Discussion

If x is added to each of 14, 12, 34 and 30, the numbers so obtained, in this order, are in proportion. What is the value of \(\sqrt{12x + 9}\) ?

Answer: B

The value of \(\sqrt{12x + 9}\) is 9.

For the numbers to be in proportion, (14 + x) : (12 + x) = (34 + x) : (30 + x). Cross-multiplying:

\[ \begin{aligned} (14 + x)(30 + x) &= (12 + x)(34 + x) \\ 420 + 44x + x^{2} &= 408 + 46x + x^{2} \\ 12 &= 2x, \quad x = 6 \end{aligned} \]

Check: 20 : 18 = 40 : 36, both equal to 10 : 9. Then

\[ \sqrt{12(6) + 9} = \sqrt{81} = 9 \]

8, 11 and 13 are the square roots of 64, 121 and 169, which would need x = 4.6, 9.3 or 13.3, none of which makes the four numbers proportional.

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