Question

Precious metals and gems are measured in troy weights in the English system: $$ \begin{aligned} 24 \text { grains } &=1 \text { pennyweight (exact) } \\ 20 \text { pennyweight } &=1 \text { troy ounce (exact) } \\ 12 \text { troy ounces } &=1 \text { troy pound (exact) } \\ 1 \text { grain } &=0.0648 \mathrm{~g} \\ 1 \text { carat } &=0.200 \mathrm{~g} \end{aligned} $$ 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?

Short answer

a. 1 troy pound is approximately 0.373 kg and 0.823 pounds avoirdupois. b. The mass of a troy ounce of gold is approximately 31.104 grams and 155.52 carats. c. The volume of a troy pound of gold is approximately 19.34 cubic centimeters.

Step-by-step solution

Convert troy pound to grams

First, we need to convert 1 troy pound to grams using the given conversion rates. We know that 1 troy pound is equal to 12 troy ounces, and 1 troy ounce is equal to 20 pennyweight, and 1 pennyweight is equal to 24 grains. We will use these conversions to find out how many grams is 1 troy pound. 1 troy pound = 12 troy ounces = 12 * 20 pennyweight = 240 pennyweight Now, we'll convert 240 pennyweight to grams. 240 pennyweight = 240 * 24 grains = 5760 grains Finally, we'll convert 5760 grains to grams. 1 grain = 0.0648 g 5760 grains = 5760 * 0.0648 g ≈ 373.248 g

Convert grams to kilograms and pounds avoirdupois

Now we have the equivalent mass of 1 troy pound in grams, we'll convert it to kilograms and pounds avoirdupois. 1 kg = 1000 g 373.248 g ≈ 0.373 kg 1 pound avoirdupois = 453.592 g 373.248 g ≈ 373.248/453.592 ≈ 0.823 pounds avoirdupois So, 1 troy pound is approximately 0.373 kg and 0.823 pounds avoirdupois. #b. Calculate the mass of a troy ounce of gold in grams and carats#

Convert troy ounce to grams

We know 1 troy ounce is equal to 20 pennyweight, and 1 pennyweight is equal to 24 grains. Using these conversions, we'll find out how many grams are in 1 troy ounce. 1 troy ounce = 20 pennyweight = 20 * 24 grains = 480 grains Now, we'll convert 480 grains to grams. 480 grains = 480 * 0.0648 g ≈ 31.104 g

Convert grams to carats

Now we have the equivalent mass of 1 troy ounce in grams, we'll convert it to carats. 1 carat = 0.200 g 31.104 g = 31.104/0.200 ≈ 155.52 carats So, the mass of a troy ounce of gold is approximately 31.104 grams and 155.52 carats. #c. Calculate the volume of a troy pound of gold#

Recall the mass of a troy pound in grams

From part (a), we calculated that the mass of 1 troy pound is approximately 373.248 g.

Use the density formula to find the volume

We know the density of gold is 19.3 g/cm³. Using the formula Density = Mass/Volume, we can find the volume of a troy pound of gold. Volume = Mass/Density Volume = 373.248 g / 19.3 g/cm³ ≈ 19.34 cm³ So, the volume of a troy pound of gold is approximately 19.34 cubic centimeters.

Key concepts

Mass Conversion

Understanding mass conversion is essential when dealing with different weight measurement systems like troy weight and the metric system. Troy weight primarily measures precious metals such as gold and silver. To convert between units, you need to understand the relationships between them.

• 1 troy pound = 12 troy ounces
• 1 troy ounce = 20 pennyweights
• 1 pennyweight = 24 grains

Given these relationships, converting a troy pound to grams involves calculating the total grains in a troy pound and then converting grains to grams. It’s critical to ensure precision in conversions, as even slight errors can lead to significant differences in weight, especially when valuing precious metals.

Metric System

The metric system is a universal system of measurement based on the powers of ten, making it simple to convert units. In the context of converting troy weight to the metric system, grams and kilograms are common units. Converting grains to grams uses the factor that 1 grain equals 0.0648 grams. After finding the weight in grams, it can easily be converted to kilograms by remembering that 1 kilogram equals 1,000 grams.
By converting troy weights to the metric system, comparisons and valuations across different contexts become more straightforward and precise for international trade and scientific research.

Density Calculation

Density is a pivotal concept in determining how much mass is contained within a given volume. It is expressed as mass per unit volume. The formula for density is \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]When calculating the volume of a troy pound of gold, knowing the density is essential. Gold has a known density of 19.3 grams per cubic centimeter (g/cm³). Using the mass obtained from your troy weight conversion, you can find the volume by rearranging the formula to \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]This calculation is crucial for understanding how much physical space the gold will occupy, helpful in crafting, storing, and managing precious metal assets.

Precious Metals

Precious metals like gold and silver hold significant economic value, often measured in troy weights. The properties of these metals, such as high density and resistance to tarnish, make them ideal for use in jewelry, coins, and industrial applications.
Knowing how to measure these metals correctly is important for valuation, trading, and legal purposes. The precision in measurement reflects in their pricing and trading; thus, converting between units must be accurate.

• Gold is often measured in carats (1 carat = 0.2 g) when priced by weight.
• Due to high density, gold has a smaller volume than many other materials of the same mass.

Volume Calculation

Volume calculation of materials like gold involves using its mass and density. Understanding the volume helps in determining the size of a storage unit needed or space occupied by metal bars or coins. The calculation process uses the formula \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]For example, with gold's density of 19.3 g/cm³ and a troy pound mass conversion to approximately 373.248 grams, we calculate the volume as roughly 19.34 cubic centimeters.
This calculation provides insights into the spatial requirements for storing gold, crucial for logistics and asset management in industries handling large quantities of precious metals.