Three numbers are in the ratio 2 : 3 : 4 and the sum of their cubes is 33957.…

When three quantities are in the ratio a : b : c, express them using a common multiplier x as ax, bx, and cx. Any additional condition given on the quantities (their sum, product, or sum of like powers) becomes an equation in x alone; solving that equation pins down x, and substituting back yields both the individual values and any required combination such as their total.

1. Let the three numbers be 2x, 3x, and 4x, using the given ratio 2 : 3 : 4 with common multiplier x.

2. Sum of their cubes: (2x)³ + (3x)³ + (4x)³ = 8x³ + 27x³ + 64x³ = 99x³.

3. Set this equal to the given total: 99x³ = 33957, so x³ = 33957 ÷ 99 = 343.

4. Taking the cube root, x = 7, since 7³ = 343.

5. The required sum of the three numbers is 2x + 3x + 4x = 9x = 9 × 7 = 63.

Substituting x = 7 back: the numbers are 14, 21, and 28, and 14³ + 21³ + 28³ = 2744 + 9261 + 21952 = 33957, which matches the given sum of cubes exactly, confirming the sum 63.