Speed Mathematics Simplified (27 page)

The remainder is the same as the divider. So our final answer digit must be raised by 1, and the problem will come out even. As we pointed out before, you raise the digit by underlining it, so the final answer is

which you read as 356.

You were promised that digit-revision would be automatic. It is. Any digit in your answer may need revising, even the first. But the time to do so is signaled to you automatically, so you do not need to watch especially for such events.

This problem demonstrates why:

We “see” the first answer digit as the result of 7 into 54, which can be only 7. Now (use your pencil and pad to help build the technique) we find the first remainder:

One: 6 x 7 is in the 40's, and 4 from 5 is 1:

Two: 6 x 7 ends in 2. 3 x 7 is in the 20's. 2 plus 2 is 4, and 4 from 4 is 0. Where a zero digit appears in the
middle
of a subtraction answer, as it does here, it is wise to put it down to avoid possible confusion:

Three: 3 x 7 ends in 1, and 1 from 7 is 6:

Perhaps you have noticed that your remainder in this case is larger than your divider. Something is wrong, and what is wrong is that your answer digit needs raising by one.

You do not have to be especially alert to this situation, however. If you didn't notice at this point, you could not help but notice as soon as you tried to get the next answer digit. You mentally bring down the 4 and divide 63 into 1064—seeing it as 7 into 106.

Such an answer digit would be over ten. There is no such digit. This, in case you missed the signal that developed when your first remainder was larger than the divider, is the STOP signal that warns you to revise your answer digit.

This is automatic division. To raise the answer digit, you merely underline it. To adjust the remainder, you merely
subtract the divider
and put down the new remainder before going on:

The underlined 7 is, of course, now 8. The 43 is the answer we get after subtracting 63 (the divider) from 106 (the too large remainder). Now we are ready to continue, with everything adjusted and correct so far.

Finish this problem yourself on your pad. It does come out even, though one other answer digit will need revision. Finishing this problem will involve just about every technique in shorthand division covered so far.

Do it now.

If your final answer did not come out to an even 768, which you read or rewrite as 869, check the appearance of your working figures against this model:

This section, in its joining together of many different techniques from earlier parts of the book into one effective but apparantly complex whole, has been perhaps the most difficult chapter to understand in one reading.

Sit back for a moment and let some of what you have done sink into your mind. Don't be discouraged if it takes several readings to understand fully what has been going on. It involves quite a new way of looking at numbers, a way really simpler than the traditional ways because much of it has been adapted from the simplest and highest-speed arithmetical system known—the modem Japanese abacus—but until you get used to it it does take some special lip-biting.

Division is the most complex of all our basic operations in arithmetic. There is simply no help for this; it is the nature of the beast.

What we have done so far is to reduce the process to the simplest series of easy steps that can possibly work. You never hold more than a digit or two in your mind at any one point; you work from left to right; you never have to carry as you multiply; and you never have to “borrow” as you subtract.

The seeming complexity at this point is inherent in the function itself. If you had never learned long division, and if somebody sat down to explain it to you, it would seem much more complex. There are many separate processes to be done, and if full accuracy is required every process must be done in full.

Except for one really minor special case, you now know everything you need to know for this rapid way to divide. Ease and speed will come with practice, which the next chapter will help to provide.

The one special case involves slashing a number not once, but twice. We hinted at this possibility when we pointed out that you slash left when you use a complement in multiplying, and you also slash left when you use a complement in subtracting. It does not come up very often, but it does come up and it is very easy to handle.

Here is the sort of problem in which you will find this necessity:

Let's begin this problem step by step, to drive the method deeper into your mind.

8 (not 74) into 50 (not 505) gives 6 as the first answer digit. Put it down, and determine the remainder:

One: 7 x 6 is in the 40's, and 4 from 5 is 1.

Two: 7 x 6 ends in 2. 4 x 6 is in the 20's. 2 plus 2 is 4, and 4 from 0 is (complement) 6 and slash.

Three: 4 x 6 ends in 4, and 4 from 5 is 1.

Our example now looks like this:

Inspection shows us that the next answer digit (8 into 61) is 7. We put it down and work out the remainder:

One: 7 x 7 is in the 40's, and 4 from 6 is 2: