The sides of a triangle are in the ratio $\frac{1}{2}:\frac{1}{3}:\frac{1}{4}$ and its perimeter is $104cm.$ The length of the longest side is ______.

The correct option is C: $48cm$

Sides are in the ratio $\frac{1}{2}:\frac{1}{3}:\frac{1}{4}$

Multiply each fraction by the L.C.M of $(2,3,4)$ i.e by $12,$ so that we will get the ratio in whole number form.

$\frac{1}{2}\times 12:\frac{1}{3}\times 12:\frac{1}{4}\times 12$

i.e. $6:4:3$
Let the sides be $6x,4x$ and $3x$

Given that $6x+4x+3x=104$ [$\because$ Perimeter $=104cm$]

$\Rightarrow 13x=104$

$\Rightarrow x=8$

$\therefore$ Longest side $=6x=6\times 8=48cm$