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Chapter 8—Measurements

8-1 The U.S. Customary System

One of the most common application of everyday math is working with measurements. We measure a lot of things, and we do it almost daily. We figure distances to places we want to go. We figure how much time it will take us to drive or walk from one place to another. And we measure amounts of things such as sugar, gasoline, and the amount of fat and protein in our favorite hamburger. Indeed, measurement is an important part of our lives.

Most people in the world today use the metric system of measurement for everyday math.  The metric system is a simple system that uses multiples of ten and units of measurement with familiar-sounding names such as meters,  grams, and liters. But the USA  is a holdout against the universal application of the metric  system. The United States attempted to adopt the metric system of measurement, but Americans refused to change from their traditional "English" system of measurement--the one based on units of measure such as ounces, pounds, inches, and feet. Today, The USA is only developed economic power in the world that embraces the U.S. Customary System (as it is now called).

 FACTOIDSThe conventional (U.S. or English) system of measurement is part of U.S. cultural heritage from the days when the thirteen colonies were under British rule. It started as a collection of Anglo-Saxon, Roman, and Norman-French weights and measures. For example, the inch represents the width of the thumb and the foot is from the length of the human foot. Tradition holds that King Henry I decreed that the yard should be the distance from the tip of his nose to the end of his thumb. Since medieval times, commissions appointed by various English monarchs have reduced the chaos of measurement by setting specific standards for some of the most important units. Some of the conventional units of measure are: inches, feet, yards, miles, ounces, pints, gallons, and pounds. Because the conventional system was not set up systematically, it contains a random collection of conversions. For example, 1 mile = 5,280 feet and 1 foot = 12 inches.

The most common units of measure in the U.S. Customary System are length (or distance), weight, and capacity. Units of time are are actually shared by the U.S. Customary and metric systems. The table below  shows the common units of measure and their abbreviations.

 Units of Length (or Distance)

inches (in or " )
feet (ft or ' )
yards (yd)
miles (mi)
 Units of Weight
ounce (oz)
pound (lb)
ton (T)
 Units of Capacity
liquid ounces (fl oz)
pint (pt)
quart (qt)
gallon (gal)
 Units of Time (Actually shared by U.S. customary and the metric systems
second (sec)
minute (min)
hour (hr)
day (da)
week (wk)
month (mo)
year (yr)

"But  what about other ordinary units such as area, volume and speed?"

Other U.S. Customary measurements, such as area, volume, and speed, are multiples or combinations of the basic units shown on the table. For example:

• Area is a combination of two distances
• Volume is a combination of three distances
• Speed is a combination of distance and time

Units of temperature are not officially associated with either the U.S. Customary or metric systems. Unofficially:

• The popular scale for temperature in the USA is degrees Fahrenheit (ºF)
• Anyone using the metric system tends to use the Celsius scale (ºC)

orking with Length and Distance in the U.S. Customary System

One of the most common units of measure deal with length, or distance. In a mathematical sense, the terms length and distance can be used interchangeably.

Units of Length or Distance
(U.S. Customary)

 Name and abbreviation inches (in or " ) feet (ft or ' ) yards (yd) miles (mi) Conversions Factors 1 ft = 12 in 1 yd = 3 ft 1 mi = 5280 ft

Converting Units of Length

Everyday measurements often make it necessary to convert between the various units of length—inches to feet,  feet back  to inches, feet to yards, miles to feet, and so on. Confusion on this matter often goes like this:

"How many feet are in 48 inches?
"Um-m-m, do I divide by 12 or by 3?  Or is it multiplying by 12 or 3? I know it's something like that!"

You can end that sort of confusion by applying what you already know about ratios.  Think of conversion factors as ratios:

 12 in or 1 ft 1 ft 12 in
 3 ft or 1 yd 1 yd 3 ft
 5280 ft or 1 mi 1 mi 5280i
12 inches per foot is the same as 1 foot per 12 inches 3 feet per yard is the same as 1 yard per 3 feet 5280 feet per mile is the same as 1 mile per 5280 feet

Here is one example. We'll work it first, then show you how and why it works.

Example 1

Problem: Convert 48in to feet.

Procedure

1. Set up the feet/inches ratio

 1ft 12in

2. Multiply by the number of inches

 1ft x 48in = 4ft 12in 1

Solution:  48in = 4ft

Notice in Step 2 how the ratios are set up. They are set up so that the units of inches cancel, thus leaving only units of feet for the product of the multiplication.

 1ft x 48in = 4ft 12in 1

Example 2

Problem: Convert 0.5mi to feet:

Procedure

1. Set up the feet/miles ratio

 5280ft 1mi

2. Multiply by the number of miles

 5280ft x 0.5mi = 2640ft 1mi 1

Solution: 0.5mi = 2640ft

Notice how Step 2 was set up so that the mi labels "cancelled," thus leaving the ft label alone in the numerator (and the solution).

More Examples

Examples and Exercises

Adding and Subtracting Units of Length

Important

Only add or subtract lengths and distances having the same units.
 3 in + 5 in = 8 in Units are the same 6½ mi – 4 mi = 2½ mi Units are the same 2 in + 3ft = ??? Units are NOT the same

When it is necessary to add or subtract lengths that are not expressed in the same units, one (or both) must be converted to identical units of length. Operations are possible only after making sure all terms are in the same units.

Example 3

Problem: Eddy walks 3 miles east and 4 miles north. What is the total distance Eddy walked.

Procedure:

1. Conceptualize the situation

distance  one + distance two = total distance

2. Express with known values

3mi + 4mi = 7mi

Solution: Eddy walked a total of 7mi

Note: There was no need to convert either of the distances.  Why not? They were expressed in the same units ...  miles.

Example 4

Problem: Find the sum of these three lengths:  2in, 8in, 1ft

Conceptualize the problem

length 1 + length 2 + length 3 = total  length

Express with the known values

2in + 8in + 1ft = ?

The units must be identical. The simplest  approach in this example is to convert the units of feet to inches.

2in + 8 in + 12in = 22in

Example

Convert 4ft 7in to all inches.

1. Convert the 4ft to inches:

 Choose the ft/in ratio that has feet in the denominator: 12in 1ft
 Multiply the conversion ratio by the number of inches: 12 in x 4 ft = 48in 1ft 1

48in + 7in = 55in

Solution: 4ft 7in = 55in

More Examples

Examples and Exercises

Multiplying and Dividing Units of Length and Distance

 Important Multiplication and division of length/distance values is valid under two conditions: The units are the same (all feet, all inches, etc.) A unit is multiplied or divided by a number that has no units

Two length are to be multiplied when determining the area of a square or rectangular object. The basic equation for the area of a rectangle is length x width. When the length and width are provided in U.S. Customary units, you have a situation where it is necessary to multiply a set of units -- both having the same unit of measure.

Example 1

A section of property in the city measure 280ft by 160ft. What is the area in square feet?

Both units are expressed in feet, so the solution is a matter of multiplying them:

Area = 280ft x 160ft = 44,800 square feet.

Example 2

A section of property measure 260ft by 120yd. What is the area in square feet?

In this example, the two terms are not expressed in the same units -- one is in feet and the other in yards. You can complete the multiplication only after making certain both terms are expressed in identical units. This example states the result should be in square feet, so the procedure is this:

Convert the 120 yd to feet, then multiply the result by 260 ft

1. Convert 120yd to feet

 Choose the ft/yd ratio that has yards in the denominator: 3 ft 1 yd
 Multiply the conversion ratio by the number of yards: 3 ft x 120yd = 360ft 1 yd 1

So 120yd = 360ft

2. Multiply the lengths of the sides

Area = 260ft x 360ft = 93,600ft2

More Examples

sq inches

Examples and Exercises

Lengths can always be multiplied and divided by unitless values. For example: 8in ÷ 2 = 4in. And 2mi x 4.5 = 9mi. The result is always in the same units as the original measurement.

More Examples

Examples and Exercises

Working with Fractions of Feet and Inches

A common ruler or tape measure divides an inch into sixteen equally spaced parts. Each part, or division, thus represents 1/16th of an inch.  Measurements for precision machines use measurements where each inch is divided into 1/32, 1/64, and even 1/128th of an inch.

 Number of Divisions Length 1 1/16 2 2/16 = 1/8 in 3 3/16 in 4 4/16 = 1/4 in 5 5/16 in 6 6/16 = 3/8 in 7 7/16 in 8 8/16 = 1/2 in 9 9/16 in 10 10/16 = 5/8 in 11 11/16 in 12 12/16 = 3/4 in 13 13/16 in 14 14/16 = 7/8 in 15 15/16 in 16 16/16 = 1 in

Math Operations with Inches
 Working with inches is simply a practical application of fractions. If you know how to add, subtract, multiply, and divide mixed fractions, you already know how to handle the mathematics of inches. Also, it can be convenient to work with mixed fractions as a sum of a whole number and a fraction. For example, let 3 5/16 = 3 + 5/16.You can see how this is used in the following examples.

Expressing Mixed Fractions of Inches as Sums

 Use this scroll bar to view all of the examples.

Examples and Exercises

 Adding and Subtracting Inches These examples and exercises will show you that you've mastered the whole idea of adding and subtracting measurements expressed in inches.

Example: Multiplying Inches
 Problem Multiply 3 5/8" times 4 Procedure Set up the problem 4 x 3 5/8" = _____ Multiply 4 x ( 3 + 5/8 ) = 12 + 20/8 Reduce 12 + 20/8 = 12 + 2 4/8 = 12 + 2 1/2 Complete the addition 12 + 2 1/2 = 14 1/2 Solution  4 x 3 5/8" = 14 1/2"

Example: Dividing Inches

 Problem Divide 8 3/8" by 4 Procedure Set up the problem 8 3/8" ÷ 4  = _____ Convert to multiplication 8 3/8" ÷ 4  = ( 8 + 3/8 ) x 1/4 Complete the multiplication ( 8 + 3/8 ) x 1/4 = 2 + 3/32 Simplify 2 + 3/32 = 2  3/32 Solution  8 3/8" ÷ 4  = 2  3/32"

Examples and Exercises

 Multiplying and Dividing Inches These examples and exercises will show you that you've mastered the whole idea of multiplying and dividing measurements expressed in inches.

Measures of Weight in the U.S. Customary System

1. Units of Weight
(U.S. Customary)

 Name and abbreviation ounce (oz) pound (lb) ton (T) Conversions Factors 1 pound (lb) = 16 ounces (oz) 1 ton (T) = 2000 pounds (lb)

Measures of Capacity in the U.S. Customary System

Units of Capacity (Volume)
(U.S. Customary)

 Name and abbreviation fluid ounces (fl oz) pint (pt) quart (qt) gallon (gal) Conversions Factors 1 cup (c) = 8 fluid ounces (fl. oz) 1 pint (pt) = 2 cups (c) 1 quart (qt) = 4 cups (c) 1 quart (qt) = 2 pints (p) 1 gallon (ga.) = 4 quarts (qt)

Measures of Time

Units Time

 Name and abbreviation second (sec) minute (min) hour (hr) day (da) week (wk) month (mo) year (yr) Conversions Factors 1 minute (min) = 60 seconds (sec) 1 hour (hr) = 60 minutes (min) 1 day (da) = 24 hours (hr) 1 week (wk) = 7 days (da) 1 year (yr) = 12 months (mo) 1 year (yr) = 365 days (da)* Notes: Every 4th year is a leap year. That year, 1 yr = 366 da The number of days in a month varies between 28 and 31 The approximate number of weeks in a year is 52