Find the third proportional to 25 and 30. [Indian Railway Gr. 'D' Exam, 2014]

### Concept & Formula For any two numbers $a$ and $b$, the third proportional $c$ satisfies the continued proportion $a : b :: b : c$. The formula for the third proportional is: $$ c = \frac{b^2}{a} $$ ### Step-by-Step Solution 1. Identify the given numbers: $a = 25$ and $b = 30$. 2. Let the third proportional be $c$. 3. Set up the proportion: $$ 25 : 30 :: 30 : c $$ $$ \frac{25}{30} = \frac{30}{c} $$ 4. Solve for $c$: $$ c = \frac{30 \times 30}{25} $$ $$ c = \frac{900}{25} $$ 5. Calculate the final value: $$ c = 36 $$ ### Exam Strategy & Shortcut Use proportional scaling. The multiplier to get from 25 to 30 is $\frac{30}{25} = \frac{6}{5}$. To find the third proportional, just multiply the second number (30) by this same multiplier: $30 \times \frac{6}{5} = 6 \times 6 = 36$. ### Common Pitfall A common mistake is calculating the mean proportional instead of the third proportional (i.e., finding $\sqrt{25 \times 30}$). Read carefully which proportional is requested. ### Final Answer Therefore, the correct answer is **36**.