Speed Mathematics Simplified (55 page)

18

MASTERING FRACTIONS

O
F ALL the specialized branches of mathematics, fractions seem to be greeted with more general panic than all the others put together.

It does not have to be so. Fractions are really not much more complicated than multiplying or dividing. Perhaps the reason for their general unpopularity is that they are taught, to an even greater extent than is true for the other processes, almost entirely by rote. The rote itself simply has to have a few more steps and rules than do whole numbers.

You can add any two whole numbers together without doing anything to them first. But not fractions. The reason
why
this is so has apparently escaped the normal teaching methods. Many people have trouble understanding why you can multiply two fractions together and get an answer smaller than either of them. If you multiply two numbers together, isn't the answer larger than either? Again—not with fractions.

Both peculiarities, along with the other peculiarities, are inherent in the true nature of fractions. Let us approach their nature with some general observations.

A fraction is, in essence, a number that cannot be expressed normally in our decimal system of digits running from 1 to 9 and then starting over. It is usually
smaller
than 1, and our counting system has no way of expressing such a quantity other than the apparently awkward form of the fraction (other than a decimal, which is merely a fraction written in another way).

A fraction, even if we have no other way to indicate it than a fraction, is however a very real number or quantity. The form in which we show it is really a fabulously ingenious and useful method of expressing any conceivable quantity
from any conceivable counting base
in terms of the number system we know.

Imagine, if you will, that our base quantity “1” is a loaf of bread. We have built up a complete arithmetic based on loaves of bread; we have units of ten loaves; we have learned by heart how to add 3 loaves to 6 loaves, to start with 8 loaves and take away 4 loaves, to imagine that one group of 2 loaves has been doubled, or multiplied by 2. But then, suddenly, one of our loaves breaks into pieces and we must account for the pieces.

This is a fraction. The loaf may have broken into “3” pieces, but we have no arithmetic with which to handle it. The only units we know are in terms of loaves of bread. Yet this “1”—this loaf—is no longer 1. It is less than 1.

How do we express the quantity represented by each of these pieces? Some genius or geniuses, centuries ago, suggested that we represent it by “1”—because it once was 1 loaf—
divided by
“3”—as if each of the pieces were now a loaf. The 3 came from 1, so the essential quantity is the one expressed by a division of 3 into 1.

We write it 1/3.

This is the basic fact about all fractions. They are real quantities, but quantities that cannot be expressed in our regular number system, so we express them in terms of divisions.

A fraction is, then, merely a division problem.

When we write the quantity 2/5, we really intend to convey the idea of a quantity that is outside our number system, and can best be expressed by dividing 2 by 5, or 2 ÷ 5, or 5
Because we wish to show it as a quantity more than a problem, we write it 2/5.

Thinking of a fraction as really a
problem in division
, which also expresses a specific quantity, may help you to gain an emotional grasp of the entire system.

Why
2 × 2
Is “Less” Than
2

One of the most baffling habits of fractions is that when you multiply two of them together, your answer is less than either of them alone. We are so accustomed to thinking of multiplication as an increasing process that this jars our basic number sense.

If you think of multiplying as counting a number a certain number of times—which is precisely what it is—the concept becomes clearer. If you count a number more than once, then the result is obviously larger than the number was. But if you count the number
less
than once, as you do when you count it only 1/3 times, for instance, then the answer must be smaller than the number was when you started. If the number you counted was less than 1 to start with, such as ¼, then the answer will obviously also be smaller than the number of times you counted it—because to get an answer as large as your “counting” number you would have to count another number at least as large as 1.

This is why multiplying by two fractions smaller than 1 gives you an answer smaller than either of the fractions. You count a number that is smaller than 1 to begin with, and you don't even count it one whole time. When you multiply ¼ × ½, you are saying in effect “count ½ exactly ¼ times.”

This fact leads us into the first natural rule for handling fractions with understanding as well as memorized rules: to multiply fractions, multiply the top numbers together for the top of the answer, and multiply the bottom numbers together for the bottom of the answer.

Note that we define this rule in terms of top numbers and bottom numbers. Arithmetic has become topheavy with special names such as “numerator” and “denominator” that confuse things more than they clarify them for most of us. If you agree, “top” and “bottom” is instantly and unmistakably clear.

Following this rule, then, count 3/5 exactly ¼ times—or, if it sounds clearer, ¼ of one time:

The top of our answer is 3, which is what we get when we multiply 3 × 1. The bottom is 20, which is produced by multiplying 5 × 4. This is what is produced when you start with a quantity expressed by dividing 5 into 3 (3/5) and count it not even once, but a number of times expressed by the division of 4 into 1 (¼).

In order to refresh your memory, try it yourself:

This is simple, naturally, but if you are at all rusty it would help to cover up the answer with your pad and write down the answer.

Multiplying the top numbers, we get 6. Multiplying the bottom numbers, we get 28. The answer is 6/28.

This is true, but 6/28 is a fairly complex fraction. Is there a simpler expression of the same quantity? 6 and 28 are both evenly divisible by 2. If we divide both the top and bottom by 2, our fraction becomes 3/14.

Think about this fact for a bit. Your memory of the rules undoubtedly tells you that it is the same, but visualize the two expressions and see if you can feel their identity.

This leads us to a general rule for all fractions:

If you multiply or divide both the top and bottom numbers of a fraction by the same number, the quantity remains unchanged.

By this rule, 6/8 is the same quantity as ¾. Is it? You know by training that it is. But can you
feel
it? As a good exercise in number sense, try expressing this quantity by 6 dots above a line with 8 dots below it. Thoughtfully connect each two adjacent dots so they become 1 line, in pairs, and note that you now have 3 lines over 4 lines. You have not changed the
relationship
of the quantities above and below the line, but you have changed the numbers.

Try a few multiplication exercises on your pad:

Work these out. They are elementary, but important.

The raw answers are, of course, 6/72, 5/60, 3/8, and 1/8. We say “raw” because some of these can be reduced to simpler terms. 6/72, for instance, is reduceable at sight to 3/86, and this in turn is clearly 1/12. Check your other answers for reduction possibilities.

Short-Cut Multiplying

If any fraction whose top and bottom numbers can be evenly divided by the same number can be reduced to a simpler form by dividing, then two fractions to be multiplied can also go through the same process even before they are multiplied.

This means that often you can do part of the reducing before you multiply, rather than after.

The secret that makes this possible is that it does not make a bit of difference in what order you multiply or divide numbers: the result will be the same. 4 × 8 × 6 is the same as 8 × 6 × 4 is the same as 6 × 4 × 8—as well as 8 × 4 × 6 and 6 × 8 × 4.

If this fact is not instinctive with you, work out each of the above multiplications and make it instinctive.

When we start out with a problem such as the first one above, we note that more than one top and bottom can be divided by the same number:

The top 3 and bottom 9 are both divisible by 3. They become, respectively, 1 and 3. So now we have: