The sides of a triangle are in the ratio $$\dfrac{1}{3} : \dfrac{1}{5} : \dfrac{1}{4}$$ and its perimeter is 141 cm. The difference between the greatest side and the smallest side is:

The given ratio of the sides of the triangle is

$$\frac{1}{3} : \frac{1}{5} : \frac{1}{4}$$

To simplify the ratio, we take the LCM of 3, 5, and 4, which is 60.

Multiplying each fraction by 60, we get:

$$\frac{1}{3}\times 60 : \frac{1}{5}\times 60 : \frac{1}{4}\times 60$$

$$20 : 12 : 15$$

So, the sides are in the ratio (20:12:15).

The sum of the ratio terms is:

$$20+12+15=47$$

Since the perimeter of the triangle is 141 cm, one ratio unit is:

$$\frac{141}{47}=3$$

Therefore, the actual sides are:

$$20\times 3=60\text{ cm}$$

$$12\times 3=36\text{ cm}$$

$$15\times 3=45\text{ cm}$$

The greatest side is 60 cm, and the smallest side is 36 cm.

Hence, the difference between them is:

$$60-36=24\text{ cm}$$

Therefore, the required difference is 24 cm.