JEE Main · 2019 · 10 Jan 2019 · Shift 1 · Units & Measurement

Re-expressing a density in a new system means rewriting one kilogram and one metre in the new units. One kilogram is $1/0.05=20$ of the new mass units, and one metre is $1/0.25=4$ new length units, so one cubic metre is $4^3=64$ new volume units. The density $128$ kg m${}^{-3}$ therefore reads $128\times \frac{20}{64}=40$ in the new system. Equivalently the numerical value scales by $\frac{(0.25\text{ m}{)}^3}{0.05\text{ kg}}$ relative to SI, again $128\times \frac{0.015625}{0.05}=40$. The figure falls from $128$ to $40$ because the chosen mass unit is large while the length unit shrinks the volume unit.