5. (15 points) The density of a body $\rho$ is defined as the ratio of the body's mass $m$ to its volume $V$. A unit of mass used in jewelry is the carat (1 carat equals 0.2 grams). A unit of length used in many countries is the inch (1 inch equals 2.54 centimeters). It is known that the density of emerald is $\rho=2.7 \text{ g} / \mathrm{cm}^{3}$. Convert this value to carats per cubic inch.
Answer: $\approx 221$ carats/inch ${ }^{3}$.

Solution. 1 gram $=\frac{1}{0.2}$ carat $=5$ carats,

1 cm $=\frac{1}{2.54}$ inch.

![](https://cdn.mathpix.com/cropped/2024_05_06_0588747f110ff428b268g-7.jpg?height=129&width=1585&top_left_y=938&top_left_x=287)

(2 points)