Three numbers are in the ratio 2 : 3 : 4. The sum of their cubes is 33957, then the numbers are :

Question

Accepted Answer

To determine why Option 1 is the correct answer, we need to verify that the numbers 14, 21, and 28 are in the ratio 2:3:4 and that their cubes sum to 33957.

1. **Verification of the Ratio:**
   - The numbers given in Option 1 are 14, 21, and 28.
   - To check the ratio, divide each number by the smallest number, which is 14:
     - $ \frac{14}{14} = 1 $
     - $ \frac{21}{14} = 1.5 $
     - $ \frac{28}{14} = 2 $
   - Simplifying the ratio 1:1.5:2 gives us 2:3:4 when multiplied by 2.

2. **Verification of the Sum of Cubes:**
   - Calculate the cubes of the numbers:
     - $ 14^3 = 2744 $
     - $ 21^3 = 9261 $
     - $ 28^3 = 21952 $
   - Sum of the cubes: $ 2744 + 9261 + 21952 = 33957 $

Since both the ratio and the sum of the cubes match the conditions given in the question, Option 1 is indeed the correct answer.

**Explanation for Incorrectness of Other Options:**

- **Option 2 (12, 18, 24):**
  - Ratio: $ \frac{12}{12} = 1 $, $ \frac{18}{12} = 1.5 $, $ \frac{24}{12} = 2 $ simplifies to 2:3:4, but let's check the cubes.
  - Cubes: $ 12^3 = 1728 $, $ 18^3 = 5832 $, $ 24^3 = 13824 $
  - Sum of cubes: $ 1728 + 5832 + 13824 = 21384 $, which does not equal 33957.

- **Option 3 (8, 12, 16):**
  - Ratio: $ \frac{8}{8} = 1 $, $ \frac{12}{8} = 1.5 $, $ \frac{16}{8} = 2 $ simplifies to 2:3:4, but let's check the cubes.
  - Cubes: $ 8^3 = 512 $, $ 12^3 = 1728 $, $ 16^3 = 4096 $
  - Sum of cubes: $ 512 + 1728 + 4096 = 6336 $, which does not equal 33957.

- **Option 4 (4, 6, 8):**
  - Ratio: $ \frac{4}{4} = 1 $, $ \frac{6}{4} = 1.5 $, $ \frac{8}{4} = 2 $ simplifies to 2:3:4, but let's check the cubes.
  - Cubes: $ 4^3 = 64 $, $ 6^3 = 216 $, $ 8^3 = 512 $
  - Sum of cubes: $ 64 + 216 + 512 = 792 $, which does not equal 33957.

Thus, Option 1 is the only choice that satisfies both the ratio and the sum of cubes condition, confirming it as the correct answer.