Find the fourth proportional to 4, 9, and 12, and the third proportional to 16…

Concept

If a, b, c, and d are in proportion, then a : b = c : d, so the fourth proportional is d = (b × c) ÷ a.

If a, b, and c are in continued proportion, then a : b = b : c, so the third proportional is c = b2 ÷ a.

Application

1. Let x be the fourth proportional to 4, 9, and 12. Then 4 : 9 = 12 : x.

2. Cross-multiplication gives 4x = 9 × 12 = 108, so x = 108 ÷ 4 = 27.

3. Let y be the third proportional to 16 and 36. Then 16 : 36 = 36 : y.

4. Cross-multiplication gives 16y = 36 × 36 = 1296, so y = 1296 ÷ 16 = 81.

Cross-check

For the first proportion, 4/9 = 12/27 because both simplify to 4/9. For the second, 16/36 = 36/81 because both simplify to 4/9.

Result

Therefore, the required fourth and third proportionals are 27 and 81, respectively.