A rectangular area of 6 square metres is to be painted on a 3 m. \times 4 m. board leaving a border of uniform width on all sides. What should be the width of the border?

Board area = 3 × 4 = 12 sq m. Painted area = 6 sq m, so border area = 6 sq m. If border width = w, inner painted dimensions are (3-2w) × (4-2w). Setting (3-2w)(4-2w) = 6 gives 12 - 14w + 4w² = 6, so 2w² - 7w + 3 = 0. Solving: (2w-1)(w-3) = 0, giving w = 0.5 m (w = 3 m is invalid as it would make inner dimensions negative).