# If three numbers are in the ratio 3 : 4 : 5 such that the sum of their cubes is 1728

If three numbers are in the ratio 3 : 4 : 5 such that the sum of their cubes is 1728, then the cube of the middle number is

- 216
- 512
- 256
- 128

**Answer: (2) 512**

**Solution:-**

The ratio of three numbers = 3: 4: 5.

Then, let three number be 3 X, 4 X and 5 X

The sum of their cubes is 1728,

Then, (3 X)^{3} + (4 X)^{3} + (5 X)^{3} = 1728

27 X^{3} + 64 X^{3} + 125 X^{3} = 1728

216 X^{3} = 1728

X^{3} = 1728/216

X^{3} = 8

Cube root on both side,

X = 2

The middle number = 4 X = 4 x 2

= 8

So, the cube of 8 = 8 x 8 x 8

= 512

Hence, the correct answer is option (2) 512.