Awesome Operations: Math Fundamentals

In This Chapter

* Reviewing the four arithmetic operations

* Manipulating fractions

* Using charts to convey and understand information

* Strategies to help you solve word problems

Math has basic operations that you need to know. These operations — addition, subtraction, multiplication, and division — make all the other math in this book possible.

The good news is that you most likely learned about basics (like counting) even before you entered school, and you learned about basic arithmetic operations in elementary school. So you've been at it for a long time.

In this chapter, I review counting and the fundamentals of the four basic arithmetic operations. Other important topics I cover here are fractions, percentages, charts and graphs, and word problems. But don't worry: None of these are mysterious.

Numbers You Can Count On

The most fundamental component of math is numbers. The first thing you do with numbers is count, and you probably started counting when you were very young. As soon as you could talk, your mother cajoled you to tell Aunt Lucy how old you were or to count from 1 to 5.

Counting was the first and most useful thing you did with math, and you still use it every day, whether you're buying oranges at the grocery store or checking the number of quarts of motor oil in a case.

TECHNICAL STUFF

Counting has been essential since people first walked the earth. In fact, the Ishango bone is a tally stick (a counting stick), and it's over 20,000 years old!

Several kinds of numbers exist. Over time, mathematicians have given them many names. The two most important kinds are whole numbers and fractions. To see a little bit about how these numbers work, use a number line, a simple display of numbers on a line (see Figure 1-1).

The numbers to the right of 0 are called natural numbers or counting numbers. Of course, they are the numbers you use to count. They're easy for anyone to work with because they represent how many of something someone has (for example, 6 apples or 3 oranges).

Over many centuries and in different cultures, people made up the number 0, which represents the lack of a quantity. The numbers to the left of 0 on the number line, negative numbers, are a harder concept to grasp. You recognize negative number in real life. For example, if your checking account is overdrawn, you have a negative balance. If someone owes you $3.00, you have "negative cash" in your pocket.

Here are the key points to know about the number line:

[check] All the numbers you see in Figure 1-1 are whole numbers, also called integers. An integer is a number with no fraction part. The word comes from Latin, and it means "untouched," so it's the whole deal.

[check] The numbers to the right of zero are positive integers. The numbers to the left of zero are negative integers.

TECHNICAL STUFF

Mathematicians (and I'm not making this up) have trouble with zero. The best they can do is attach it to the positive integers and label the group non-negative integers.

[check] The number line stretches to the left and right, to infinity and beyond (as Buzz Lightyear says).

[check] Decimals (such as 0.75) and regular fractions (such as 3/5) are only a part of a whole number. They all have a place somewhere on the number line. They fit in between the integers. For example 2.75 "fits" between 2 and 3 on the number line, because it's greater than 2 but less than 3.

Reviewing the Four Basic Operations

To do any sort of math, you need to know your math basics. The four basic operations — addition, subtraction, multiplication, and division — let you take care of all kinds of real life math. But what's also very important is that those same basic math operations allow you to handle fractions and percentages, which come up all the time in ordinary math tasks. Later (in Chapter 2), these operations form the basis for managing algebra equations and geometry.

The core operations are addition and subtraction. You very likely know what they are and how they work. Multiplication and division are "one step up" from addition and subtraction. The following sections give you a quick review of these four operations.

Addition

Addition is a math operation in which you combine two or more quantities to get (usually) a larger quantity. Addition was probably the first math you ever did.

You can add numbers (called the operands) in any order. This property (that is, the ability to perform the operation in any order) is called commutativity.

21 + 31 + 41 + 51 = 144

is equal to

51 + 41 + 31 + 21 = 144

No matter in what order you add the operands, the sum still equals 144.

Subtraction

Subtraction is a math operation in which you take away the value of one number from another, resulting in (usually) a smaller quantity.

In subtraction, the order of the operands is important. You can't rearrange the numbers and get the same answer. For example, 77 – 22 (which equals 55) is not the same as 22 – 77 (which equals -55).

Multiplication

Think of multiplication as repeated addition. For example, you likely know that 3 4 = 12, but you can also get there by adding 3 four times:

3 + 3 + 3 + 3 = 12

The technique also works for large numbers. For example, 123 × 7 = 738 is equivalent to this:

123 + 123 + 123 + 123 + 123 + 123 = 738

But who wants to do all that adding?

Here's the best advice for multiplication:

[check] For little numbers, know your multiplication table. It's easy, up to 10 x 10.

[check] For big numbers, use a calculator.

As with addition, you can multiply the numbers in a list in any order. The expression 3 × 4 is the same as the expression 4 × 3. Both equal 12.

Division

Division is essentially "multiple subtraction." In a simple problem such as 12 ÷ 4 = 3, you can get the result by subtracting 3 four times from the number 12.

12 ÷ 3 = 4 with no remainder

is equal to 12 – 3 – 3 – 3 – 3 = 0 (4 subtractions with no remainder)

REMEMBER

In division, the order of the operands is important. You can't rearrange them and end up with the same answer.

Finagling Fractions

Fractions take several forms, but in real life, the forms you deal with are common fractions and decimal fractions.

A common fraction has two parts. The numerator is the top number, and the denominator is the bottom number. You don't have to learn these words, however. Just think "top number" and "bottom number."

numerator/denominator

What do you do with fractions? Arithmetic operations and conversions, that's what.

TECHNICAL STUFF

A common fraction is sometimes called a simple fraction or a vulgar fraction. The vulgar fraction isn't really rude; vulgar is just another word for common (from the Latin vulgus, meaning "common people").

Getting familiar with types of fractions

Like the popular ice cream parlor, fractions come in several flavors. Not 31 flavors, however. For this book, you have to remember only a few fraction types:

[check] Proper fraction: In a proper fraction, the numerator is smaller than the denominator (for example, 4/9).

[check] Improper fraction: In an improper fraction, the numerator is larger than the denominator (for example, 9/4). Think "Honey, does this numerator make my fraction look big?"

[check] Mixed fraction: A mixed fraction is a combination of a whole number and a fraction. Here's an example of a mixed fraction:

1 3/4

[check] Decimal fraction: A decimal fraction uses a decimal point (for example, 0.23, 1.75, or $47.25).

TIP

Decimals are fractions, too, even though they don't look like the other types of fractions. Look at this: 0.75 is a decimal. But what does that really mean? It means 75/100.

Reducing fractions

Here's fair warning: Doing fraction math often produces "clumsy" fractions. By clumsy, I mean unwieldy proper fractions (48/60, for example) and bad-looking improper fractions (37/16, for example). They are handy during the calculations but are very inconvenient as final answers.

You turn a clumsy fraction into something lovely to behold by reducing it.

Reducing proper fractions

You reduce proper fractions by finding a number that the numerator and denominator share and then separating it out. This tactic is called factoring, and multiplication rules allow you to do it. For example, for the fraction 48/60, you "break out" the common factor 12 in both the numerator and denominator:

48/60 = [4/5] × [12/12]

48/60 = 4/5 × 12/12

48/60 = 4/5 × 1

48/60 = 4/5

REMEMBER

When a fraction has the same numerator and denominator, it's equal to 1. Hence, 12/12 becomes 1.

Another way of describing this is to say, "You reduce a proper fraction by dividing the top and bottom numbers by the same number."

Reducing improper fractions

To reduce an improper fraction, you break it into whole numbers and a remaining, smaller fraction. To do this, you divide the top number by the bottom number, and then you use the whole number and the remaining fraction to form a mixed fraction. Here's an example:

49/16 = [16 + 16 + 16 + 1]/16

49/16 = [16/16] + [16/16] + [16/16] + [1/16]

49/16 = 1 + 1 + 1 + [1/16]

49/16 = 3 1/16

Adding, subtracting, multiplying, and dividing fractions

Fractions are just numbers. Like integers, you can add, subtract, multiply, and divide them. Before you panic, keep in mind that you perform these math calculations on fractions all the time. Don't believe me? Think about money.

At first, dollars and cents don't look like fractions because they're in decimal form. But they are fractions, for sure. To look at the details, take a gander at the following sections.

Addition

To add two fractions, the fractions must have the same denominator (also called a common denominator). After the denominators are the same, you add fractions simply by adding the numerators.

When the denominators aren't the same, you need to make them the same. You can't directly add 1/2 pie to 1/4 pie to get 3/4 pie, for example. You need to convert the 1/2 pie into quarters (2/4 pie). Figure 1-2 shows what adding pieces of pie looks like.

Getting the denominators the same is easy because you're allowed to multiply both the top number and the bottom number by the same number. In the pie example, you multiply both numerator and denominator of the fraction 1/2 by 2:

x = 1/2 × 2/2

x = [1×2]/2×2]

x = 2/4

After you have all operands in 1/4 pie units, adding 2/4 and 1/4 to get 3/4 is easy. (Remember that the denominator stays the same when you add the numerators.)

Subtraction

To subtract two fractions, the fractions must have a common denominator (just as they must in addition); then you simply perform the operation on the numerators.

If the denominators aren't the same, you need to make them the same before you can subtract. For example, you can't directly subtract 1/4 pie from 1 whole pie (which in fraction form is 1/1) to get 3/4 pie because the denominators are different. Again, you need to convert the whole pie into quarters, and you do that by multiplying the numerator and denominator by 4 to get 4/4 pie. Then you can do the subtraction:

x = 1/1 × 4/4

x = [1×4]/[1×4]

x = 4/4

After all the operands are in 1/4 pie units, subtracting 1/4 from 4/4 to get 3/4 is easy. (Remember that the denominator stays the same when you subtract the numerators.)

x = 4/4 - 1/4

x = 3/4

Multiplication

Compared to adding and subtracting fractions, multiplying fractions is easy. Just multiply the numerators, multiply the denominators, and then reduce.

x = 3/5 × 6/7

x = [3×6]/[5×7]

x = 18/35

The answer is 18/35. When possible, try to reduce the result. In this case, you can't reduce 18/35 at all.

Division

Here's the secret to dividing fractions: Invert and multiply. That is, flip the second fraction so that the numerator is on the bottom and the denominator is on the top, and then multiply as you would any other fraction.

Say you want to divide 1/4 by 2. (Note: The fraction form of a whole number is that number over 1.) The answer is obviously 1/8. Not so obvious, you say? Here's how you get the answer:

x = 1/4 ÷ 2

x = 1/4 ÷ 2/1

x = 1/4 × 1/2

x = 1/8

You follow the same process when you want to divide a fraction by a fraction:

x = 1/4 ÷ 1/3

x = 1/4 × 3/1

x = 3/4

Notice that dividing by a fraction yields a higher result than dividing by a whole number.

TIP

You can't divide by 0. It's mathematically impossible. The old saying is, "Never divide by zero! It's a waste of time, and it annoys the zero."

Converting fractions

The handiest fraction conversions are turning common fractions into decimal fractions and turning decimal fractions into common fractions.

Turning a common fraction into a decimal fraction

To turn a common fraction into a decimal fraction, just divide the denominator into the numerator. A number like 4/5 easily turns into 0.80 when you divide 4 by 5.

Don't be surprised or alarmed if some division doesn't come out "even." For example, the decimal equivalent of 1/3 is 0.333333333 (and the 3s go on forever). If you see a sale item marked "33% off," it's been reduced by 33 percent or about 1/3. If the item is marked "20% off," it's been reduced by 20/100, or 1/5. (See the section "Processing Percentages" for the lowdown on how to work with percentages.)

Turning a decimal fraction into a common fraction

To turn a decimal fraction into a common fraction, just express the decimal as a fraction and reduce the fraction.

A decimal with one decimal place (0.6, for example) needs a fraction with 10 in the denominator. A decimal with two decimal places (0.25, for example) needs a fraction with 100 in the denominator, and so forth. Here are some examples:

0.6 = 6/10

0.71 = 71/100

0.303 = 303/1000

Notice that the number of zeroes in the denominator is the same as the number of decimal places in the decimal fraction.

For example, say you want to convert 0.375 into a fraction. Here's how you'd go about it:

x = 375/1000

x = [3×125]/[8×125]

x = 3/8 × 125/125

x = 3/8 × 1

x = 3/8

In this example, when you "factor out" 125 from both the numerator and denominator, the result is the common fraction 3/8. See the section "Reducing proper fractions" for details on factoring.

Processing Percentages

A percentage is a fraction whose denominator never changes. It's always 100. A number like 33 percent, for example, refers to 33 parts in 100, or 33/100, or 0.33. You see percentages written as "33%" and "33 percent." No matter how it's written, it's just another way of saying "thirty-three parts in one hundred."

TECHNICAL STUFF

Percent and per cent means "per centum," which is from the Latin phrase meaning "by the hundred." So a percentage always refers to a number of parts out of 100.

Percentages are especially handy for comparing two quantities. For example, if one beer contains 5.5 percent alcohol and another contains 12 percent alcohol, you can be sure that the "high octane" beer has a lot more punch.

Percentages also let you compare values to an arbitrary standard. Nutrition labels are a good example They compare items in food, such as dietary fiber, cholesterol, or vitamins and minerals, to the Dietary Reference Intake (DRI) nutrition recommendations used by the United States and Canada.

REMEMBER

A percentage is a dimensionless proportionality, meaning that it doesn't have a physical unit. Fifty percent of a length is still 50 percent, whether you're talking about feet or light years.