A question of Quantitative aptitude and Logic in School Level

Question

I need help in a question asked by my younger brother in his aptitude exam of Secondary level.

Question's Image ->

I thought answer should be B

Answer is given (A)

I can't reason how it must be independent of both d and n.

Can anyone please explain.

Answer 1

Since the balls fill the cylinder snugly and from end to end, the cylinder must have an internal diameter of $d$ and height of $nd$, so its volume is

$$\pi\left(\frac{d}2\right)^2\cdot nd=\frac{\pi nd^3}4\;.$$

The total volume of the balls is

$$n\cdot\frac43\pi\left(\frac{d}2\right)^3=\frac{\pi nd^3}6\;.$$

The volume fraction occupied by the melted balls is therefore

$$\frac{\frac{\pi nd^3}6}{\frac{\pi nd^3}4}=\frac23\;,$$

which is independent of both $n$ and $d$.

One can arrive at this conclusion without actually doing the calculations by realizing that both the internal volume of the cylinder and the total volume of wax are proportional to $nd^3$, so the volume fraction of the melted wax is just the ratio of the proportionality constants.