Precious metals and gems are measured in troy weights in the English system: $$\begin{aligned} 24 \text { grains } &=1 \text { pennyweight (exact) } \\ 20 \text { pennyweight } &=1 \text { troy ounce (exact) } \\ 12 \text { troy ounces } &=1 \text { troy pound (exact) } \\ 1 \operatorname{grain} &=0.0648 \mathrm{g} \\ 1 \text { carat } &=0.200 \mathrm{g} \end{aligned}$$ a. The most common English unit of mass is the pound avoirdupois. What is 1 troy pound in kilograms and in pounds? b. What is the mass of a troy ounce of gold in grams and in carats? c. The density of gold is 19.3 \(\mathrm{g} / \mathrm{cm}^{3} .\) What is the volume of a troy pound of gold?

First, let's convert 1 troy pound to grams using the conversion factors. According to the given information: \(1 \text{ troy pound} = 12 \text{ troy ounces}\) \(1 \text{ troy ounce} = 20 \text{ pennyweight}\) \(1 \text{ pennyweight} = 24 \text{ grains}\) \(1 \text{ grain} = 0.0648 \text{ grams}\) Now, let's convert 1 troy pound to grams: \(1 \text{ troy pound} = 12 \times 20 \times 24 \times 0.0648 \text{ grams}\) Calculating that, we get: \(1 \text{ troy pound} = 373.24 \text{ grams}\) Now, we will convert grams to kilograms and pounds. We know that: 1 kg = 1000 grams 1 pound = 453.592 grams So, 1 troy pound in kilograms and pounds is given by: \(\frac{373.24 \text{ grams}}{1000} \approx 0.373 \text { kg}\) \(\frac{373.24 \text { grams}}{453.592} \approx 0.822 \text{ pounds}\)

Let's convert 1 troy ounce to grams and carats. We know the following conversion for troy ounce to grams from the previous step: \(1 \text{ troy ounce} = 20 \times 24 \times 0.0648 \text{ grams}\) Calculating that, we get: \(1 \text{ troy ounce} = 31.1035 \text{ grams}\) Now, let's convert 1 troy ounce to carats. We know the following conversion: \(1 \text{ carat} = 0.200 \text{ grams}\) So, 1 troy ounce will be equal to: \(\frac{31.1035 \text{ grams}}{0.200} = 155.5175 \text{ carats}\)

We already know the mass of 1 troy pound of gold in grams from step a, which is 373.24 grams. Given the density of gold is 19.3 \(\text{g}/\text{cm}^{3}\), we can find the volume as follows: Volume = Mass / Density \( \text{Volume} = \frac{373.24 \text{ grams}}{19.3 \text { g}/\text{cm}^{3}} \) Calculating that, we get: \(\text{Volume} \approx 19.33 \text{ cm}^{3}\)