Basic Math and Pre-Algebra For Dummies (33 page)

 To find the GCF of a set of numbers, list all the factors of each number, as I show you in “Generating a number's factors.” The greatest factor appearing on every list is the GCF. For example, to find the GCF of 6 and 15, first list all the factors of each number.

• Factors of 6: 1, 2, 3, 6
  
• Factors of 15: 1, 3, 5, 15
  

Because 3 is the greatest factor that appears on both lists, 3 is the GCF of 6 and 15.

As another example, suppose you want to find the GCF of 9, 20, and 25. Start by listing the factors of each:

• Factors of 9: 1, 3, 9
  
• Factors of 20: 1, 2, 4, 5, 10, 20
  
• Factors of 25: 1, 5, 25
  

In this case, the only factor that appears on all three lists is 1, so 1 is the GCF of 9, 20, and 25.

Even though multiples tend to be larger numbers than factors, most students find them easier to work with. Read on.

The preceding section, “Finding Fabulous Factors,” tells you how to find all the factors of a number. Finding all the factors is possible because factors of a number are always less than or equal to the number itself. So no matter how large a number is, it always has a finite (limited) number of factors.

Unlike factors, multiples of a number are greater than or equal to the number itself. (The only exception to this is 0, which is a multiple of every number.) Because of this, the multiples of a number go on forever — that is, they're infinite. Nevertheless, generating a partial list of multiples for any number is simple.

 To list multiples of any number, write down that number and then multiply it by 2, 3, 4, and so forth.

For example, here are the first few positive multiples of 7:

• 7
   14 21 28 35 42
  

As you can see, this list of multiples is simply part of the multiplication table for the number 7. (For the multiplication table up to 9 × 9, see Chapter
3
.)

 The least common multiple (LCM) of a set of numbers is the lowest positive number that's a multiple of every number in that set.

For example, the LCM of the numbers 2, 3, and 5 is 30 because

• 30 is a multiple of 2 (2 × 15 = 30).
  
• 30 is a multiple of 3 (3 × 10 = 30).
  
• 30 is a multiple of 5 (5 × 6 = 30).
  
• No number lower than 30 is a multiple of all three numbers.
  

 To find the LCM of a set of numbers, take each number in the set and jot down a list of the first several multiples in order. The LCM is the first number that appears on every list.

 When looking for the LCM of two numbers, start by listing multiples of the larger number, but stop this list when the number of multiples you've written equals the smaller number. Then start listing multiples of the lower number until one of them matches the first list.

For example, suppose you want to find the LCM of 4 and 6. Begin by listing multiples of the higher number, which is 6. In this case, list only four of these multiples because the lower number is 4.

• Multiples of 6:

  6, 12, 18, 24, …

Now start listing multiples of 4:

• Multiples of 4:

  4, 8, 12, …

Because 12 is the first number to appear on both lists of multiples, 12 is the LCM of 4 and 6.

This method works especially well when you want to find the LCM of two numbers, but it may take longer if you have more numbers.

Suppose, for instance, that you want to find the LCM of 2, 3, and 5. Start with the two largest numbers — in this case, 5 and 3 — and keep listing numbers until you have one or more matching numbers:

• Multiples of 5:	5, 10, 15, 20, 25, 30, …
  Multiples of 3:	3, 6, 9, 12, 15, 18, 21, 24, 27, 30, …

The only numbers repeated on both lists are 15 and 30. In this case, you can save yourself the trouble of making the last list because 30 is obviously a multiple of 2, and 15 isn't. So 30 is the LCM of 2, 3, and 5.

Part III

 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

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!

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.

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
).

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.