Many of the fractions in this chapter appear as stacked fractions. A stacked fraction looks like this:
It is written straight up and down.
In your everyday activities, you often see frac- tions written unstacked or inline. They look like this:
1/7
The inline representation of a fraction might be convenient for casual writing and some reports, but it’s not the best way to represent fractions in math operations.
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✓ 0/8 represents a cake with 8 pieces, and you have none of them. A frac- tion with 0 in the numerator is a legitimate fraction.
In Figure 8-2, the shaded portion is 0/8 of the cake. No shading! That’s nothing. You have zero, nada, nil, cipher, goose egg, bupkis.
✓ 1⁄2/8 represents a cake with 8 pieces, and you have 1⁄2 of one piece.
In Figure 8-3, the shaded portion is 1⁄2/8 of the cake. This must be what dieting looks like.
✓ 8/8 represents a cake with 8 pieces, and you have all 8 of them.
In Figure 8-4, the shaded portion is 8/8 of the cake. That’s the whole cake, and you’re very lucky — or very greedy.
✓ 1/1 represents a cake with 1 piece, and you have the only piece. As with 8/8, you have the whole cake.
Figure 8-2:
Cake with
0⁄8 shaded.
Figure 8-3:
Cake with
1⁄2/8 shaded.
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Chapter 8: Fun with Fractions
In Figure 8-5, the shaded portion is 1⁄1 of the cake. Now you can tell people, “I only took one piece.”
✓ 15/8 is a little trickier. It represents a cake with 8 pieces per cake and you have 15 pieces. That means the 15 pieces come from more than one cake. At first, this may not make a lot of sense, unless you’re a caterer with 15 guests to feed.
In Figure 8-6, the shaded portion is 15/8 of the cake. If no one asks for seconds, this setup will serve all 15 guests, with one piece left over.
Figure 8-4:
Cake with 8⁄8
shaded.
Figure 8-5:
Cake with
1⁄1 shaded.
Figure 8-6:
Cake with
15⁄8 shaded.
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Defining denominators
The denominator can be any number, even a fraction, but with one big excep- tion. The denominator can never be 0 (zero). You can have one-fifth of some- thing or one-third of something, but you can’t have one-zeroeth of anything.
The top number can be 0, but the bottom number can’t because it makes no sense, mathematically. Look at these examples of denominators:
✓ In the fraction 7/8, 8 is the denominator.
✓ Lab techs may use the fraction 50/1,000 to indicate part of a liter. It rep- resents 50 milliliters.
✓ Machinists can use 6/1,000 as another way of expressing 6 mils (since a mil is 1/1,000 of an inch).
✓ The following example shows a denominator of 1⁄2. Fractions in the denominator (as well as the numerator) are allowed, but usually you convert the whole fraction to another fraction that’s easier to work with.
Numerators can be anything — but unlike numerators, denominators can never be 0.
Dealing with special cases
A denominator of 0 can be called a special case, because it messes up the math and isn’t possible. You come across several other special cases in frac- tion math, though, that aren’t toxic.
Three cases — 1 as the denominator, 0 as the numerator, and the same number as both numerator and denominator — are special, because you use them all the time to solve math with fractions in the fastest and easiest way. The fourth case — fractions representing cents in a dollar — is special because transactions involving fractions of dollars are common in business and personal finance.
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Chapter 8: Fun with Fractions
✓ When 1 is the denominator: Any fraction where 1 is the bottom number (for example, 34⁄1) is equal to the top number. The 1 is known as the invis- ible denominator. A math rule: Any number divided by 1 is equal to that number.
✓ When 0 is the numerator: Any fraction with 0 as the top number (for example, 0⁄17) is equal to zero. If you have 0 parts of something made of 17 parts, you have 0 parts. A math rule: When any number is divided into 0, the result is 0.
✓ When the numerator and the denominator are the same, as in 1/1, 3/3, 12/12, and so on: A math rule: When any number is divided into itself, the result is 1. (These fractions are called equivalent fractions, and you find out more about them later in this chapter.)
✓ Dollars and cents: You know that 100 cents make up one U.S. dollar (and about sixty other countries in the world also divide their currency into cents). You also know that a cent (a penny) is 1/100 of a dollar. If you have 37 cents in your pocket, you have 37/100 of a dollar. Currency can be expressed as a fraction where the denominator (100) never changes.
Tackling the Different Types of Fractions
Knowing the names of the different types of fractions is a big help to you in this chapter and when doing math in the real world. You even use one of them in everyday life. In the following sections, we discuss three concepts in detail:
✓ Proper and improper fractions: Fractions have two important and obvi- ous forms: A proper fraction’s denominator is larger than the numerator, and an improper fraction’s denominator is smaller than the numerator.
(“Honey, does this denominator make my fraction look big?”) ✓ Mixed numbers: Whole numbers are from Mars. Fractions are from
Venus. A mixed number is what you get when you combine them.
✓ Ratios: A ratio is a fraction, sort of, much like your Uncle Willie is a relative until he misbehaves at Thanksgiving dinner. A ratio shows two numerators that add up to be a denominator, which may make them deceiving.
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Does the preceding list cover every type of fraction? Nope! As proof that mathe- maticians don’t get out much, you may (or may not) want to know that there are also unit fractions, dyadic fractions, Egyptian fractions, continued fractions, and partial fractions. Some of these are special cases (unit and dyadic), some are archaic (Egyptian), and some are used only in higher mathematics (continued and partial).
Proper and improper fractions
Proper and improper have nothing to do with good or bad behavior. When a fraction has a numerator smaller than its denominator (smaller number on top), it’s a proper fraction.
When a fraction has a denominator smaller than its numerator (larger number on top), it’s an improper fraction.
So what’s so important about this? When you encounter fractions in your work, they will often be part of a mixed number (see the following section).
You usually convert the mixed number into an improper fraction so that your calculations are faster and easier to do. Then after doing the math, you con- vert the result back into a mixed number. This occurs a lot with carpenters, electricians, and workers laying flooring or carpeting.
Mixed numbers
When a whole number and a fraction appear together, such as in
it’s called a mixed number.
The number represents 2 whole units and 7/8 of another unit.
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Chapter 8: Fun with Fractions
In various occupations, you find mixed numbers used a lot.
✓ A carpenter may cut a piece of plywood to be 243⁄4 inches x 48 inches.
✓ A carpetlayer may trim a roll of carpet to fit in a room that is 1111⁄2
inches wide.
✓ An electrician may cut EMT (conduit) to a length of 275⁄8 inches.
A mixed fraction example
You have a piece of electrical metal tubing that is 8 feet, 9 inches long. How long is that in inches?
The length 8 feet, 9 inches may not appear to be a mixed fraction, but it is.
The 8 feet is the number of whole feet; the 9 inches is a fraction of a foot. There are 12 inches in a foot, so 9 inches is 9/12 of a foot.
To do the conversion, you begin with the following expression:
8 feet 9 inches
The whole unit of feet isn’t useful; you need to convert the units as described in Chapter 6. There are 12 inches in a foot. You can convert the whole number (8 feet) to inches with this simple expression:
The result is 96 inches. Add that to 9 inches from the problem.
96 inches + 9 inches = 105 inches The answer is 105 inches.
Another mixed fraction example
In a medical office, a person’s height is often measured and recorded in inches (for example, 69 inches). Express this height as a mixed number (containing feet and inches).
An inch is 1/12 of a foot, so 69 inches is 69/12 of a foot.
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Break the fraction into two fractions. Notice that 60⁄12 of a foot easily converts to 5 feet. The 9/12 of a foot is 9 inches.
Yet another mixed fraction example
You work in industrial or assisted living food service. You have 15 cups of pecans. You need cup of pecans to make one dozen muffins. How many dozens can you make?
Start with the mixed fraction 151⁄2.
Start by turning the whole number into a fraction. How many 1/2 cups are in 15 cups? Multiply by 1, but in the form of an equivalent fraction (2/2).
Fifteen cups has 30 half-cups in it. Now add the converted whole number to the fraction (1/2).
You have 31 half-cups. Because you need one half-cup to make a dozen muffins, you can bake 31 dozen muffins.
Ratios
Fractions are related to ratios. A ratio is a way to compare two quantities relative to each other. A ratio is expressed as two numbers separated by a colon (:). For example, 3:4.
In a vinaigrette dressing, the classic ratio of oil to vinegar is three parts oil to one part vinegar, written as 3:1. You typically describe this as “a 3-to-1 mix- ture.” This ratio of ingredients is true whether you’re mixing ounces of ingre- dients at home or gallons in a large cafeteria.
A ratio is not as straightforward as a fraction. It describes the parts, but not the whole, so don’t automatically trust ratios without understanding what they really represent.
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Chapter 8: Fun with Fractions
For example, in small two-cycle engines, you want to mix fuel with oil for lubrication. Common mix ratios are 12:1, 16:1, 24:1, and 32:1. The ratio 12:1 tells you to mix 12 parts of gasoline with 1 part of oil. Here, the result is a total of 13 parts: 12 of gas and 1 of oil. So the final mix is 12/13 oil and 1/13 gas. Notice that these fractions are similar, but not identical, to the ratio. By using the ratio alone, you end up with the mixture you need, but knowing the exact fraction can also be valuable.
The same is true in professional mixology (bartending), where cocktails are expressed as ratios as well as whole and fractional quantities. The classic dry martini uses 21⁄2 ounces gin and 1/2 ounce dry vermouth. This recipe is a ratio of 21⁄2:1/2. Because fractions in ratios are a little clumsy, double both sides of the ratio and you get a 5:1 ratio of gin to vermouth.
You can turn ratios into fractions. Consider the cyberdating success ratio developed in 2005 by the University of Bath. For every 100 cyberdates (dates with people who met online), 94 went on to date each other again and 6 did not. This figure is a 94:6 ratio. You can express the proportion of people dating again as 94/100 and those not dating again as 6/100.
Performing Math Operations with Fractions
Adding, subtracting, multiplying, and dividing regular numbers isn’t hard.
Why should fractions be any harder? They aren’t. Any difficulties are usually with the bottom number, and that’s easy to fix.
Mathematicians use some complicated math processes for fractions (for example, the rationalization of monomial and binomial square roots in the denominators of fractions), but you aren’t likely to need them in your work.