Speed Mathematics Simplified (43 page)

In finding the first line of the working figures for the breakdown example, you do not copy the problem itself. You should be able to work without copying the problem with any single-digit multiplier if you have been working conscientiously on your no-carry multiplication. The second line of working figures, of course, is merely the first line doubled—two places to the right.

Since you are beginning to find it more and more natural to multiply by any single digit, you can extend your breakdowns into any number of tenths or hundredths. The number 558 might, at first sight, not show any exciting breakdown possibilities. 58 bears no reasonably simple proportion to 500. 558 is 42 less than 600. The key is to look at the 42 and the 6 in 600, and note—6 × 7 is 42. So you can multiply any number by 558 by first multiplying with 600 and then subtracting
—which you do by multiplying the first product by 7 and putting down the answer two places to the right.

Choosing Multipliers

In all of our demonstrations so far, we have broken down the bottom number of the problem—the one normally considered to be the multiplier. Except when problems are set up for us in this fashion, there is of course no real “multiplier” and “number multiplied.” In actual business or personal life, we simply need to multiply two numbers together, and it is up to us to decide which we will treat as the multiplier.

The reason this fact is worth special attention is that you can break down
either
number of a multiplication. As you start a particular problem, glance at both numbers for breakdown possibilities. The one you break down becomes your multiplier.

For example, you might face the problem 69 × 58. A quick look at 69 shows you that it can be broken down into 70, minus 1. 58 can be broken down, but not nearly as easily. So pick 69 as your multiplier.

Mixed through the various examples so far have been two
different methods of breakdown. One is the special case called “rounding off and adjusting,” in which you choose a convenient round number and then add or subtract the other number or a simple multiple of it to adjust. 69 is an example of this. So might be 68, since it is easier to multiply by 70 and then subtract twice the other number than it is to multiply it first by 6 and then by 8 and then add.

The second method is rounding off and adjusting by a fraction of the product of your first multiplication, rather than by the other number. For 63, you multiply by 70 and subtract
of the product. For 392, you multiply by 400 and subtract
of the product.

This difference should be crystal clear. In the first case, your difference is adjusted in terms of the number multiplied. In the second case, your difference is adjusted in terms of the product of your first multiplication.

Here, in order to make the difference very specific, is the same number broken down in each way:

Which of the two breakdowns is better? Once again, neither. It depends on the other number and on the methods you yourself find easiest to handle. Either breakdown, you note, permits you to substitute a simple doubling for a multiplication by 8.

You can push the breakdown technique to impressive extremes. The nearest convenient one-digit multiplier may not be the next even ten or hundred at all; it may be two or more away. 1860, for instance, can be broken down so that you multiply by 60 and add 30 times the product. 328 can be-become: multiply by 8 and add 40 times the result.

This field of sophisticated breakdowns is fascinating, but it is too involved to be treated fully here. If you enjoy the idea, you can doodle for hours and find a breakdown for
almost any number you may try. As genuinely useful short cuts, however, the more abstruse applications are questionable. You would spend more time breaking down your multiplier than the whole problem would take in simplified arithmetic. Number sense, again, is the real key. If you cannot “see” a relationship at one or two glances, then the short cut is not a real short cut for you.

The most useful ground rules for the two types of breakdowns are these:

If the answer to either of these questions is yes, then breakdown can save you work and time in solving the problem. If the answer seems to be no, then another short cut may be in order.

Answering these two questions rapidly is the way to break down problems quickly and easily. See how many sensible breakdowns you can find in these multipliers:

Each of these numbers can be broken down in a way that will
save you work. Some of them save you quite a bit of work; others let you add or subtract instead of multiplying by a high digit; still others reduce the number of lines of working figures. Try them on your pad before you check your reactions against the proposed breakdowns that follow.

In some cases, more than one breakdown is possible. We will give only the one that seems simplest and most generally useful. Since the breakdowns are of both types, we will use the shorthand N to mean that adjustment is in terms of the other number, and P to mean that adjustment is in terms of the product of the first multiplication.

For instance, our breakdown for the first number—58—is given as 60 – 2N. This means you multiply by 60 and then subtract the other number, doubled. The breakdown for 72 is given as 80 –
P, which means you multiply the other number by 80 and then subtract
of the product.

Here are the breakdowns: