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
- (a) 20 ✓ UPSC's answer
- (b) 12
- (c) 10
- (d) 7
Why the answer is (a)
• All tiles are 3′ wide, so only their lengths matter: 3a + 7b + 11c = 100.
• Maximum number: use as many 3′ tiles as possible. 100 leaves remainder 1 on division by 3, and one 7′ tile fixes it: 7 + 31 × 3 = 100, i.e. 32 tiles. So x = 32.
• Minimum number: use as many 11′ tiles as possible. 9 × 11 = 99 leaves 1 (impossible); 8 × 11 = 88 leaves 12 = 4 × 3, i.e. 12 tiles. 11 tiles or fewer cannot work, since 3n + 4b + 8c = 100 then has no whole-number solution. So y = 12.
• x − y = 32 − 12 = 20: (a).
Why the other options are wrong
- (b) 12
- 12 is the minimum number y itself, not x − y.
- (c) 10
- 10 does not follow from x = 32 and y = 12.
- (d) 7
- 7 does not follow from x = 32 and y = 12.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2026, held on 24 May 2026. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.