The correct answer is: B. One-fourth of the boys play neither cricket nor football.
Let $x$ be the total number of boys in the class. Then, 3/4 of the boys play football, so $3x/4$ play football. One-half of the boys play cricket, so $x/2$ play cricket. One-fourth of those who play cricket do not play football, so $x/8$ play cricket but not football. Therefore, $x/8$ of the boys play neither cricket nor football.
Here is a table that summarizes the information:
| Activity | Number of boys |
|—|—|
| Football | $3x/4$ |
| Cricket | $x/2$ |
| Both cricket and football | $x/8$ |
| Neither cricket nor football | $x/8$ |
Option A is incorrect because it states that two-thirds of the boys play only football. However, only 3/4 of the boys play football, and not all of those boys play only football.
Option C is incorrect because it states that one-third of the boys play neither cricket nor football. However, only $x/8$ of the boys play neither cricket nor football.
Option D is incorrect because it states that one-eighth of the boys play neither cricket nor football. However, this is the correct answer.
Option E is incorrect because it states that two-fifths of the boys play only football. However, only 3/4 of the boys play football, and not all of those boys play only football.