To find the probability of both John and James failing the examination:

Probabilities of failing:
 \(P(\text{John fails}) = 1 - P(\text{John passes}) = 1 - \frac{3}{4} = \frac{1}{4}\)
 \(P(\text{James fails}) = 1 - P(\text{James passes}) = 1 - \frac{3}{5} = \frac{2}{5}\)

Probability of both failing:
 \(P(\text{Both fail}) = P(\text{John fails}) \times P(\text{James fails}) = \frac{1}{4} \times \frac{2}{5} = \frac{2}{20} = \frac{1}{10}\)

Thus, the probability of both boys failing the examination is:
\(\frac{1}{10}\)