Three cubes of a metal whose edges are in the ratio 3:4:5 are melted and converted into a single cube whose diagonal is 12 3 cm. Find the edges of the three cubes.

Let the edges of the three cubes be in the ratio 3:4:5.
So, let the edges be 3x, 4x, and 5x.

Volume of a cube = edge³
Total volume of the three cubes = (3x)³ + (4x)³ + (5x)³
= 27x³ + 64x³ + 125x³ = 216x³

The three cubes are melted and formed into one new cube.
The diagonal of the new cube is given as 12√3 cm.

The formula for the diagonal of a cube = √3 × edge
So,
12√3 = √3 × edge
⇒ edge = 12 cm

Volume of the new cube = 12³ = 1728 cm³

Now, equate the volumes:
216x³ = 1728
⇒ x³ = 8
⇒ x = 2

Now, calculate the edges of the original cubes:
First cube: 3x = 3 × 2 = 6 cm
Second cube: 4x = 4 × 2 = 8 cm
Third cube: 5x = 5 × 2 = 10 cm

Hence the edges of the three cubes are 6 cm, 8 cm, and 10 cm.