The ratio between the third proportional of $12$ and $30$ and mean proportional of $9$ and $25$ is

### Concept & Formula This problem requires two separate proportional calculations before finding their final ratio: 1. Third proportional $c$ to numbers $a, b$: $c = \frac{b^2}{a}$ 2. Mean proportional $x$ to numbers $p, q$: $x = \sqrt{p \times q}$ ### Step-by-Step Solution 1. Calculate the third proportional to 12 and 30. Let it be $c$. Here, $a = 12$ and $b = 30$. $$ c = \frac{30^2}{12} = \frac{900}{12} = 75 $$ 2. Calculate the mean proportional to 9 and 25. Let it be $x$. Here, $p = 9$ and $q = 25$. $$ x = \sqrt{9 \times 25} = \sqrt{225} = 15 $$ 3. Find the requested ratio of $c$ to $x$: $$ \text{Ratio} = \frac{c}{x} = \frac{75}{15} $$ 4. Simplify the ratio by dividing both numbers by their common factor of 15: $$ \frac{75}{15} = \frac{5}{1} \implies 5 : 1 $$ ### Exam Strategy & Shortcut You don't have to multiply out large numbers. For the third proportional: $\frac{30 \times 30}{12} = \frac{30 \times 10}{4} = \frac{300}{4} = 75$. For the mean proportional: instead of finding $\sqrt{225}$, simply multiply the roots independently: $\sqrt{9} \times \sqrt{25} = 3 \times 5 = 15$. Then quickly deduce $75 : 15 = 5 : 1$. ### Common Pitfall Finding the ratio in the reverse order (Mean Proportional : Third Proportional), which would result in $15 : 75 \implies 1 : 5$. Always read the question carefully to ensure the correct order of the final ratio. ### Final Answer Therefore, the correct answer is **$5 : 1$**.