Courtyard Tiles and a Scale Drawing: Area Versus Length
The courtyard behind Lincoln Middle School is a rectangle 14 m long and 9 m wide. The whole courtyard is paved with square tiles that are 0.5 m on each side, except for a circular fountain in the middle with a radius of 2.5 m; no tiles are laid under the fountain. The PTA has approved a scale drawing of the courtyard for a new welcome sign, using a scale of 1 cm to 1 m. A parent volunteer has sketched the courtyard as 14 cm by 9 cm and the fountain as a circle with a 2.5 cm radius. A second volunteer, looking at the sketch, says that because the scale is 1 to 100, the fountain covers 1/100 of the drawing's area and therefore 1/100 of the courtyard. The tile supplier sells tiles only in boxes of 20 and recommends adding 10% extra for cuts and breakage. The PTA needs a tile order and a corrected explanation before Tuesday's meeting. Use π ≈ 3.14.
1. How many tiles must actually be laid in the courtyard? Explain the rounding decision.
Read the worked answer
Tile area = 0.5 × 0.5 = 0.25 m².
Courtyard area = 14 × 9 = 126 m², which would be 126 ÷ 0.25 = 504 tiles with no fountain.
Fountain area = πr² ≈ 3.14 × 2.5² = 3.14 × 6.25 = 19.625 m², which is 19.625 ÷ 0.25 = 78.5 tiles' worth of space.
Tiles needed = 504 − 78.5 = 425.5.
Half a tile cannot be bought, and the tiles bordering the circle must be cut from whole tiles, so round up to 426 tiles. Rounding down to 425 would leave a paved gap; the fractional part is a signal that cuts are required, not that the count can be truncated.
2. The supplier adds 10% for cuts and breakage and sells only in boxes of 20. What should the PTA order, and how many spare tiles will that leave?
Read the worked answer
Apply the 10% to the 426 tiles needed: 426 × 1.10 = 468.6, and since tiles are whole, plan for 469 tiles.
Convert to boxes: 469 ÷ 20 = 23.45, so 24 boxes are required, giving 24 × 20 = 480 tiles.
Checking the alternative: 23 boxes is 460 tiles, which is 9 short of the 469 the allowance calls for, so 23 boxes is not enough even before counting mistakes during installation.
The spare after installation is 480 − 426 = 54 tiles, which absorb breakage and the offcuts around the fountain. Selling the leftover boxes back or saving them for future repairs is a reasonable follow-up decision.
3. Correct the volunteer's claim about the fountain covering 1/100 of the area, using area computations from the drawing and from the courtyard.
Read the worked answer
A linear scale compares lengths, and area scales by the square of the length factor. With 1 cm representing 1 m, lengths shrink by a factor of 1/100, so areas shrink by 1/100 × 1/100 = 1/10,000 - not 1/100.
Check with numbers. On the sketch: fountain area = 3.14 × 2.5² = 19.625 cm²; courtyard area = 14 × 9 = 126 cm²; ratio = 19.625 ÷ 126 ≈ 0.156, about 15.6%. In reality: fountain 19.625 m² of 126 m², which is also about 15.6%.
The two ratios agree, so the sketch itself is correct - only the reasoning about 1/100 is wrong. They agree numerically because at this scale 1 cm² on the drawing corresponds to exactly 1 m² of courtyard; that coincidence does not make the area factor 1/100.