A circle is divided into four sectors such that the proportions of their areas are 1 : 2 : 3 : 4. What is the angle of the largest sector ?

The area of a sector of a circle is directly proportional to the angle it subtends at the center. The total angle at the center of a circle is 360°.

Given that the proportions of the areas of the four sectors are 1 : 2 : 3 : 4, the sum of the parts is:
1 + 2 + 3 + 4 = 10 units

Since the total angle of 360° corresponds to 10 units, the value of one unit is:
360° ÷ 10 = 36°

The largest sector corresponds to the largest proportion in the ratio, which is 4 units. Therefore, the angle of the largest sector is calculated as:
4 × 36° = 144°

Thus, the angle of the largest sector is 144°, making option D the correct answer.