Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)
Part IV. Picturing and Measuring - Graphs, Measures, Stats, and Sets
Chapter 15. How Much Have You Got? Weights and Measures
In This Chapter
Using units for non-discrete measurement
Discovering differences between the English and metric systems
Estimating and calculating English and metric system conversions
In Chapter 4, I introduce you to units, which are items that can be counted, such as apples, coins, or hats. Apples, coins, and hats are easy to count because they're discrete — that is, you can easily see where one ends and the next one begins. But not everything is so easy. For example, how do you count water — by the drop? Even if you tried, exactly how big is a drop?
Units of measurement come in handy at this point. A unit of measurement allows you to count something that isn't discrete: an amount of a liquid or solid, the distance from one place to another, a length of time, the speed at which you're traveling, or the temperature of the air.
In this chapter, I discuss two important systems of measurement: English and metric. You're probably familiar with the English system already, and you may know more than you think about the metric system. Each of these measurement systems provides a different way to measure distance, volume, weight (or mass), time, and speed. Next, I show you how to estimate metric amounts in English units. Finally, I show how to convert from English units to metric and vice versa.
Examining Differences between the English and Metric Systems
The two most common measurement systems today are the English system and the metric system.
Most Americans learn the units of the English system — for example, pounds and ounces, feet and inches, and so forth — and use them every day. Unfortunately, the English system is awkward for use with math. English units such as inches and fluid ounces are often measured in fractions, which (as you may know from Chapters 9 and 10) can be difficult to work with.
The metric system was invented to simplify the application of math to measurement. Metric units are based on the number 10, which makes them much easier to work with. Parts of units are expressed as decimals, which (as Chapter 11 shows you) are much friendlier than fractions.
Yet despite these advantages, the metric system has been slow to catch on in the U.S. Many Americans feel comfortable with English units and are reluctant to part with them. For example, if I ask you to carry a 20-lb. bag for one-fourth of a mile, you know what to expect. However, if I ask you to carry a bag weighing 10 kilograms half a kilometer, you may not be sure.
In this section, I show you the basic units of measurement for both the English and metric systems.
If you want an example of the importance of converting carefully, you may want to look to NASA — they kind of lost a Mars orbiter in the late 1990s because an engineering team used English units and NASA used metric to navigate!
Looking at the English system
The English system of measurement is most commonly used in the United States (but, ironically, not in England). Although you're probably familiar with most of the English units of measurement, in the following list, I make sure you know the most important ones. I also show you some equivalent values that can help you do conversions from one type of unit to another.
· Units of distance: Distance — also called length — is measured in inches (in.), feet (ft.), yards (yd.), and miles (mi.):
· Units of fluid volume: Fluid volume (also called capacity) is the amount of space occupied by a liquid, such as water, milk, or wine. I discuss volume when I talk about geometry in Chapter 16. Volume is measured in fluid ounces (fl. oz.), cups (c.), pints (pt.), quarts (qt.), and gallons (gal.):
Units of fluid volume are typically used for measuring the volume of things that can be poured. The volume of solid objects is more commonly measured in cubic units of distance, such as cubic inches and cubic feet.
· Units of weight: Weight is the measurement of how strongly gravity pulls an object toward Earth. Weight is measured in ounces (oz.), pounds (lb.), and tons.
Don't confuse fluid ounces, which measure volume, with ounces, which measure weight. These units are two completely different types of measurements!
· Units of time: Time is hard to define, but everybody knows what it is. Time is measured in seconds, minutes, hours, days, weeks, and years:
The conversion from days to years is approximate because Earth's daily rotation on its axis and its yearly revolution around the sun aren't exactly synchronized. A year is closer to 365.25 days, which is why leap-years exist.
I left months out of the picture because the definition of a month is imprecise — it can vary from 28 to 31 days.
· Unit of speed: Speed is the measurement of how much time an object takes to move a given distance. The most common unit of speed is miles per hour (mph).
· Unit of temperature: Temperature measures how much heat an object contains. This object can be a glass of water, a turkey in the oven, or the air surrounding your house. Temperature is measured in degrees Fahrenheit (°F).
Looking at the metric system
Like the English system, the metric system provides units of measurement for distance, volume, and so on. Unlike the English system, however, the metric system builds these units using a basic unit and a set of prefixes.
Table 15-1 shows five important basic units in the metric system.
Table 15-1 Five Basic Metric Units
Measure Of |
Basic Metric Unit |
Distance |
Meter |
Volume (capacity) |
Liter |
Mass (weight) |
Gram |
Time |
Second |
Temperature |
Degrees Celsius (°C) |
For scientific purposes, the metric system has been updated to the more rigorously defined System of International Units (SI). Each basic SI unit correlates directly to a measurable scientific process that defines it. In SI, the kilogram (not the gram) is the basic unit of mass, the kelvin is the basic unit of temperature, and the liter is not considered a basic unit. For technical reasons, scientists tend to use the more rigidly defined SI, but most other people use the looser metric system. In everyday practice, you can think of the units in Table 15-1 as basic units.
Table 15-2 shows ten metric prefixes, with the three most commonly used in bold and italicized (see Chapter 14 for more information on powers of ten).
Large and small metric units are formed by linking a basic unit with a prefix. For example, linking the prefix kilo- to the basic unit meter gives you the kilometer, which means 1,000 meters. Similarly, linking the prefix milli- to the basic unit liter gives you the milliliter, which means 0.001 (one thousandth) of a meter.
Here's a list giving you the basics:
· Units of distance: The basic metric unit of distance is the meter (m). Other common units are millimeters (mm), centimeters (cm), and kilometers (km):
· Units of fluid volume: The basic metric unit of fluid volume (also called capacity) is the liter (L). Another common unit is the milliliter (mL):
Note: One milliliter is equal to 1 cubic centimeter (cc).
· Units of mass: Technically, the metric system measures not weight, but mass. Weight is the measurement of how strongly gravity pulls an object toward Earth. Mass, however, is the measurement of the amount of matter an object has. If you traveled to the moon, your weight would change, so you would feel lighter. But your mass would remain the same, so all of you would still be there. Unless you're planning a trip into outer space or performing a scientific experiment, you probably don't need to know the difference between weight and mass. In this chapter, you can think of them as equivalent, and I use the word weight when referring to metric mass.
The basic unit of weight in the metric system is the gram (g). Even more commonly used, however, is the kilogram (kg):
Note: 1 kilogram of water has a volume of 1 liter.
· Units of time: As in the English system, the basic metric unit of time is a second (s). For most purposes, people also use other English units, such as minutes and hours.
For many scientific purposes, the second is the only unit used to measure time. Large numbers of seconds and small fractions of sections are represented with scientific notation, which I cover in Chapter 14.
· Units of speed: For most purposes, the most common metric unit of speed (also called velocity) is kilometers per hour (km/hr). Another common unit is meters per second (m/s).
· Units of temperature (degrees Celsius or Centigrade): The basic metric unit of temperature is the Celsius degree (°C), also called the Centigrade degree. The Celsius scale is set up so that, at sea level, water freezes at 0°C and boils at 100°C.
Scientists often use another unit — the kelvin (K) — to talk about temperature. The degrees are the same size as in Celsius, but 0 K is set at absolute zero, the temperature at which atoms don't move at all. Absolute zero is approximately equal to –273.15°C.
Estimating and Converting between the English and Metric Systems
Most Americans use the English system of measurement all the time and have only a passing acquaintance with the metric system. But metric units are being used more commonly as the units for tools, footraces, soft drinks, and many other things. Also, if you travel abroad, you need to know how far 100 kilometers is or how long you can drive on 10 liters of gasoline.
In this section, I show you how to make ballpark estimates of metric units in terms of English units, which can help you feel more comfortable with metric units. I also show you how to convert between English and metric units, which is a common type of math problem.
When I talk about estimating, I mean very loose ways of measuring metric amounts using the English units you are familiar with. In contrast, when I talk about converting, I mean using an equation to change from one system of units to the other. Neither method is exact, but converting provides a much closer approximation (and takes longer) than estimating.
Estimating in the metric system
One reason people sometimes feel uncomfortable using the metric system is that, when you're not familiar with it, estimating amounts in practical terms is hard. For example, if I tell you that we're going out to a beach that's mile away, you prepare yourself for a short walk. And if I tell you that it's 10 miles away, you head for the car. But what do you do with the information that the beach is 3 kilometers away?
Similarly, if I tell you that the temperature is 85°F, you'll probably wear a bathing suit or shorts. And if I tell you it's 40°F, you'll probably wear a coat. But what do you wear if I tell you that the temperature is 25°C?
In this section, I give you a few rules of thumb to estimate metric amounts. In each case, I show you how a common metric unit compares with an English unit that you already feel comfortable with.
Approximating short distances: 1 meter is about 1 yard (3 feet)
Here's how to convert meters to feet: . But for estimating, use the simple rule that 1 meter is about 1 yard (that is, about 3 feet).
By this estimate, a 6-foot man stands about 2 meters tall. A 15-foot room is 5 meters wide. And a football field that's 100 yards long is about 100 meters long. Similarly, a river with a depth of 4 meters is about 12 feet deep. A mountain that's 3,000 meters tall is about 9,000 feet. And a child who is only half a meter tall is about a foot and a half.
Estimating longer distances and speed
Here's how to convert kilometers to miles: . For a ballpark estimate, you can remember that 1 kilometer is about a mile. By the same token, 1 kilometer per hour is about mile per hour.
This guideline tells you that if you live 2 miles from the nearest supermarket, then you live about 4 kilometers from there. A marathon of 26 miles is about 52 kilometers. And if you run on a treadmill at 6 miles per hour, then you can run at about 12 kilometers per hour. By the same token, a 10-kilometer race is about 5 miles. If the Tour de France is about 4,000 kilometers, then it's about 2,000 miles. And if light travels about 300,000 kilometers per second, then it travels about 150,000 miles per second.
Approximating volume: 1 liter is about 1 quart (1/4 gallon)
Here's how to convert liters to gallons: . A good estimate here is that 1 liter is about 1 quart (a gallon consists of about 4 liters).
Using this estimate, a gallon of milk is 4 quarts, so it's about 4 liters. If you put 10 gallons of gasoline in your tank, it's about 40 liters. In the other direction, if you buy a 2-liter bottle of cola, you have about 2 quarts. If you buy an aquarium with a 100-liter capacity, it holds about 25 gallons of water. And if a pool holds 8,000 liters of water, it holds about 2,000 gallons.
Estimating weight: 1 kilogram is about 2 pounds
Here's how to convert kilograms to pounds: . For estimating, figure that 1 kilogram is equal to about 2 pounds.
By this estimate, a 5-kilogram bag of potatoes weighs about 10 pounds. If you can bench-press 70 kilograms, then you can bench-press about 140 pounds. And because a liter of water weighs exactly 1 kilogram, you know that a quart of water weighs about 2 pounds. Similarly, if a baby weighs 8 pounds at birth, he or she weighs about 4 kilograms. If you weigh 150 pounds, then you weigh about 75 kilograms. And if your New Year's resolution is to lose 20 pounds, then you want to lose about 10 kilograms.
Estimating temperature
The most common reason for estimating temperature in Celsius is in connection with the weather. The formula for converting from Celsius to Fahrenheit is kind of messy:
Instead, use the handy chart in Table 15-3.
Table 15-3 Comparing Celsius and Fahrenheit Temperatures
Celsius (Centigrade) |
Description |
Fahrenheit |
0° |
Cold |
32° |
10° |
Cool |
50° |
20° |
Warm |
68° |
30° |
Hot |
86° |
Any temperature below 0°C is cold, and any temperature over 30°C is hot. Most of the time, the temperature falls in this middling range. So now you know that when the temperature is 6°C, you want to wear a coat. When it's 14°C, you may want a sweater — or at least long sleeves. And when it's 25°C, head for the beach!
Converting units of measurement
Many books give you one formula for converting from English to metric and another for converting from metric to English. People often find this conversion method confusing because they have trouble remembering which formula to use in which direction.
In this section, I show you a simple way to convert between English and metric units that uses only one formula for each type of conversion.
Here’s a nice pair that’s easy to remember: 16°C is about 61°F.
Understanding conversion factors
When you multiply any number by 1, that number stays the same. For example, 36 × 1 = 36. And when a fraction has the same numerator (top number) and denominator (bottom number), that fraction equals 1 (see Chapter 10for details). So when you multiply a number by a fraction that equals 1, the number stays the same. For example:
If you multiply a measurement by a special fraction that equals 1, you can switch from one unit of measurement to another without changing the value. People call such fractions conversion factors.
Take a look at some equations that show how metric and English units are related (all conversions between English and metric units are approximate):
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·
·
·
Because the values on each side of the equations are equal, you can create
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·
When you understand how units of measurement cancel (which I discuss in the next section), you can easily choose which fractions to use to switch between units of measurement.
Canceling units of measurement
When you're multiplying fractions, you can cancel any factor that appears in both the numerator and the denominator (see Chapter 9 for details). Just as with numbers, you can also cancel out units of measurement in fractions. For example, suppose you want to evaluate this fraction:
You already know that you can cancel out a factor of 2 in both the numerator and the denominator. But you can also cancel out the unit gallons in both the numerator and the denominator:
So this fraction simplifies as follows:
· = 3
Converting units
When you understand how to cancel out units in fractions and how to set up fractions equal to 1 (see the preceding sections), you have a foolproof system for converting units of measurement.
Suppose you want to convert 7 meters into feet. Using the equation 1 meter = 3.26 feet, you can make a fraction out of the two values, as follows:
Both fractions equal 1 because the numerator and the denominator are equal. So you can multiply the quantity you're trying to convert (7 meters) by one of these fractions without changing it. Remember that you want the meters unit to cancel out. You already have the word meters in the numerator (to make this clear, place 1 in the denominator), so use the fraction that puts 1 meter in the denominator:
Now cancel out the unit that appears in both the numerator and the denominator:
At this point, the only value in the denominator is 1, so you can ignore it. And the only unit left is feet, so place it at the end of the expression:
Now do the multiplication (Chapter 11 shows how to multiply decimals):
It may seem strange that the answer appears with the units already attached, but that's the beauty of this method: When you set up the right expression, the answer just appears.
You can get more practice converting units of measurement in Chapter 18, where I show you how to set up conversion chains and tackle word problems involving measurement.