Page 1Introduction to College Math
Section 4.1: Linear Measurements
Measurement conversion is a necessary skill since the same set of units is often not used throughout a calculation. The basis of measurement conversion is the unit fraction.
A unit fraction is a fraction that has a value of 1. The catch is that 1 has infinitely many “aliases”. For example,
All of these represent unit fractions, since the numerator is the same amount as the denominator.
When multiplying a quantity by a unit fraction (which has a value of one), the value of the quantity remains unchanged. That is for any quantity Q:
The trick is to use the “proper” aliases to cancel the units you don’t want and get the units you do want. For example, to convert 18 in to ft we use the unit faction containing the ratio of the number of inches in a foot as follows:
As you can see, the number of inches is placed in the denominator of the unit fraction so that it appears in both the numerator and denominator and as a result cancels out.
Section 5.1.1: Conversions within the English System
As mentioned at the beginning of the chapter we will look at conversions within the English System and the Metric System. In this section, as in the example above, we will continue to look at conversions within the English System.
Example:Convert 5 gallons to quarts.
Solution: First, we must identify the necessary conversion factor between gallons and quarts. Since there are 4 quarts in 1 gallon, we set up a unit fraction using these values. We place the 1 gallon in the denominator of our unit fraction as follows because we are trying to “get rid of” the gallon units.
Thus 5 gallons is equivalent to 20 quarts.
As an aide in setting up conversion calculations, a set of equivalent measurements is presented in the appendix at the end of this chapter.
There are times when we wish to convert measurements that are in decimal form to fractional form. The same technique using unit fractions works.
Example: Round 0.567 in to the nearest 32nd of an inch.
Solution:Since we are trying to convert to 32nd of an inch, the key is to use the unit fraction .
0.567 inches is eighteen thirty-seconds to the nearest thirty-seconds of an inch.
Example: Round 0.434 in to the nearest 64th of an inch.
Solution:Since we are trying to convert to 64th of an inch, the key is to use the unit fraction .
So 0.434 in is seven sixteenth’s of an inch to the nearest 64th of an inch.
For some problems we do not have a conversion factor that directly gives us the number of one type of unit in terms of the other. In these cases we may need to multiply by more than one unit fraction to get us from one unit to the other.
Example:Convert 190 ounces to gallons.
Solution:
190 ounces is or which is approximately 1.5 gallons.
In other more complicated conversions we may need to use more than one unit fraction because we have a measurement involving some type of rate where we need to convert both units involved in the rate.
Example:Convert 100 feet per second to miles per hour correct to one decimal place.
Solution:In this problem, we not only multiply by the unit fraction to convert the feet units to miles, but we must also multiply by the unit fractions that convert seconds to hours. As we do this we must make sure to set up the unit fractions so that the units we wish to “get rid of” are diagonal from each other so that they cancel.
Section 4.1.2: The Metric System
One of the consequences of the French Revolution of 1789 was the development of the metric system of measurement. This system was designed to replace the earlier French system, which like its English counterpart had its origins in medieval society and royal institutions. Three features make the metric system very attractive. First, it is built on powers of 10, just like our decimal number system. Every unit is a multiple of 10 of some other unit. Thus, “strange” English multipliers like 3, 12, and 16 are banished! Second, a deliberate effort was made to coordinate different measures. For example, the fundamental unit of volume, the liter symbolized by L, is simply related to the fundamental unit of length, the meter symbolized by m, through the equation 1 m3 = 1000 L . Contrast this with the English system where
1 gal = 231 in3 = 0.134 ft3. The third advantage of the metric system is that it is “universal”. It can be used with any kind of measurement in the same way.
It is interesting to note that the metric system was so well accepted and in place when electrical measurements began some 150 years ago, only metric units were developed. The customary electric units we are all know, the volt ( V ), amp ( A ), and ohm () are all metric.
The metric system uses a two-part representation for all measurements. Each unit of measurement contains the base unit which is determined by what we are trying to measure. The important metric base units are as follows:
Base Unit / Symbol / What it Measuresmeter / m / length
seconds / s / time
gram / g / mass
liter / L / volume
watt / w / power
joule / j / energy
newton / N / force
hertz / Hz / frequency
ampere / A / electric current
volt / V / electric potential
ohm / / resistance
farad / F / electric capacitance
henry / H / electric inductance
Other units smaller or larger than the base unit are created by multiplying or dividing the base unit by powers of 10. The names of each of these smaller/larger units are identified by attaching a prefix to the name of the base unit. The first character or prefix indicates the power of 10 of the number of the base units being used.
Prefix / Symbol / Power of Ten / Number per Base Unitnano / n / /
micro / / /
milli / m / /
centi / c / /
deci / d / /
deca / da / /
hecto / h / /
kilo / k / /
mega / M / /
giga / G / /
Examples of Metric Units:
Centimeter (cm): Centimeters are a smaller unit than the meter. Centimeters are of a meter. This means there are centimeters in one meter.
Kilohertz (kHz): Kilohertz are a larger unit than the hertz. A kilohertz is hertz. This means that there is of a kilohertz in one hertz.
Milligrams (mg): Milligrams are a smaller unit than the gram. Milligrams are of a gram. This means there are milligrams in one gram.
Conversions within the Metric System:
Conversions within the metric system simply require the shifting of the decimal point. As with conversions within the English System, we can use the idea of unit fractions to help us convert between units in the Metric System. We create unit fractions by using what the prefix means in terms of the base unit.
Example:Convert 0.264 kilograms to grams.
Solution:
Notice that in the example above, we shift the decimal point three places to the right since we are multiplying by
Example:Convert 175 millimeters to centimeters.
Solution:
Notice in this example, we end up shifting 1 decimal place to the left since we need to divide by 10.
Example:Convert 0.00059 kA to mA.
Solution:
Notice in this example, we end up shifting the decimal point 6 places to the right since we are multiplying by
Section 4.1.3: Performing Operations with Measurements
In many situations we may need to perform operations with measurements. We may need to add measurements to find the perimeter around an object; we may need to subtract to find out how much we have left; we may want to multiply measurements to find area or the amount of work required. In this section we take a look at how to perform these operations.
Section 4.1.3a: Addition and Subtraction of Measurements:
It is not sensible to add or subtract measurements of different kinds of things. For example,
15 lb + 7 ft is a meaningless operation. This is just the old adage that it’s impossible to add apples and oranges! To add or subtract measurements requires the same kind of quantities,
as in 9 feet + 8 feet = 17 feet . Note, we just add the numbers and carry the factor of the unit. This is just the distributive property discussed in the Algebra Chapter. In some problems there may be two different units involved in each measurement. In these problems the key is to add the like type of units.
Example:Perform the indicated operation: 9 lb 8 oz + 4 lb 9 oz
Solution:
However since 17 oz is more than 1 pound’s worth of ounces, we do not leave the answer in this form. We take 16 of the ounces and convert them to one pound. This leaves us with one ounce. We then add one pound to our current total number of pounds and we write the remaining number of ounces.
Our final answer is: 14 lb 1 oz.
What do we do, however, if the units of the measurements we are asked to add or subtract are not the same? We must do unit conversions so that both measurements have the same units.
Example:Perform the indicated operation: 8 ft + 36 in.
Solution:The quantities to be added are both lengths so the operation makes sense, but we can’t actually perform the addition until we get the units to agree. We must either get both units to be inches or both to be feet.
Performing the indicated operation using feet:
We must first convert the 36 inches to feet as follows:
Now we perform the operation using 3 feet in place of the 36 inches.
Thus 8 ft + 36 in is 11 feet.
Performing the indicated operation using inches:
First we must convert 8 feet to inches as follows:
Now we perform the operation using 96 inches in place of 8 feet.
Thus an alternate answer to this problem is that 8 ft + 36 in is equal to 132 inches.
Example: Perform the indicated operation:
Solution:First we convert the 4 feet to inches.
Next we replace the 4 ft by 48 in and do the subtraction.
We can write the answer in one of two ways. We can write the answer as 34 inches; however, it may be preferable to have the answer in terms of feet and inches. To convert this into feet and inches we divide 34 by 12 and we get 2 full feet (24 inches) and write the remaining number of inches.
Thus an alternate answer to this problem is 2 feet and 10 inches.
Example:Perform the indicated operation:
Solution:If we set up the problem vertically as follows, we notice that we do not have enough pounds in the first measurement to do the subtraction.
As a result, we must either borrow or convert all of the units to pounds and then do the subtraction. When borrowing, we borrow 1 from the tons column and convert it over to its equivalent value of 2000 lb. We add the 2000 lb to the number of pound we already have and do the subtraction.
Thus the answer is 2 tons 1025 lb.
Section 4.1.3b Multiplying and Dividing Measurements:
While adding or subtracting different kinds of measurements is impossible, multiplying or dividing measurements is always possible. As mentioned above there are two parts to every measurement. If we are asked to multiply/ divide two measurements we not only multiply/ divide the number parts of the measurements, we also multiply/ divide the unit parts of each measurement.
Example:Perform the indicated operation:
Solution: where a is is a “foot pound” which is a measure of either energy or torque.
Example:Perform the indicated operation:
Solution:
Example:Perform the indicated operation:
Solution:
Example:Perform the indicated operation: .
Solution: