Discussion
The ratio of the third proportional to 12 and 24 to the mean proportional between 9 and 36 is:
Answer: C
The ratio is 8 : 3. The third proportional to 12 and 24 is the number c with 12 : 24 = 24 : c, and the mean proportional between 9 and 36 is the square root of their product.
\[ c = \frac{24 \times 24}{12} \]
\[ c = 48 \]
\[ \text{mean proportional} = \sqrt{9 \times 36} \]
\[ \sqrt{324} = 18 \]
\[ 48 : 18 = 8 : 3 \]
The options 3 : 5, 7 : 5 and 9 : 4 come from mixing up the two proportionals, for example using the mean of 12 and 24 or the third proportional of 9 and 36, and none of them simplifies from 48 : 18.
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