Precious metals and gems are measured in troy weights in the...

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?

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?

Key Concepts

Volume Calculation via Density

Calculating volume via density involves rearranging the density formula to solve for volume, that is, volume = mass/density. This method is particularly useful when given a fixed mass of a substance and its density, as in determining the physical space occupied by a known weight of gold, thus linking mass measurements directly to spatial dimensions.

Troy Weight System

The Troy weight system is a traditional system of mass measurement that is primarily used for precious metals such as gold, silver, and gems. It is distinct from more common modern measurements, featuring units such as grains, pennyweights, troy ounces, and troy pounds, with each unit having an exact relationship with the others. This system is particularly important in industries where precise weights are crucial, like jewelry and bullion trading.

Avoirdupois Weight System

The Avoirdupois weight system is the modern system of weights traditionally used in everyday commerce and is based on pounds and ounces. It differs from the Troy system in its subdivision and conversion factors, making it the common benchmark for body weights, food quantities, and other general measurements. Understanding the differences between these systems is crucial for accurate conversions and comparisons.

Unit Conversion

Unit conversion involves changing a quantity expressed in one set of units to another using conversion factors. This concept is essential when working across different measurement systems, such as converting from the Troy weight system to the Avoirdupois system or to metric units, ensuring that the quantity remains consistent and accurate irrespective of the units used.

Metric Conversion

Metric conversion refers to translating measurements from one unit within the metric system to another, or from non-metric systems into metric units like grams or kilograms. Since many scientific and international applications rely on the metric system, it is important to accurately convert weights and measures, such as determining the mass of precious metals in grams or kilograms based on their original measurement in the Troy system.

Density

Density is a physical property that relates the mass of a substance to its volume, typically expressed in units like grams per cubic centimeter (g/cm³). This concept is critical in problems where the volume of a material needs to be determined from its mass, or vice versa, by applying the formula density = mass/volume, and is widely used in fields such as metallurgy and material science.

Transcript

00:01 Hi, so in order to solve this problem, what we're going to do is to employ the conversion factors that we can see here.

00:07 So we want to convert one troy pound into grams, into kilograms, and into pounds.

00:17 So first of all, in order to convert this into kilograms, the easiest way to do it is to trace back the conversion factors that are given to us and get the result in grams.

00:27 So first of all, one troy pound is going to be, we're going to put the equivalence as the denominator, and this is equal to 12 troy ounces.

00:44 Next up, one troy ounce is going to be equal to 20 pennyweights, i'm going to abbreviate it as penny w for sake of space, and then one pennyweight is equal to 24 grains, and one grain is equal to 0 .0648 grams.

01:24 So after using these conversion factors, you're going to realize that the units continuously cancel out until we get the result in grams.

01:36 In this case, the result is going to be 12 times 20 times 24 times 0 .0648, corresponding to 373 .248 grams.

01:52 Now, this converting into kilograms, we can use a conversion factor, 1000 grams is going to be equal to one kilogram.

02:06 So after division, we're going to get 0 .373.

02:11 We can round up this value, let's say that instead of having 0 .248 grams, we can round it up as 25.

02:21 So we can round it up like this.

02:25 If you want to give the full result, simply do not round up that quantity, and we can put the full value that we found from the calculation.

02:36 So 248 kilograms.

02:41 Next up, what we want to do is, this is the value of one troy pound into kilograms, so now we want to know it in pounds.

02:48 So for pounds, we're going to use the conversion factor 248.

02:55 We put the quantity that we want to convert and the conversion from kilograms to pounds...

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