Set Counting — Sports Participation in a Class Three-fourths of the boys play football, one-half play cricket, and one-fourth of the cricket players do not play football. What fraction of boys play neither cricket nor football?

Introduction / Context:
This is a classic two-set (football and cricket) counting problem. We are given overall fractions for football and cricket participation and an additional constraint about a portion of cricket players not playing football. We must find the fraction who play neither game.

Given Data / Assumptions:

• Total boys = 1 (use unit method for fractions).
• Football (F) = 3/4 of boys.
• Cricket (C) = 1/2 of boys.
• Among cricket players, 1/4 do not play football ⇒ C-only = (1/4) * (1/2) = 1/8.

Concept / Approach:
Use set identities: total = F-only + C-only + both + neither. Also, F = F-only + both, and C = C-only + both. Solve systematically.

Step-by-Step Solution:

Verification / Alternative check:
Check totals: 3/8 (F-only) + 3/8 (both) = 3/4 = F, OK. 1/8 (C-only) + 3/8 (both) = 1/2 = C, OK. Complement gives 1/8 neither, consistent.

Why Other Options Are Wrong:

• Two-third only football / Two-fifth only football: incompatible with F = 3/4 and computed overlap.
• One-fourth or One-third neither: contradict the computed 1/8 neither.

Common Pitfalls:
Forgetting that “one-fourth of cricket players do not play football” applies to cricket subset only, or double-counting the intersection.

Final Answer:
One-eighth of the boys play neither cricket nor football.