Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)
Part III. Parts of the Whole: Fractions, Decimals, and Percents
To discover a great way to solve percent problems, go to www.dummies.com/extras/basicmathandprealgebra.
In this part…
· Work with basic fractions, improper fractions, and mixed numbers
· Add, subtract, multiply, and divide fractions, decimals, and percents
· Convert the form of a rational number to a fraction, a decimal, or a percent
· Use ratios and proportions
· Solve word problems that involve fractions, decimals, percentages
Chapter 9. Fooling with Fractions
In This Chapter
Looking at basic fractions
Knowing the numerator from the denominator
Understanding proper fractions, improper fractions, and mixed numbers
Increasing and reducing the terms of fractions
Converting between improper fractions and mixed numbers
Using cross-multiplication to compare fractions
Suppose that today is your birthday and your friends are throwing you a surprise party. After opening all your presents, you finish blowing out the candles on your cake, but now you have a problem: Eight of you want some cake, but you have only one cake. Several solutions are proposed:
· You can all go into the kitchen and bake seven more cakes.
· Instead of eating cake, everyone can eat celery sticks.
· Because it's your birthday, you can eat the whole cake and everyone else can eat celery sticks. (That idea was yours.)
· You can cut the cake into eight equal slices so that everyone can enjoy it.
After careful consideration, you choose the last option. With that decision, you've opened the door to the exciting world of fractions. Fractions represent parts of a thing that can be cut into pieces. In this chapter, I give you some basic information about fractions that you need to know, including the three basic types of fractions: proper fractions, improper fractions, and mixed numbers.
I move on to increasing and reducing the terms of fractions, which you need when you begin applying the Big Four operations to fractions in Chapter 10. I also show you how to convert between improper fractions and mixed numbers. Finally, I show you how to compare fractions using cross-multiplication. By the time you're done with this chapter, you'll see how fractions really can be a piece of cake!
Slicing a Cake into Fractions
Here's a simple fact: When you cut a cake into two equal pieces, each piece is half of the cake. As a fraction, you write that as . In Figure 9-1, the shaded piece is half of the cake.
Illustration by Wiley, Composition Services Graphics
Figure 9-1: Two halves of a cake.
Every fraction is made up of two numbers separated by a line, or a fraction bar. The line can be either diagonal or horizontal — so you can write this fraction in either of the following two ways:
The number above the line is called the numerator. The numerator tells you how many pieces you have. In this case, you have one dark-shaded piece of cake, so the numerator is 1.
The number below the line is called the denominator. The denominator tells you how many equal pieces the whole cake has been cut into. In this case, the denominator is 2.
Similarly, when you cut a cake into three equal slices, each piece is a third of the cake (see Figure 9-2).
Illustration by Wiley, Composition Services Graphics
Figure 9-2: Cake cut into thirds.
This time, the shaded piece is one-third — — of the cake. Again, the numerator tells you how many pieces you have, and the denominator tells you how many equal pieces the whole cake has been cut up into.
Figure 9-3 shows a few more examples of ways to represent parts of the whole with fractions.
Illustration by Wiley, Composition Services Graphics
Figure 9-3: Cakes cut and shaded into
In each case, the numerator tells you how many pieces are shaded, and the denominator tells how many pieces there are altogether.
The fraction bar can also mean a division sign. In other words, signifies 3 ÷ 4. If you take three cakes and divide them among four people, each person gets of a cake.
Knowing the Fraction Facts of Life
Fractions have their own special vocabulary and a few important properties that are worth knowing right from the start. When you know them, you find working with fractions a lot easier.
Telling the numerator from the denominator
The top number in a fraction is called the numerator, and the bottom number is called the denominator. For example, look at the following fraction:
In this example, the number 3 is the numerator, and the number 4 is the denominator. Similarly, look at this fraction:
The number 55 is the numerator, and the number 89 is the denominator.
Flipping for reciprocals
When you flip over a fraction, you get its reciprocal. For example, the following numbers are reciprocals:
is its own reciprocal
Using ones and zeros
When the denominator (bottom number) of a fraction is 1, the fraction is equal to the numerator by itself. Conversely, you can turn any whole number into a fraction by drawing a line and placing the number 1 under it. For example,
When the numerator and denominator match, the fraction equals 1. After all, if you cut a cake into eight pieces and you keep all eight of them, you have the entire cake. Here are some fractions that equal 1:
When the numerator of a fraction is 0, the fraction is equal to 0. For example,
The denominator of a fraction can never be 0. Fractions with 0 in the denominator are undefined — that is, they have no mathematical meaning.
Remember from earlier in this chapter that placing a number in the denominator is similar to cutting a cake into that number of pieces. You can cut a cake into two, or ten, or even a million pieces. You can even cut it into one piece (that is, don't cut it at all). But you can't cut a cake into zero pieces. For this reason, putting 0 in the denominator — much like lighting an entire book of matches on fire — is something you should never, never do.
Mixing things up
A mixed number is a combination of a whole number and a proper fraction added together. Here are some examples:
A mixed number is always equal to the whole number plus the fraction attached to it. So means means , and so on.
Knowing proper from improper
When the numerator (top number) is less than the denominator (bottom number), the fraction is less than 1:
Fractions like these are called are called proper fractions. Positive proper fractions are always between 0 and 1. However, when the numerator is greater than the denominator, the fraction is greater than 1. Take a look:
Any fraction that's greater than 1 is called an improper fraction. Converting an improper fraction to a mixed number is customary, especially when it's the final answer to a problem.
An improper fraction is always top heavy, as if it's unstable and wants to fall over. To stabilize it, convert it to a mixed number. Proper fractions are always stable.
Later in this chapter, I discuss improper fractions in more detail when I show you how to convert between improper fractions and mixed numbers.
Increasing and Reducing Terms of Fractions
Take a look at these three fractions:
If you cut three cakes (as I do earlier in this chapter) into these three fractions (see Figure 9-4), exactly half of the cake will be shaded, just like in Figure 9-1, no matter how you slice it. (Get it? No matter how you slice it? You may as well laugh at the bad jokes, too — they're free.) The important point here isn't the humor, or the lack of it, but the idea about fractions.
Illustration by Wiley, Composition Services Graphics
Figure 9-4: Cakes cut and shaded into
The fractions are all equal in value. In fact, you can write a lot of fractions that are also equal to these. As long as the numerator is exactly half the denominator, the fractions are all equal to — for example,
These fractions are equal to , but their terms (the numerator and denominator) are different. In this section, I show you how to both increase and reduce the terms of a fraction without changing its value.
Increasing the terms of fractions
To increase the terms of a fraction by a certain number, multiply both the numerator and the denominator by that number.
For example, to increase the terms of the fraction by 2, multiply both the numerator and the denominator by 2:
Similarly, to increase the terms of the fraction by 7, multiply both the numerator and the denominator by 7:
Increasing the terms of a fraction doesn't change its value. Because you're multiplying the numerator and denominator by the same number, you're essentially multiplying the fraction by a fraction that equals 1.
One key point to know is how to increase the terms of a fraction so that the denominator becomes a preset number. Here's how you do it:
1. Divide the new denominator by the old denominator.
To keep the fractions equal, you have to multiply the numerator and denominator of the old fraction by the same number. This first step tells you what the old denominator was multiplied by to get the new one.
For example, suppose you want to raise the terms of the fraction so that the denominator is 35. You're trying to fill in the question mark here:
Divide 35 by 7, which tells you that the denominator was multiplied by 5.
2. Multiply this result by the old numerator to get the new numerator.
You now know how the two denominators are related. The numerators need to have the same relationship, so multiply the old numerator by the number you found in Step 1.
Multiply 5 by 4, which gives you 20. So here's the answer:
Reducing fractions to lowest terms
Reducing fractions is similar to increasing fractions, except that it involves division rather than multiplication. But because you can't always divide, reducing takes a bit more finesse.
In practice, reducing fractions is similar to factoring numbers. For this reason, if you're not up on factoring, you may want to review this topic in Chapter 8.
In this section, I show you the formal way to reduce fractions, which works in all cases. Then I show you a more informal way you can use when you're more comfortable.
Reducing fractions the formal way
Reducing fractions the formal way relies on understanding how to break down a number into its prime factors. I discuss this in detail in Chapter 8, so if you're shaky on this concept, you may want to review it first.
Here's how to reduce a fraction:
1. Break down both the numerator (top number) and the denominator (bottom number) into their prime factors.
For example, suppose you want to reduce the fraction . Break down both 12 and 30 into their prime factors:
2. Cross out any common factors.
As you can see, I cross out a 2 and a 3 because they're common factors — that is, they appear in both the numerator and the denominator:
3. Multiply the remaining numbers to get the reduced numerator and denominator.
You can see now that the fraction reduces to :
As another example, here's how you reduce the fraction :
This time, cross out two 2s from both the top and the bottom as common factors. The remaining 2s on top and the 5s on the bottom aren't common factors. So the fraction reduces to .
Reducing fractions the informal way
Here's an easier way to reduce fractions when you get comfortable with the concept:
1. If the numerator (top number) and denominator (bottom number) are both divisible by 2 — that is, if they're both even — divide both by 2.
For example, suppose you want to reduce the fraction . The numerator and the denominator are both even, so divide them both by 2:
2. Repeat Step 1 until the numerator or denominator (or both) is no longer divisible by 2.
In the resulting fraction, both numbers are still even, so repeat the first step again:
3. Repeat Step 1 using the number 3, and then 5, and then 7, continuing testing prime numbers until you're sure that the numerator and denominator have no common factors.
Now, the numerator and the denominator are both divisible by 3 (see Chapter 7 for easy ways to tell if one number is divisible by another), so divide both by 3:
Neither the numerator nor the denominator is divisible by 3, so this step is complete. At this point, you can move on to test for divisibility by 5, 7, and so on, but you really don't need to. The numerator is 3, and it obviously isn't divisible by any larger number, so you know that the fraction reduces to .
Converting between Improper Fractions and Mixed Numbers
In “Knowing the Fraction Facts of Life,” I tell you that any fraction whose numerator is greater than its denominator is an improper fraction. Improper fractions are useful and easy to work with, but for some reason, people just don't like them. (The word improper should've tipped you off.) Teachers especially don't like them, and they really don't like an improper fraction to appear as the answer to a problem. However, they love mixed numbers. One reason they love them is that estimating the approximate size of a mixed number is easy.
For example, if I tell you to put of a gallon of gasoline in my car, you probably find it hard to estimate roughly how much that is: 5 gallons, 10 gallons, 20 gallons?
But if I tell you to get gallons of gasoline, you know immediately that this amount is a little more than 10 but less than 11 gallons. Although is the same as , knowing the mixed number is a lot more helpful in practice. For this reason, you often have to convert improper fractions to mixed numbers.
Knowing the parts of a mixed number
Every mixed number has both a whole number part and a fractional part. So the three numbers in a mixed number are
· The whole number
· The numerator
· The denominator
For example, in the mixed number , the whole number part is 3 and the fractional part is . So this mixed number is made up of three numbers: the whole number (3), the numerator (1), and the denominator (2). Knowing these three parts of a mixed number is helpful for converting back and forth between mixed numbers and improper fractions.
Converting a mixed number to an improper fraction
To convert a mixed number to an improper fraction, follow these steps:
1. Multiply the denominator of the fractional part by the whole number, and add the result to the numerator.
For example, suppose you want to convert the mixed number to an improper fraction. First, multiply 3 by 5 and add 2:
2. Use this result as your numerator, and place it over the denominator you already have.
Place this result over the denominator:
So the mixed number equals the improper fraction . This method works for all mixed numbers. Furthermore, if you start with the fractional part reduced, the answer is also reduced (see the earlier “Increasing and Reducing Terms of Fractions” section).
Converting an improper fraction to a mixed number
To convert an improper fraction to a mixed number, divide the numerator by the denominator (see Chapter 3). Then write the mixed number in this way:
· The quotient (answer) is the whole-number part.
· The remainder is the numerator.
· The denominator of the improper fraction is the denominator.
For example, suppose you want to write the improper fraction as a mixed number. First, divide 19 by 5:
Then write the mixed number as follows:
This method works for all improper fractions. And as is true of conversions in the other direction, if you start with a reduced fraction, you don't have to reduce your answer (see “Increasing and Reducing Terms of Fractions”).
Cross-multiplication is a handy little technique to know. You can use it in a few different ways, so I explain it here and then show you an immediate application.
To cross-multiply two fractions, follow these steps:
1. Multiply the numerator of the first fraction by the denominator of the second fraction and jot down the answer.
2. Multiply the numerator of the second fraction by the denominator of the first fraction and jot down the answer.
For example, suppose you have these two fractions:
When you cross-multiply, you get these two numbers:
You can use cross-multiplication to compare fractions and find out which is greater. When you do so, make sure that you start with the numerator of the first fraction.
To find out which of two fractions is larger, cross-multiply and place the two numbers you get, in order, under the two fractions. The larger number is always under the larger fraction. In this case, 14 goes under and 12 goes under . The number 14 is greater than 12, so is greater than .
For example, suppose you want to find out which of the following three fractions is the greatest:
Cross-multiplication works only with two fractions at a time, so pick the first two — and — and then cross-multiply:
Because 27 is greater than 25, you know now that is greater than . So you can throw out .
Now do the same thing for and :
Because 33 is greater than 30, is greater than . Pretty straightforward, right? And that set of steps is all you have to know for now. I show you a bunch of great things you can do with this simple skill in the next chapter.
Making Sense of Ratios and Proportions
A ratio is a mathematical comparison of two numbers, based on division. For example, suppose you bring 2 scarves and 3 caps with you on a ski vacation. Here are a few ways to express the ratio of scarves to caps:
The simplest way to work with a ratio is to turn it into a fraction. Be sure to keep the order the same: The first number goes on top of the fraction, and the second number goes on the bottom.
In practice, a ratio is most useful when used to set up a proportion — that is, an equation involving two ratios. Typically, a proportion looks like a word equation, as follows:
For example, suppose you know that both you and your friend Andrew brought the same proportion of scarves to caps. If you also know that Andrew brought 8 scarves, you can use this proportion to find out how many caps he brought. Just increase the terms of the fraction so that the numerator becomes 8. I do this in two steps:
As you can see, the ratio 8:12 is equivalent to the ratio 2:3 because the fractions are equal. Therefore, Andrew brought 12 caps.