A rectangular floor measures 4 m in length and 2·2 m in breadth. Tiles of size 140 cm by 60 cm have to be laid such that the tiles do not overlap. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. What is the maximum number of tiles that can be accommodated on the floor ?
- (a) 6
- (b) 7
- (c) 8
- (d) 9 ✓ UPSC's answer
Why the answer is (d)
• Floor: 400 cm × 220 cm = 88,000 cm²; tile: 140 cm × 60 cm = 8,400 cm². Area alone would allow at most 10 tiles, but the shapes do not allow that many.
• Simple layouts give only 8, e.g. six tiles laid lengthwise in a 280 × 180 block plus two upright tiles in the remaining 120 cm strip.
• Nine fit with a mixed layout. Measuring from the top-left corner: three tiles lie lengthwise in the right-hand strip from 260 to 400 cm (stacked 0–60, 60–120 and 120–180 cm).
• In the remaining 260 × 220 cm area, six tiles interlock: upright at 0–60 × 0–140 and 60–120 × 20–160; lengthwise at 0–140 × 160–220 and 120–260 × 20–80; upright at 140–200 × 80–220 and 200–260 × 80–220.
• So the maximum is 9, option (d).
Why the other options are wrong
- (a) 6
- Even simple layouts fit 8 tiles, so 6 is too few.
- (b) 7
- Even simple layouts fit 8 tiles, so 7 is too few.
- (c) 8
- A mixed layout fits 9 tiles, so 8 is not the maximum.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.