3.4 Continued Proportion, Mean Proportional

Three quantities a, b, c are in continued proportion if a : b = b : c, i.e., b² = ac. Here b is called the mean proportional between a and c.

Mean proportional = √(ac)

Example 21: Find the mean proportional between 4 and 25. Mean proportional = √(4 × 25) = √100 = 10.

Example 22: Find the mean proportional between 9 and 16. √(9 × 16) = √144 = 12.

Example 23: If 8 is the mean proportional between 4 and x, find x. 8² = 4 × x → 64 = 4x → x = 16.

Example 23A: Find the mean proportional between 12 and 27. √(12 × 27) = √324 = 18. (Recognising 12×27=324=18² instantly saves time — always check if the product is a perfect square before reaching for a calculator-style square root.)

Example 23B (additional, decimal values): Find the mean proportional between 0.4 and 0.9. √(0.4 × 0.9) = √0.36 = 0.6.

Example 23C (additional, reverse-solve for the first term): If the mean proportional between x and 8 is 12, find x. 12² = x × 8 → 144 = 8x → x = 18.