Tiling the Assembly Hall on a Fixed Budget
A secondary school in Rajshahi is tiling the floor of a new assembly hall measuring 12.4 m by 8.4 m. The supplier offers two sizes of square tile: 25 cm at 28 taka each, and 40 cm at 62 taka each. Laying charges are 18 taka per 25 cm tile and 30 taka per 40 cm tile. The mason will cut tiles where necessary along the walls, but every cut tile counts as a full tile bought, and offcuts cannot be reused. The head teacher has a total budget of 60,000 taka covering both tiles and laying, and wants the floor finished with no gaps. A parent claims the 25 cm tile is cheaper because a metre divides evenly into 25 cm pieces, so nothing is wasted. The supplier adds one condition: orders of 600 tiles or more receive an 8% discount on the tile price only, never on laying charges. The order must be placed on Sunday, and the school cannot buy extra tiles later at the same rate.
1. Calculate the number of tiles required and the total cost for each option, applying the discount rule where it applies.
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40 cm option. 1240 ÷ 40 = 31 exactly and 840 ÷ 40 = 21 exactly, so no cutting is needed: 31 × 21 = 651 tiles. Tiles: 651 × 62 = 40,362 taka. Laying: 651 × 30 = 19,530 taka. Subtotal 59,892 taka. Since 651 ≥ 600, the tile price is discounted: 40,362 × 0.92 = 37,133.04 taka, giving 56,663.04 taka.
25 cm option. 1240 ÷ 25 = 49.6, so 50 tiles across; 840 ÷ 25 = 33.6, so 34 tiles down. That is 50 × 34 = 1,700 tiles, although the true floor area needs only 104.16 m² ÷ 0.0625 m² = 1,666.56 tiles, so about 33 tiles are bought and wasted. Tiles: 1,700 × 28 = 47,600 taka; discounted to 47,600 × 0.92 = 43,792 taka. Laying: 1,700 × 18 = 30,600 taka. Total 74,392 taka.
The 40 cm option fits the budget; the 25 cm option exceeds it by 14,392 taka.
2. Which tile should the school order, and how much can the laying charge rise before that decision breaks down?
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Order the 40 cm tile: 56,663.04 taka leaves 3,336.96 taka of headroom under the 60,000 taka budget, while the 25 cm option cannot be paid for at all.
The headroom is spent only on laying, so the tolerable rise per tile is 3,336.96 ÷ 651 ≈ 5.13 taka, i.e. the laying rate could climb from 30 to about 35.10 taka per tile before the budget is breached. Any larger rise forces a redesign of the plan, not a switch to 25 cm tiles, which remain about 17,700 taka dearer.
A second sensitivity matters. If the 8% discount were withdrawn, the 40 cm total becomes 59,892 taka - still inside the budget, but with only 108 taka to spare, so a single broken box or a price rise of even 1% would sink it. The decision is therefore sound but not comfortable; the school should confirm the discount and the laying rate in writing before ordering, because the margin, not the tile size, is the real risk.
3. Test the parent's claim that the 25 cm tile wastes nothing because a metre divides evenly into 25 cm pieces.
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The claim confuses the unit with the room. It is true that 1 m = 100 cm is divisible by 25 cm, but the hall is not a whole number of metres in the way that matters for tiling: its sides are 1240 cm and 840 cm. 1240 ÷ 25 = 49.6 and 840 ÷ 25 = 33.6, so the 25 cm grid fails to close on both dimensions and tiles must be cut along two walls, not one. The 40 cm grid, by contrast, divides both 1240 and 840 exactly, which is why it needs no cutting at all.
Waste also grows with the number of pieces, not only with the leftover area: 1,700 tiles mean 1,700 joints, more cutting, more breakage and roughly two and a half times the laying labour of 651 tiles. The parent is right that the smaller tile has the lower unit price (28 < 62 taka), but the correct comparison is cost per finished square metre: about 714 taka/m² for the 25 cm option against about 544 taka/m² for the 40 cm option. Unit price is the wrong metric; total laid cost is the right one.