There are three types of rectangular tiles: 3′ × 3′, 3′ × 7′ and 3′ × 11′. An area of rectangular shape of dimensions 3′ × 100′ is to be covered using these tiles without breaking them. If x and y are the maximum and minimum numbers of tiles of various sizes, respectively, that can be used to cover the area exactly, then x − y is
Answer & explanation
Answer: (a) 20
Every tile is 3′ wide, so the tiles lie in one row and their lengths must add up to exactly 100: 3a + 7b + 11c = 100. Using mostly 3′ tiles gives at most 32 tiles; using mostly 11′ tiles gives at least 12, so x − y = 20.
- The strip is 3′ wide and every tile is 3′ on one side (a 7′ or 11′ side cannot fit across), so 3a + 7b + 11c = 100.
- Maximum: 100 = 3 × 33 + 1 is not possible with 3′ tiles alone; 3 × 31 + 7 = 100 gives 32 tiles — the most.
- Minimum: 11 × 8 + 3 × 4 = 100 gives 12 tiles (so does 11 × 4 + 7 × 8).
- 11 tiles cannot work: with a + b + c = 11, the equation becomes 8c + 4b = 67, and the left side is even. Fewer tiles fail the same way or fall short in length.
- x − y = 32 − 12 = 20.
Remember · Covering a strip: turn it into an equation of lengths, then push towards the smallest (max count) or largest (min count) pieces.
Question and answer: UPSC's provisional GS Paper II (2026, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·