Explainer: Converting and Comparing Length: Customary Units

In this explainer, we will learn how to convert customary units of length, such as feet, inches, yards, and miles, and compare them to solve real-world problems.

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, and masses, for example, by providing references.

The dimensions of the different objects that make the world can be very different. The diameter of an atom is much shorter than our height, which is much shorter than the diameter of Earth, which is much shorter than the diameter of the universe. Describing all these lengths with a single unit is possible, but it would not be the best choice. Even when confining ourselves to lengths we may need in our everyday life, it is more convenient to use different units in order to have measurements neither too small nor too big.

For instance, inches are very useful to measure the length of small objects like a computer screen or a picture frame, dimensions of a piece of furniture, and so forth. To measure bigger lengths, such as the dimensions of a room, feet might be more convenient. Yards will be used for short distances you may walk, while miles are used for larger distances, such as the distance between two cities.

Now, inches, feet, yards, and miles are all different length units, and they are clearly defined. Luckily, there is a whole number of inches in a foot, a whole number of feet in a yard, and a whole number of yards in a mile, as shown below.

Definition: Main Customary Units of Length

  • 12=1inchesfoot;
  • 3=1feetyard;
  • 1,760=1yardsmile.

The values given above form ratios: the measures of a given length in inches and feet are always in a 121 ratio, those in feet and yards are in a 31 ratio, and so on. 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 3 feet correspond to one yard. How many feet are then in 5 yards? The situation can be illustrated on a double-line diagram.

We see that, for each yard, there are 3 feet. Hence, there are 5 groups of 3 feet in 5 yards. The corresponding calculation is 5×3=15.

We simply need to multiply the number of yards by the number of feet there is in one yard.

Let us look now at a conversion between a smaller unit (inches) and a larger unit (foot). We know that 12 inches correspond to one foot. What are 30 inches in feet then? The question can be represented on a double-line diagram.

Here, there are mainly two ways to envision this situation. First, we can see that every group of 12 inches makes one foot, and so we need to find how many groups of 12 inches there are in 30 inches to find the number of feet (this is the number of groups). The corresponding calculation is 30÷12=212.

We can also use the ratio 121 between inches and feet, which is equivalent to saying that the measures of a given length in inches and feet are proportional. The double-line diagram can therefore be used to visualize how to use the 121 ratio to find the number of feet  30 inches correspond to.

The diagram is useful when you are not sure in which way to use the 121 ratio. The diagram then allows you to reason that a length is expressed in a smaller unit (here inches) with a larger number and, hence, that you need to divide 30 by 12 to find the number of feet.

In both of our examples, we have used the fact that the measurements of a given length expressed in two different units are always in the same ratio. For inches and feet, this ratio is 121, meaning that there are 12 inches in a foot, or that a group of 12 inches is one foot. This ratio can be written also in the form 121,inchesfoot which can be understood as 12 inches in a (or per) foot, or 112,footinches which means that each group of 12 inches is one foot.

Looking back at our first example where we converted five yards into feet, we can write what we have done for the conversion using the ratio of feet to yards written in this way; namely, 5×31.yardsfeetyard

Writing the units shows the logic of our calculation: there are 3 feet for each yard, so we need to multiply the number of yards by 3 to convert yards into feet.

In our second example, we did 30×112;inchesfootinches that is, we worked out the number of groups of 12 inches in 30 inches.

Now, if we look at the units in both calculations, we may discover how units can help us write calculations that make sense. Let us compare 30×112=212inchesfootinchesft and 30×121=360/.inchesinchesfootinft

In the first calculation, we do get the right result, a length in feet, but that is not the case in the second calculation (there is no meaning attached to the result of the second equation). The units in any calculation involving measurements do combine together, in the same way as when you multiply 2 inches by 3 inches you find an area of 6 square inches. In our first calculation, we see that the inches cancel out because we divide 30 inches by 12 inches, which gives 212.

Note that this method of looking at the units can always be applied and not only when dealing with unit conversion.

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

Example 1: Converting Miles into Yards

Complete the following: 4=miyd.

Answer

Here, we want to convert 4 miles into yards. We know that there are 1,760 yards in one mile.

So, there will be 4 times 1,760 yards in 4 miles; that is, 4×1,760=7,040.

Hence, 4=7,040miyd.

We can visualize this on a double-line diagram.

The calculation can also be written with making the units apparent, using the ratio 1,7604yardsmiles: 4×1,7604=7,040.milesyardsmilesyards

Example 2: Converting Yards into Miles

Fill in the blanks: 72,160=ydmi.

Answer

We want to convert 72,160 yards into miles. Mile is a larger unit than yard, and, indeed, each group of 1,760 yards makes one mile, so we need to find how many groups of 1,760 yards there are in 72,160 yards. The calculation is then 72,160÷1,760=41.

The calculation can be written with making the units apparent, using the ratio 11,760mileyards: 72,160×11,760=41.yardsmileyardsmiles

Example 3: Converting Inches into a Whole Number of Feet and Inches

Fill in the blanks: 155=inftin.

Answer

We want to convert 155 inches into a number of feet and inches. Recall that 12 inches make one foot. So, we need to find how many groups of 12 inches there are in 155 inches and find the remainder.

We have 155÷12=12,.remainder11

Hence, 155=1211.inftin

Example 4: Converting Feet into Miles

Fill in the blanks: 15,840=ftmi.

Answer

We want to convert 15,840 feet into miles. For this, we need to know how many feet make one mile. Recall that 1,760 yards make 1 mile and 3 feet make 1 yard. So, there are 1,760×3=5,280 feet in 1,760 yards, that is, in 1 mile.

Now, to convert 15,840 feet into miles, we need to find how many groups of 5,280 feet there are in 15,840 feet. This is given by 15,840÷5,280=3.

Hence, 15,840=3.ftmi

Example 5: Converting Feet into Inches and to Yards

If Matthew threw his javelin 102 feet during practice. Determine, in inches and yards, how far he threw his javelin.

Answer

Matthew threw his javelin at a distance of 102 feet. We want to convert this distance first into inches and then into yards. Recall that there are 12 inches in each foot. Hence, 102×121=1,224.ftinftin

The yard, by contrast, is a greater unit than the foot: 3 feet make a yard. We need therefore to find how many groups of 3 yards there are in 102 feet. This is given by 102÷3 or 102×13.ftydft

Hence, 102=34.ftyd

Key Points

  1. The dimensions of the different objects that make the world can be very different. Describing all these lengths with a single unit is possible, but it would not be the best choice since we would end up with very large or very small measurements.
  2. In the customary system, the main units of length are the inch, the foot, and the yard; their relative values are
    • 12=1inchesfoot;
    • 3=1feetyard;
    • 1,760=1yardsmile.
  3. The values given above form ratios: the measures of a given length in inches and feet are always in a 121 ratio, those in feet and yards are in a 31 ratio, and so on.
  4. We can use a double-line diagram to represent these ratios to help us convert between these units.

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