Speed Mathematics Simplified (47 page)

Now divide this product by the top of the fraction. In practice you would do it at sight:

That is all there is to it. Without dividing by anything more difficult than the single digit 3, you know that 29700 divided by 75 is 396.

Reversing Aliquots

If you will turn back for a moment to the table of exact aliquots, you will note that several of them are really simpler in their decimal form than they are in their fractional form.

A later chapter will cover fractions and decimals. If this special application of their interchangeability in terms of aliquots is at all confusing, it might be a good idea to refresh your memory with that chapter first.

The fraction
, for instance, has the aliquot form .8. The decimal form of
is .4.

This fact makes possible a reverse short cut whenever you must deal in fractions that are more simply expressed in decimals. Rather than suffer through the fraction, use the simpler form.

One example should illustrate this sufficiently. Consider this problem:

This problem would traditionally be solved by multiplying 3 × 287 and then dividing the product by 5. But it is far, far easier to multiply 287 × .6:

The important lesson in this reverse-aliquot approach is that no
single
method is always best. The point is to learn awareness of the many different ways of accomplishing the same result, and to be on the lookout for the easiest and quickest in each particular case.

Sometimes it will be breakdown. Sometimes it will be the use of an aliquot. And sometimes it will be the use of factors—quite a different short cut.

15

FACTORS

M
OST of us remember, from our school days, the word “factors.” Chances are you have not encountered the word or the process since. Instead of considering them merely an exercise for students, however, we will show how they can short-cut many problems in multiplication and division.

A factor means, basically, a maker or doer. The word has many applications in English. In mathematics it means one of two or more numbers which, multiplied together, produce the number in question.

6 has two factors: 2 and 3. 2 and 3 are factors of 6 because 2 × 3 gives 6.

Almost three-quarters of all numbers are factorable. That is, they can be broken down into two or more other numbers which, multiplied together, produce the number you started with. Of the first hundred numbers (from 1 to 100) only 26 are prime numbers. Prime numbers are those that cannot be factored.

1 and 2 are both prime numbers, because they cannot be factored. It is true that 1 × 1 is 1, but we do not consider 1 to be a legitimate factor. It would not be of any use to us in short-cut mathematics, in any event. 3 is also prime. But 4 can be factored into 2 and 2, because 2 × 2 is 4.

Before going into the ways of factoring numbers, let us show
the exciting possibilities in their use. They are a powerful short cut because they can save major steps in multiplying and dividing.

To multiply by a factorable number, multiply first by one of its factors and then multiply the result by the other. Where is the short cut? Watch:

In order to use factors, we first find a number that can be factored. Even though in real-life situations you will look at both parts of a multiplication rather than arbitrarily decide that one of them is the multiplier, it is usually quicker to consider the shorter of the two numbers the multiplier.

In this case, 56 is the multiplier. Can it be factored? Can you think of two other numbers that, multiplied together, produce 56? Your knowledge of the multiplication tables should snap the factors 7 and 8 into your mind.

The factor short cut in multiplying any number by 56, then, is to multiply first by 7, then the result of that multiplication by 8. Compare the two ways:

The two examples may
look
about equally time-consuming. But note than in the usual way you multiply first by 5, then by 6, then add the two products to get your final answer. In the factor method you still multiply by two digits—7 and then 8—but you never add any partial products at all. You save roughly one-third the work.

Let's do another before you try one on your own. Check this problem for factor possibilities:

If you have “seen” the factors of 28 at a glance, let us compare methods again:

Once more, we managed to skip entirely the step of adding two lines of partial products. Multiply by 4, then by 7, and you have the final answer.

Try this one by yourself. Cover the answer with your pad as you work: