Find the two numbers whose mean proportion is $12$ and the third proportional is $324$. (R.R.B., 2006)

### Concept & Strategy Let the two unknown numbers be $a$ and $b$. We are given two conditions: 1. The mean proportion of $a$ and $b$ is $\sqrt{ab} = 12 \implies ab = 144$. 2. The third proportional to $a$ and $b$ is $\frac{b^2}{a} = 324 \implies b^2 = 324a$. Instead of solving complex algebraic cubic equations, the fastest and most reliable method is to back-solve by testing the multiple-choice options against these two conditions. ### Step-by-Step Solution 1. Let's solve algebraically first to prove the concept. We have two equations: (i) $a \times b = 144 \implies a = \frac{144}{b}$ (ii) $b^2 = 324a$ 2. Substitute $a$ from equation (i) into equation (ii): $b^2 = 324 \times \left(\frac{144}{b}\right)$ 3. Multiply both sides by $b$: $b^3 = 324 \times 144$ 4. Break into prime factors to find the cube root easily: $324 = 18^2 = (2 \times 3^2)^2 = 2^2 \times 3^4$ $144 = 12^2 = (2^2 \times 3)^2 = 2^4 \times 3^2$ $b^3 = (2^2 \times 3^4) \times (2^4 \times 3^2) = 2^6 \times 3^6$ 5. Take the cube root by dividing exponents by 3: $b = 2^2 \times 3^2 = 4 \times 9 = 36$ 6. Find $a$ using equation (i): $a = \frac{144}{36} = 4$ The numbers are exactly 4 and 36. ### Exam Strategy & Shortcut **Option Elimination (Back-Solving):** Condition 1: Mean proportion is 12, so the product of the numbers must be $12^2 = 144$. Test options: (a) $6 \times 8 = 48$ (Incorrect) (b) $4 \times 36 = 144$ (Possible) (c) $3 \times 24 = 72$ (Incorrect) Condition 2: Third proportional is 324. Let's test option (b) for this condition: $\frac{36^2}{4} = \frac{1296}{4} = 324$. This matches perfectly. ### Common Pitfall Trying to solve $b^3 = 324 \times 144 = 46656$ without factorization. Finding the cube root of 46656 manually is extremely time-consuming and prone to computational errors. Back-solving from the options is much safer. ### Final Answer Therefore, the correct answer is **$4$ and $36$**.