we need to use the Beer's Law equation, which is:
A = ε l c
where A is the absorbance, ε is the molar extinction coefficient, l is the length (or thickness) of the cell, and c is the concentration.
From your question, we know the values of l (10 cm) and c (0.05 M).
In UV spectroscopy, we can measure the intensity of light absorbed by a solution either in units of absorbance or %-transmittance. Absorbance and %-Transmittance are inversely proportional to each other. For example, 100% T equals a value of 0 (zero) absorbance. If %-Transmittance is known, we can calculate A using the following equation:
A = 2 - log(%-T)
The question states that the intensity of light was reduced by 1/4. If we assume that intensity of the starting light = 100%, then 1/4 of 100% = 25%. This is the value of %-T. Now use the above equation to solve for A.
A = 2 - log(25%) = 2 - log(25) = 2 - 1.40 = 0.60 (FYI - Absorbance has no units!)
Now rearrange the Beer's Law equation to solve for ε.
ε = A / lc = 0.60 / (10 cm x 0.05 M) = 1.2 M-1 cm-1