If 12, \(2x\) and 27 are in continued proportion, then the value of \(x\)
9
Three numbers a, b and c are in continued proportion when \(a : b = b : c\). In that case the middle term b is called the mean proportional of a and c, and it satisfies \(b^{2} = a \times c\). Here \(a = 12\), \(b = 2x\) and \(c = 27\), so the square of the middle term must equal the product of the two outer terms.
Working:
\(12 : 2x = 2x : 27\)
\((2x)^{2} = 12 \times 27\)
\(4x^{2} = 324\)
\(x^{2} = 81\)
\(x = 9\) (taking the positive value)
Check: \(2x = 18\), and \(12 : 18 = 2 : 3 = 18 : 27\)
If \(x = 6\), then \(2x = 12\) and \(12 : 12\) is not equal to \(12 : 27\). If \(x = 8\), then \(2x = 16\) and \(16^{2} = 256\), which is not 324. If \(x = 11\), then \(2x = 22\) and \(22^{2} = 484\), which is also not 324. Only \(x = 9\) makes the middle term the mean proportional of 12 and 27.
Hence, 9.
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