Three numbers are in the ratio 2 : 3 : 4. If the sum of their cubes is 21384,…

Concept: When three quantities are in a fixed ratio a : b : c, they share one common multiplier x, so they can be written as ax, bx and cx. Every quantity is then carried by a single unknown, and a symmetric expression such as the sum of their cubes collapses to (a3 + b3 + c3)x3, which turns the condition into one cubic equation in x.

Application: The given ratio is 2 : 3 : 4 and the sum of the cubes is 21384.

1. Let the three numbers be 2x, 3x and 4x, where x > 0.

2. Write the given condition: (2x)3 + (3x)3 + (4x)3 = 21384.

3. Cube each ratio part: 8x3 + 27x3 + 64x3 = 21384.

4. Add the like terms: 99x3 = 21384.

5. Divide both sides by 99: x3 = 21384 ÷ 99 = 216.

6. Take the cube root: x = 6, because 63 = 216.

7. The largest number is the 4-part of the ratio: 4x = 4 × 6 = 24.

Cross-check: With x = 6 the three numbers are 12, 18 and 24, and 12 : 18 : 24 reduces to 2 : 3 : 4 as required. Their cubes are 1728, 5832 and 13824, and 1728 + 5832 + 13824 = 21384, which matches the given total, so the largest number is 24.