# Explainer: Converting and Comparing Mass: Customary Units

In this explainer, we will learn how to convert between ounces, pounds, and tons and apply unit conversion to solve real-world problems using decimals and fractions.

Units are used whenever we measure a physical quantity—for instance, a length, an area, a volume, a mass, or a temperature. Units allow for comparison between lengths, areas, volumes, masses, and so on, by providing references.

Mass is a measure of the quantity of matter in an object. In everyday language, we often use the word “weight” instead. Weight and mass are very closely related, and that is the reason why there is this confusion in everyday language. In math and science, however, it is important to talk about mass.

The main customary units of mass are, from the smallest to the largest, the ounce (oz), the pound (lb), and the ton (T). They are mutually defined as given below.

### Definition: Main Customary Units of Mass

• 16 ounces (oz) = 1 pound (lb);
• 2,000 pounds (lb) = 1 ton (T).

The values given above form ratios: the measure of the mass of a given object in ounces and pounds is always in a 16 : 1 ratio, and that in pounds and tons is in a 2,000 : 1 ratio. We are going now to see how to use a double-line diagram to represent these ratios and how this will help us convert between these units.

We know that 16 ounces correspond to one pound. How many ounces then correspond to 3 pounds? The situation can be illustrated on a double-line diagram.

We see that each pound corresponds to 16 ounces. Hence, there are three groups of 16 ounces in three pounds. The corresponding calculation is

We simply need to multiply the number of pounds by the number of ounces there are in one pound.

Let us look now at a conversion between a smaller unit (pound) and a larger unit (ton). We know that 2,000 pounds correspond to one ton. What are 4,800 pounds in tons then? The question can be represented on a double-line diagram.

Every group of 2,000 pounds makes a ton, so the question essentially is to find how many groups of 2,000 pounds there are in 4,800 pounds. The corresponding calculation is

We can also use the double-line diagram to visualize the proportional relationship between the pounds and tons with a proportionality factor of 2,000. The double-line diagram allows us to use the ratio 2,000 : 1 in the right way. Indeed, in this case, we see that we need to divide by 2,000 to convert from a smaller to a larger unit. Whenever we have a doubt of how we convert from one unit to the other, it is a good idea to use a double-line diagram.

In both of our examples, we have used the fact that the measures of a given mass expressed in two different units are always in the same ratio. We can also use these ratios directly. For ounces and pounds, this ratio is 16 : 1, meaning that there are 16 ounces in a pound or that a group of 16 ounces is one pound. This ratio can be written also in the form

which can be understood as there are 16 inches in a (or per) pound, or which means that each group of 16 ounces is one pound.

In our first example, we converted 3 pounds into ounces:

Writing the units in this calculation shows its logic: there are 16 ounces for each pound, so we need to multiply the number of pounds by 16 to convert pounds into ounces.

When we convert from a smaller unit to a larger one, then the expected result is a smaller number. A mass of 4,800 pounds is 2.4 tons. In this example, we calculated that is, we worked out the number of groups of 2,000 pounds in 4,800 pounds.

Now, if we look at the units in both calculations, we may discover how units can help us write calculations that make sense. Exactly in the same way as when you multiply 2 inches by 3 inches you find an area of 6 square inches, the units in any calculation involving measurements do combine together. In our first previous calculation, we see that the pounds cancel out because we divide 4,800 pounds by 20,800 pounds, which gives 2.4, and we multiply this by 1 ton, which yields 2.4 tons.

Now, if we look at the units in both calculations, we may discover how units can help us write calculations that make sense. Exactly in the same way as when you multiply 2 inches by 3 inches you find an area of 6 square inches, the units in any calculation involving measurements do combine together. In our previous calculation, we see that the pounds cancel out because we divide 4,800 pounds by 2,000 pounds, which gives 2.4, and we multiply this by 1 ton, which yields 2.4 tons.

Let us now compare with

This equation does not make any sense! We are clearly not converting pounds to tons here.

Note that this method of looking at the units is actually much more general than the sole topic of unit conversion.

Let us look now at some examples to see how unit conversion reasoning is applied.

In the next couple of examples, we are going to see as well how to convert a measure given as a mixed number.

### Example 1: Converting Tons into Pounds

How many pounds are in tons?

We want to convert tons to pounds. Recall that 1 ton is 2,000 pounds. So, we need to multiply tons by 2,000 pounds per ton to convert it to pounds:

To carry out this calculation, we need to find . One-fifth of 2,000 is 2,000 divided by five, that is, 400. So, three-fifths of 2,000 is three times 400, that is, 1,200.

(Note that , 0.6 being of 1.)

We finally find

Our answer is T = 21,200 lb.

### Example 2: Converting Tons into Ounces

How many ounces are there in tons?

To convert tons to ounces, we need to know how many ounces there are in one ton. For this, let us use a triple-line diagram.

Since one ton is 2,000 pounds, and one pound is 16 ounces, we find that one ton is 32,000 ounces.

Now, we want to convert tons to ounces. We need to multiply by 32,000. We can visualize the result on a double-line diagram.

We find that a fifth of 32,000 is 6,400 (notice that dividing by 5 is the same as dividing by 10 and multiplying by 2), and so two-fifths is 12,800. So, one ton is 32,000 ounces and of a ton is , so tons is 44,800 ounces. We could also of course simply multiply 1.4, which is the same as , by 32,000 to convert tons to ounces.

Our answer is that there are 44,800 ounces in tons.

In the next example, we are going to convert ounces to pounds.

### Example 3: Converting Ounces into Pounds

How many pounds are there in 34 oz?

We want to convert 34 ounces into pounds. Recall that there are 16 ounces in one pound. Hence, we need to divide 34 by 16 to find how many groups of 16 ounces there are (each group being a pound).

We see easily that there are two groups of 16, with a remainder of 2. The number of pounds will then be expressed as a mixed number with 2 as a whole number. Since the measures of a given mass in ounces and pounds are proportional (this is why we can use a double-line diagram), the fraction of the remainder of 2 over 16 will be the same as the fractional part of our result.

Hence, we find that 34 ounces are pounds.

The fraction can be simplified by 2:

Our answer is that there are lb in 34 oz.

Let us look at two examples where we want to convert a measure in a smaller unit to a whole number of a larger unit and a remainder in the smaller unit.

### Example 4: Converting Pounds to a Whole Number of Tons and Pounds

.

We want to convert 5,788 pounds to tons and pounds. This means we are looking for the whole number of tons in 5,788 pounds, expressing the remainder in pounds. Recall that one ton is 2,000 pounds. The whole number of tons in 5,788 pounds is therefore the whole number of groups of 2,000. We can write

Hence, there are two groups of 2,000, that is, 2 tons, and 1,788 pounds in 5,788 pounds.

Our answer is 5,788 lb = 2 T  1,788 lb.

### Example 5: Converting Ounces into a Whole Number of Pounds and Ounces

Daniel received a package that weighed 478 ounces. Which of the following is equivalent to the weight of the package?

1. 29 lb and 14 oz
2. 39 lb and 10 oz
3. 29 lb and 449 oz
4. 23 lb and 455 oz
5. 23 lb and 18 oz

Daniel’s package weighs 478 ounces. We want to convert this mass (remember that weight is used in everyday language, but mass is the proper name for measures in ounces) to a whole number of pounds and a remainder in ounces. Since 16 ounces make a pound, we are looking here for the whole number of groups of 16 in 478.

We find that

There are 29 group of 16 ounces, that is, 29 pounds, and further 14 ounces in 478 ounces. Hence, our answer is that 478 ounces is equivalent to 29 lb and 14 oz.

### Key Points

1. Mass is a measure of the quantity of matter in an object. In everyday language, we often use the word “weight” instead. Weight and mass are very closely related, and that is the reason why there is this confusion in everyday language. In math and science, however, it is important to talk about mass.
2. The main customary units of mass are, from the smallest to the largest, the ounce (oz), the pound (lb), and the ton (T). They are mutually defined as given below:
• 16 ounces (oz) = 1 pound (lb);
• 2,000 pounds (lb) = 1 ton (T).
3. The values given above form ratios: the measure of the mass of a given object in ounces and pounds is always in a 16 : 1 ratio, and that in pounds and tons is in a 2,000 : 1 ratio.
4. We can use a double-line diagram to represent these ratios to help us convert between these units.