Ratio and Proportion

Concept:Find the third proportional and the mean proportional, then express their ratio in simplest terms.Explanation:Let $x$ be the third proportional to $12$ and $30$.For three numbers in proportion: $12 : 30 = 30 : x$.Thus $12 \times x = 30 \times 30$, so $12x = 900$.Therefore $x = \frac{900}{12} = 75$.Let $y$ be the mean proportional between $9$ and $25$.For two numbers, mean proportional satisfies: $9 : y = y : 25$.So $y^2 = 9 \times 25 = 225$.Thus $y = \sqrt{225} = 15$.Now ratio of third proportional to mean proportional is $75 : 15$.Simplify by dividing both terms by $15$: $\frac{75}{15} : \frac{15}{15} = 5 : 1$.Answer:$5 : 1$, which matches option B.