Tiling a Rectangular Terrace with Mixed-Size Tiles
A rectangular terrace measures 4.8 m by 3.6 m. The owner wants to cover it completely with square tiles. Two tile sizes are available: 40 cm × 40 cm tiles costing 6.50 leva each, and 60 cm × 60 cm tiles costing 13 leva each. Tiles cannot be cut; whole tiles only. The 60 cm tiles must be laid in a grid aligned with the terrace edges. A builder claims that the 60 cm tiles are cheaper overall because fewer tiles are needed. Delivery is free for orders over 200 leva, otherwise a 15 leva delivery fee applies. The owner also wants a 10 cm wide border strip of a different colour along the two long sides only, made from the same 40 cm tiles cut in half.
1. Determine how many tiles of each size fit exactly, and compute the total cost of each option including delivery. Which option is cheaper?
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Convert: 4.8 m = 480 cm, 3.6 m = 360 cm.
40 cm tiles: 480 ÷ 40 = 12 along the length, 360 ÷ 40 = 9 along the width. Both are whole numbers, so 12 × 9 = 108 tiles fit exactly. Cost = 108 × 6.50 = 702 leva. This exceeds 200 leva, so no delivery fee. Total = 702 leva.
60 cm tiles: 480 ÷ 60 = 8, but 360 ÷ 60 = 6. Both whole, so 8 × 6 = 48 tiles fit exactly. Cost = 48 × 13 = 624 leva. Also above 200 leva, no delivery fee. Total = 624 leva.
So the 60 cm option is cheaper by 702 − 624 = 78 leva, and the builder's claim about fewer tiles is also correct (48 < 108).
2. The border strip runs along the two long sides, each 480 cm long and 10 cm wide. Using 40 cm tiles cut in half (each half is 40 cm × 20 cm), check whether the strip can be covered with whole half-tiles and find how many 40 cm tiles must be bought for the border.
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Each long side strip is 480 cm × 10 cm. A half-tile is 40 cm × 20 cm, which is wider than 10 cm, so a half-tile cannot lie flat inside a 10 cm strip without trimming. Cutting a 40 × 20 half into a 40 × 10 strip uses exactly half of it, so each 40 cm tile yields 4 strips of 40 cm × 10 cm (two halves, each split again).
Per long side: 480 ÷ 40 = 12 strips of 40 cm length needed. Two sides: 24 strips. Each 40 cm tile provides 4 strips, so 24 ÷ 4 = 6 tiles are needed for the border.
This is a judgement point: the problem says tiles are cut in half, but a half-tile still does not fit a 10 cm width, so a further cut is required. The count of 6 tiles is correct only under that second cut.
3. Suppose the owner instead insists on no cutting at all, including the border. Explain what changes and whether the 60 cm option still works.
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Without cutting, the border cannot be made from 40 cm tiles at all, because no whole 40 cm tile is 10 cm wide. The border requirement becomes impossible under the no-cut rule, so the comparison changes: either the border is dropped or a different material is used.
The main floor still works for both sizes because 480 and 360 are both divisible by 40 and by 60. So the 60 cm option remains valid for the floor at 624 leva, and the 40 cm option at 702 leva. The judgement is that the border constraint, not the floor arithmetic, decides feasibility: a plan can be numerically cheaper yet fail a stated constraint.