The market value of beams, made of a rare metal, has a unique property: the market value of any such beam is proportional to the square of its length. Due to an accident, one such beam got broken into two pieces having lengths in the ratio 4:9. Considering each broken piece as a separate beam, how much gain or loss, with respect to the market value of the original beam before the accident, is incurred?

Given that the beam got broken into two pieces, with length in a ratio of 4 : 9.

Let the lengths of the new beams be $$4x$$ and $$9x$$ respectively.

So, the length of the original beam is $$13x$$.

Now, given the value is proportional to the square of its length.

Value of the original beam = $$k\left(13x\right)^2=169kx^2$$, where $$k$$ is the constant of proportionality.

The value of new beams is $$k\left(4x\right)^2+k\left(9x\right)^2=16kx^2+81kx^2=97kx^2$$

Hence, the gain/loss with respect to the original beam is $$169kx^2-97kx^2=72kx^2$$

In percentage terms, Loss % = $$\dfrac{72kx^2}{169kx^2}\times100=42.60\%$$

Hence, the answer is 42.60% loss