Speed Mathematics Simplified (50 page)

One other key to watch for in picking your first-division factor is to see if either of the factors of your divider is also a factor of the number divided. If it is, you know that the first division by this factor must come out even.

Now try one division by factors on your own:

Cover the answer with your pad until you have finished.

The factors of 72 are recognizable at sight: 8 and 9. Since the number divided is even, we will start with the even factor, then divide that result by the other factor. The illustration will be in condensed style. See if your working agrees with this:

Division with Three Factors

Just as it often pays to use three factors in multiplying, so you can use three factors in division to speed up and simplify your work.

In multiplying, you use each of the three (or more) factors in turn to avoid adding lines of partial products. In dividing, you use each of the factors in turn to avoid the extra complications of dividing with a number of two or more digits—where you can. When one of the factors has to be of two or more digits, you will still find the division simpler than dividing with a still longer number.

Suppose we factor this problem:

No matter how we tackle it, this is admittedly one of the divisions most of us hate to face.

First, see if the divider can be factored. 567 has a digit sum of 0, so we know at once that 9 is a factor. 9 into 567 (without writing down the problem, of course) gives us the other factor, 63. 63 we recognize as 7 × 9. So 567 has three one-digit factors: 9, 9, 7.

Since all the factors are odd, you may as well start with the smallest. Stack your working as we did before, and the factor solution looks like this:

What factoring really accomplished in this case, as you can see, was to reduce the solution to three divisions of single digits each, rather than one division by a number of three digits.

It is time now to try a three-factor division yourself. Cover the answer with your pad until you have finished this problem:

The first step is to see if the divider can be factored. The digit sum is 8, so it is not divisible by 9 or 3. It is even, so it is divisible by 2, but moreover the last two digits are divisible by 4, so 4 is a factor. We start by factoring it into 4 and 56. Since 56 is even, we double the 4 and cut the 56 in half: factors, 8 and 28.

We prefer to work with single-digit factors if we can, so we further factor the 28 to 4 and 7. All the factors we need for 224, then, are 4, 7, and 8.

Since the number divided is even, let's start with the 4 and work up:

That's all there is to it. That rather fearsome division problem is reduced, thanks to factoring, to three quick and simple single-digit divisions.

Sometimes you can factor a number into one one-digit factor and one two-digit factor, but cannot further factor the two-digit factor at all. You may still save time by using these two factors, though, just as you can in multiplication. Dividing by a two-digit number is so much easier than dividing by a three-digit number (even in the shorthand method) that it will probably pay you to use the factors—especially since you have already gone to the trouble of factoring the divider.

Every time you use factors, you will become fonder of them. They are the third major area of short-cut conversions, following naturally after breakdown and aliquots.

Now we will take up the fourth type of conversion.

16

PROPORTIONATE CHANGE

T
HE fourth generally useful type of short cut has no traditional name. Because the phrase most accurately describes what we do, we will call it proportionate change.

The technique is simply that: proportionate change. You change one number of a problem into a simpler form in
any
way you wish (double it, triple it, cut it to one-third, or whatever) but change the other number
in proportion
so the essential relationship remains the same.

For instance:

The conversion here should be quite obvious. One glance at the problem shows us that 45 can be converted into a one-digit (plus 0) number by doubling. So we double it, without hesitation, to the more easily handled 90.

The
proportionate
part of the rule is simpler for division than it is for multiplication. In division, you do to the number divided precisely the same thing you did to the divider. In multiplication, you do to the other number exactly the opposite of whatever you did to the first number.

In the example above, you double 45 to 90. To keep the proportion, you now double the 180 to 360. The answer, simply by inspection, is obviously 4—9 into 36.

Try one yourself:

Start by examining the divider for any simple change that will convert it to a one-digit number—plus a 0 if necessary. Doubling 35 changes it to 70. Do the same thing to the number divided, which changes it to 420. Most of us would find it rather difficult to “see” the answer to 35
, but the answer to 70
should be a matter of reading
ea
and
sy
as “easy.”