Speed Mathematics Simplified (52 page)

Here are both workings:

When you change your multiplier by cutting it in half or into another fraction, then you compensate by multiplying the other number or the answer by the same amount. Again, it is just the reverse of your compensation in division.

In order to master the point thoroughly, it would not hurt to work out all three forms of this problem. It will help you “feel” the identity of the end results, no matter how the numbers were twisted and turned in working out those results.

Proportionate change is especially valuable in dealing with fractions of all kinds. Even when a proportionate change cannot reduce one of the numbers you must handle to a single digit, it can often simplify it to a remarkable degree.

Would you rather divide by 4 1/3—or by 13, and multiply the answer by 3?

Is it easier to multiply by 6 5/8—or by 53, and divide the answer by 8? For an even more dramatic example, you would prefer to handle 6 ¼ as 25—and compensate by multiplying
by 4 (in division) or dividing by 4 (in multiplication).

Try out the idea on these numbers:

If you multiply each of these numbers by the quantity that will convert it into a whole number, you get the following results (The multiplier is in parentheses.):

In each of the above cases, of course, you compensate in the other number or in the answer with the same multiplier. Do whichever comes more easily.

Run through one whole problem now. Cover the answer with your pad as you work this out with proportionate change:

Surely it is much simpler with the short cut than without it. We double 7 ½ to make 15, which we see at once goes into 300 exactly 20 times—times 2 is 40. Or 15 goes into 600 precisely 40 times.

Now do a multiplication with this technique. Move your pad over the answer and solve this problem with proportionate change:

Use whichever proportionate change suits you best, but do it before checking with the answer.

The logical conversion for 1 ¾ is to multiply it by 4 and divide the other number or the answer by the same factor. Your answer either way is 497. The two workings are these:

Proportionate change is very largely a special application of factoring, and contains some elements of aliquots as well—as you have no doubt observed. It is such a special application, particularly in compensating in the other number and in often making a number larger, that it is classically considered a separate short-cut method.

As an exercise in number sense, consider the essential identity of doubling 35 to make 70 and factoring it into 7 and 5. In multiplication, if you double 35 to 70, you divide the other number or the answer by 2 in order to compensate. Now, dividing by 2 is an aliquot approach to multiplying by 5, is it not? And we picked up an extra 0 when we doubled 35 to 70—which corresponds to the seemingly missing 0 if we consider a division by 2 to be an aliquot for 5.

The various short cuts overlap and are overlapped by the others in many respects. The basic number relationships remain constant; we are merely using different conversions to make those relationships more visible and easier to handle.

17

CHOOSING AND COMBINING SHORT CUTS

Y
OU have learned and practiced the four most generally useful short cuts. There are others, but they are quite specialized. The most complete assortment can be found in the books listed in the bibliography. With the four short cuts you have learned, however, you can convert a great deal of your multiplying and dividing into simpler forms.

Review all together in one place the four different approaches:

BREAKDOWN For one of the numbers to be multiplied, use a round number if this permits an adjustment with an easy fraction of the other number
or
of the result of the first step. 39 becomes 40 – 1; 45 becomes 50 –
the first product.

ALIQUOTS When one of the numbers is an even fraction of a ten-base, use the fraction instead of the number. 25 is treated as ¼ of 100.

FACTORS When one of the numbers can be factored, multiply or divide by each of the factors in turn. 63 is treated as 9, then 7.

PROPORTIONATE CHANGE When one of the numbers can be simplified by doubling or halving it (or any other
such change), use the simpler form and compensate the other number or the answer. 35 becomes 70, with a compensating factor of 2.

Many numbers can be short-cut with not just one, but with two or more of these methods. 45, for instance, can be factored (5 × 9), broken down (50 less
product), or changed (90, compensate with 2). An interesting exercise is to locate one number to which all of these methods can apply. One such number is 125. Witness the various short-cut handlings of the number 125:

BREAKDOWN: 100, plus ¼ product.

ALIQUOTS: 1/8 of 1,000.

FACTORS: 5 × 5 × 5.

PROPORTIONATE CHANGE: quadruple to 500.

For real mastery of short cuts, try to get a feel for the real
identity
of these four apparantly different relationships. One of the techniques will work out in even-number terms for a number that none of the others might handle in this way, but essentially they are all merely different expressions of the same fundamental situation.

The aliquot approach to 125, for instance, is to take 1/8 of 1,000. The proportionate change approach is to use 500, and compensate by a factor of 4. 500 is just half of 1,000, and 4 is just half of 8. The relationships are the same; only the facets we choose to see in any one case appear to be different.