Question

The unit "troy ounce" is often used for precious metals such as gold \((\mathrm{Au})\) and platinum \((\mathrm{Pt})(1\) troy ounce \(=\) \(31.103 \mathrm{~g}\) ). (a) A gold coin weighs 2.41 troy ounces. Calculate its mass in grams. (b) Is a troy ounce heavier or lighter than an ounce \((1 \mathrm{lb}=16 \mathrm{oz} ; 1 \mathrm{lb}=453.6 \mathrm{~g})\) ?

Short answer

(a) The gold coin weighs 74.953 grams. (b) A troy ounce is heavier than a regular ounce.

Step-by-step solution

Understanding the Conversion Factor for Part (a)

Recognize that 1 troy ounce is equivalent to 31.103 grams. This is the given conversion factor that we will use to convert the weight of the gold coin from troy ounces to grams.

Calculating Mass of Gold Coin in Grams

Multiply the weight of the gold coin in troy ounces by the conversion factor. \ \[ \text{Mass in grams} = 2.41 \text{ troy ounces} \times 31.103 \frac{\text{grams}}{\text{troy ounce}} \] \ Calculate: \[ 2.41 \times 31.103 = 74.95343 \text{ grams} \] \ Thus, the mass of the gold coin is approximately 74.953 grams.

Understanding the Conversion for Part (b)

Recognize that 1 regular ounce is obtained from the conversion between pounds and regular ounces. Since 1 pound is 16 ounces and 1 pound is also 453.6 grams, we find that 1 regular ounce is: \ \[ \text{1 regular ounce} = \frac{453.6}{16} \approx 28.35 \text{ grams} \]

Comparing the Two Ounces

Compare the gram equivalences to see which is heavier. Since 1 troy ounce is 31.103 grams and 1 regular ounce is approximately 28.35 grams, we see: \ \[ 31.103 \text{ grams per troy ounce} > 28.35 \text{ grams per regular ounce} \] \ Thus, a troy ounce is heavier than a regular ounce.

Key concepts

Troy Ounce

A troy ounce is a unit of mass that is primarily used to weigh precious metals such as gold, silver, and platinum. It differs from the more commonly known avoirdupois ounce that we use in our daily lives.

• One troy ounce is precisely equal to 31.103 grams.
• The troy ounce originated in the Middle Ages and was used across various parts of Europe.
• This unit is retained in modern times due to its historical use for valuing and trading precious metals.

Remember that when dealing with tasks involving troy ounces, it's important to use the exact conversion rate to ensure accuracy in calculations. This is crucial when calculating the mass of jewelry, coins, or any precious material.

Metric System

The metric system is a standardized system of measurement used around the world for scientific and everyday measurement needs. It offers a uniform way to convert and calculate weights or other units.

• The base unit for mass in the metric system is the gram.
• All other units of mass are derived from the gram, using prefixes like kilo (1000 grams), milli (0.001 grams), etc.

For example, converting troy ounces to grams or grams to kilograms involves simple multiplication or division. The advantage of the metric system is its straightforwardness and universality, simplifying international communication and commerce.

Mass Calculation

Mass calculation is the process of determining the amount of matter in an object. For coins, such as a gold coin, calculating its mass involves specific conversion factors.Consider a gold coin weighing 2.41 troy ounces. To find its mass in grams:1. Use the conversion factor that 1 troy ounce equals 31.103 grams.2. Multiply the weight in troy ounces by this conversion rate: \[ \text{Mass in grams} = 2.41 \times 31.103 = 74.953 \text{ grams} \]This provides the precise mass of the gold coin in grams. Such calculations are crucial in fields such as chemistry, physics, or any trade involving precious metals.

Comparing Measurements

Comparing measurements is essential when dealing with different units of mass. In our case, we compare the troy ounce and the regular ounce to understand their differences.

• A troy ounce is 31.103 grams.
• A regular (avoirdupois) ounce is approximately 28.35 grams, derived by dividing 453.6 grams by 16 (since there are 16 ounces in a pound).

From these values, it's clear that a troy ounce is heavier than a regular ounce. This understanding helps in accurately interpreting and comparing goods weighed in different units, whether in commerce or scientific research.