Painting a Room: Openings, Two Coats, and Waste Allowance
A homeowner is painting a rectangular room that measures 14 feet by 11 feet with an 8-foot ceiling, and she is painting the four walls only, not the ceiling or floor. The room has one door measuring 3 feet by 6.75 feet and two identical windows, each 2.5 feet by 4 feet. She wants two coats everywhere and plans to buy 10% more than the calculated amount to cover touch-ups and roller waste. The paint she likes covers 350 square feet per gallon and is sold only in full one-gallon cans at $32.50 per can. Sales tax is 7% on paint.
1. How many gallons must she buy, and what will the paint cost including tax?
Read the worked answer
Wall area: perimeter = 2(14 + 11) = 50 ft; × 8 ft height = 400 sq ft.
Subtract openings: door = 3 × 6.75 = 20.25 sq ft; windows = 2 × (2.5 × 4) = 20 sq ft; total openings = 40.25 sq ft.
Net wall area: 400 − 40.25 = 359.75 sq ft.
Two coats: 359.75 × 2 = 719.5 sq ft.
Add 10%: 719.5 × 1.10 = 791.45 sq ft.
Gallons: 791.45 ÷ 350 ≈ 2.26 → must round up to 3 gallons (cans are whole).
Cost: 3 × $32.50 = $97.50; with tax, $97.50 × 1.07 = $104.325 ≈ $104.33.
The judgement points are subtracting the openings before doubling for coats, applying the waste allowance, and rounding cans up rather than to the nearest whole number.
2. The homeowner decides to paint the ceiling the same color. How many gallons are needed now, and how does the can count change relative to the walls-only job?
Read the worked answer
Ceiling area: 14 × 11 = 154 sq ft.
New net area: 359.75 + 154 = 513.75 sq ft.
Two coats: 513.75 × 2 = 1,027.5 sq ft.
Add 10%: 1,027.5 × 1.10 = 1,130.25 sq ft.
Gallons: 1,130.25 ÷ 350 ≈ 3.23 → round up to 4 gallons.
So the ceiling adds 154 sq ft of surface (about 44% more than the walls alone) and one extra can. Note the non-linear effect of rounding: the raw need rose from 2.26 to 3.23 gallons, but the purchased amount rose from 3 to 4 cans, so the marginal cost of the ceiling ($32.50 plus tax) is not proportional to the marginal area.
3. The store also sells a premium paint covering 400 square feet per gallon at $36.75 per can. Using the walls-only job (791.45 sq ft to cover), which paint is cheaper and by how much?
Read the worked answer
Standard paint (350 sq ft/gal): 791.45 ÷ 350 ≈ 2.26 → 3 cans; 3 × $32.50 = $97.50.
Premium paint (400 sq ft/gal): 791.45 ÷ 400 ≈ 1.98 → 2 cans; 2 × $36.75 = $73.50.
Premium is cheaper by $97.50 − $73.50 = $24.00 before tax (tax scales both by 1.07, so the ranking and the $24 gap are unchanged; after tax the gap is $24.00 × 1.07 = $25.68).
The judgement call is that a higher price per can does not mean higher total cost: because the can count is a whole number, a small coverage improvement that crosses a can boundary produces a large saving. Roughly 1.98 gallons of need means 2 cans of premium versus 3 of standard.