A certain shade of grey paint is obtained by mixing 3 parts of white paint…

Concept: When a fixed quantity of mixture must be prepared in a stated ratio, split the total using that ratio to find how much of each ingredient is actually required. If the ingredients can only be bought in fixed-size units (half-gallon or full-gallon cans here), each ingredient's purchase must then be rounded up, on its own, to the smallest combination of those units that covers its own requirement — a surplus of one color can never substitute for a shortfall in the other.

1. The white-to-black ratio is 3 : 5, so out of every 8 parts of the grey mixture, 3 parts are white paint and 5 parts are black paint.

2. For 2 gallons of mixture, the white paint required = 2 × 3/8 = 3/4 gallon, and the black paint required = 2 × 5/8 = 5/4 gallons (1.25 gallons).

3. Paint is sold only in half-gallon or full-gallon cans, so the amount purchased of each color must be the smallest multiple of half a gallon that is at least that color's own requirement above.

4. White paint's requirement of 3/4 gallon (0.75) exceeds one half-gallon can (0.5) but is at most one full-gallon can (1.0), so exactly one full-gallon can is needed.

5. Black paint's requirement of 5/4 gallons (1.25) exceeds one full-gallon can (1.0) but is at most one full-gallon plus one half-gallon can (1.5), so one full-gallon can plus one half-gallon can is needed.

6. Total = 1 gallon (white) + 1.5 gallons (black) = 2.5 gallons.

Cross-check: Buying only 2 gallons total (matching the mixture's total volume) would fail, since white alone needs more than a half-gallon can (ruling out sharing leftover capacity between colors) and black alone needs more than a full-gallon can — so 2.5 gallons, not 2 gallons, is the true minimum, confirming the can-by-can analysis above.

Least amount of paint that must be purchased = 2 ½ gallons (one full-gallon can of white + one full-gallon can and one half-gallon can of black).