Basic Math and Pre-Algebra For Dummies (40 page)

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.

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.

Chapter 10

In This Chapter

Looking at multiplication and division of fractions

Adding and subtracting fractions in a bunch of different ways

Applying the four operations to mixed numbers

In this chapter, the focus is on applying the Big Four operations to fractions. I start by showing you how to multiply and divide fractions, which isn't much more difficult than multiplying whole numbers. Surprisingly, adding and subtracting fractions is a bit trickier. I show you a variety of methods, each with its own strengths and weaknesses, and I recommend how to choose which method will work best, depending on the problem you have to solve.

Later in the chapter, I move on to mixed numbers. Again, multiplication and division won't likely pose too much of a problem because the process in each case is almost the same as multiplying and dividing fractions. I save adding and subtracting mixed numbers for the very end. By then, you'll be much more comfortable with fractions and ready to tackle the challenge.

One of the odd little ironies of life is that multiplying and dividing fractions is easier than adding or subtracting them — just two easy steps and you're done! For this reason, I show you how to multiply and divide fractions before I show you how to add or subtract them. In fact, you may find multiplying fractions easier than multiplying whole numbers because the numbers
you're working with are usually small. More good news is that dividing fractions is nearly as easy as multiplying them. So I'm not even wishing you good luck — you don't need it!

Everything in life should be as simple as multiplying fractions. All you need for multiplying fractions is a pen or pencil, something to write on (preferably not your hand), and a basic knowledge of the multiplication table. (See Chapter
3
for a multiplication refresher.)

 Here's how to multiply two fractions:

1. Multiply the numerators (the numbers on top) to get the numerator of the answer.
   
2. Multiply the denominators (the numbers on the bottom) to get the denominator of the answer.
   

For example, here's how to multiply
 :

Sometimes when you multiply fractions, you have an opportunity to reduce to lowest terms. (For more on when and how to reduce a fraction, see Chapter
9
.) As a rule, math people are crazy about reduced fractions, and teachers sometimes take points off a right answer if you could've reduced it but didn't. Here's a multiplication problem that ends up with an answer that's not in its lowest terms:

Because the numerator and the denominator are both even numbers, this fraction can be reduced. Start by dividing both numbers by 2:

Again, the numerator and the denominator are both even, so do it again:

This fraction is now fully reduced.

 When multiplying fractions, you can often make your job easier by canceling out equal factors in the numerator and denominator. Canceling out equal factors makes the numbers that you're multiplying smaller and easier to work with, and it also saves you the trouble of reducing at the end. Here's how it works:

• When the numerator of one fraction and the denominator of the other are the same, change both of these numbers to 1s. (See the nearby sidebar for why this works.)
  
• When the numerator of one fraction and the denominator of the other are divisible by the same number, factor this number out of both. In other words, divide the numerator and denominator by that common factor. (For more on how to find factors, see Chapter
  8
  .)
  

For example, suppose you want to multiply the following two numbers: