The probability that A speaks truth is $\frac{4}{5}$ while the probability that B speaks truth is $\frac{3}{4}$. The probability that A and B contradict each other when asked to relate a fact is

$\frac{7}{20}$

Probability that A speaks truth i.e. P(A) =$\frac{4}{5}$

Probability that A does not speak truth i.e. $P(\bar{A}) = 1 -\frac{4}{5} = \frac{1}{5}$

Probability that B speaks truth i.e. P(B) = $\frac{3}{4}$.

Probability that B does not speak truth i.e. $P(\bar{B}) = 1 -\frac{3}{4} = \frac{1}{4}$

Probability that A and B contradict each other

$= P(A) .  P(\bar{B})+P(\bar{A})P(B) = \frac{4}{5}.\frac{1}{4}+ \frac{3}{4}.\frac{1}{5}=\frac{7}{20}$

Hence (C) is the correct answer.