Chapter 9 - Building Speed In Division

The last exercise in the last chapter was a stunner. It was, just from the quantity of digits to be handled, the most tedious situation you are likely to face in arithmetic. The numbers go on and on, and if you need a full answer there is simply no way to avoid dealing with every single digit.

Most division is much simpler.

First of all, we seldom need to work out any division problem of this length in such detail. Largely because American business has become accustomed (and wisely so) to dealing in rounded-off numbers, you would most likely find such a problem rounded off to start with. Second, the only reason we had to use every single digit was to get a fully accurate remainder. We would have got precisely the same whole-number answer by cutting down the divider from five to three digits.

Remember the two reasons for working from left to right: it is more natural, and it is also self-estimating. Turn back to the last problem for a moment, then compare the final working with this version of the same problem:

This is really the same problem—but a rounded-off version of it. We rounded it off before beginning by doing two things:

First, we rounded off the divider to three digits because we saw simply by inspection that the whole-number answer would be in three digits. In rounding off, the first three digits (361) became 362 because the following digit is more than 5.

Second, we dropped the same number of digits from the number divided as we did from the divider. This ensures that our answer will not be ten times too big or ten times too small.

Rounding off such a problem is obviously faster as well as simpler. For most purposes, it is quite accurate enough. You will note that the remainder is not the same, and sometimes the last digit itself might be off by one or two points in value—but we are still more accurate than a slide rule.

When you need a very quick estimate, you can carry this even further. If you care only about the first two digits of the answer, then round off your divider to two digits and cross out as many digits in the number divided as you did in the divider.

In this event, the full divider 36182 becomes 36. The number divided, 18936824, becomes the far more manageable 18937.

Your solution now looks like this:

Notice that the third digit of your answer is no longer accurate at all. But your first two digits are.

You would seldom simplify a problem to quite this extent, because the possible error is ten to twenty per cent, but it is a useful device to know when speed rather than perfect accuracy is required for a very fast approximation.

Try rounding off one sample to make sure you have the idea firmly in mind—especially the proper handling of the number divided. Reduce this example to a form suitable for a three-digit answer:

As we pointed out before, an answer of which the first two digits are correct, and the third digit is one more or one less than it should be, can never be more than one per cent wrong, and may be as little as one-tenth of one per cent wrong. The least error is 998 when it should be 999.

Your rounding off of the above problem should look like this:

In this case, both final digits were raised by one in the rounding off, because both following digits were 5 or more. The only new element in this kind of rounding off is establishing the proper size of the number divided. Since we dropped two digits from the divider in this case, we also dropped two digits in the number divided.

Other than some of the short cuts in the last part of this book, which apply to many (though not all) problems, this is about all there is to know about estimating in division. The most important elements, as you can see from the work so far, are your quickness and confidence with the basic digit combinations in dividing, multiplying, and subtracting, and your mastery of the one-two-three of shorthand division.

It is now time to brush up on your vocabulary.

Division is, after all, only multiplication done backwards.

Instead of "seeing" 6 x 7 as 42, we learn to see 6 / 42 as 7 ... or 7 / 42 as 6. Just as it is in adding, subtracting, and multiplying, the best medicine for this is repetition.
 
Keep in mind that the following practice section is not to be done as a simple division drill. Go slowly and carefully, making every effort to "see" the answer rather than the problem. It may help to say aloud the answer in each case, shoving the problem as far back in your mind as you can.

Once again, you are practicing to see h and e as "he"— not as "h and e spell 'he.'" You can do this with numbers just as you can with letters if you spend a reasonable amount of time at it.

Try to read through these just as if they were words, seeing the words rather than the letters:

This is quite a new bit of practice for most of us. Even though division is merely inverted multiplying, it is the basic process on which the average person has spent less "drill" time learning his tables than on any other. Yet, for quick working of short division (or even long division), there is no substitute for knowing them backwards and forwards.

Your confidence and accuracy with any method of speed mathematics are based entirely on your confidence and accuracy with the individual digit combinations. No technique can be very helpful in your daily mathematical needs unless you can do it—with confidence and accuracy.

Improve your handling of division now by practicing the rest of the possible combinations. As always, work at seeing only the answer—not the problem:

That is the whole series. There are no other combinations.

There is, though, an important variation. When you stop to think about it, division is the only one of the four processes in which you usually have an approximate answer.

When you add, you get one specific result. 9 plus 6 is always (whether you add it or whether you subtract a complement and record a ten) 15.

When you subtract, there is no question about it. 8 from 13 is always 5.

When you multiply, 4 times 7 is always 28. There are no if s, and's, or but's about it.

But what about 8 / 3 1 ?

Your instinct or number sense or practice at division tells you that the answer to this is "almost 4." But it is not 4. No matter how close it is, you still cannot get four 8's into 31.

It is so close, of course, that you can get 3 and 87/100's 8's into 31. But you still do not get 4.

You will get 3 +. Your answer will approach 4 as you work out the remainder in decimal or fractional form, but your first digit has to be 3. This is because our methods of writing numbers include ways to write 3 plus a fraction, but not 4 minus a fraction.
 
The thought is worth considering because quick and efficient division requires us to "see" 8 into 31 as 3.

The usual process for many of us is to take a stab at the closest answer, then (consciously or unconsciously) multiply it out in our minds to see if it checks out, and revise our trial digit when required.

The automatic digit-finding technique of shorthand division (dividing by only the first digit of the divider, increased in value by one) solves a large part of the problem. The second half of the battle, however, is to learn to "see" an approximate division, such as the one above, cleanly and properly at first glance. This means knowing at sight that 8 / 47 is "5," even though the final answer will be much closer to 6.

Here is some practice on this, which will pay in faster and easier dividing. Work at these tables with the objective of "seeing" only the first answer digit. Do not worry about whether the eventual answer will be 3.001 or 3.999. In both cases, you start with 3.

Start now:

This group includes over half the possibilities. Perhaps you have seen the nature of the practice you are now doing. Each of the division examples you did contains a number divided just 1 less in quantity than one which would call for a higher first-answer digit. For instance, 3 / 14 is practically 5—but you start with 4.

Learning to "see" what we might call the breaking point of each answer digit cannot help but ease and speed up your
automatic division. Once you have learned to "see" 3/14 as 4 rather than "almost 5," you should have no trouble reading 3 / 13 as 4 also. If you recognize 8 / 71 as 8 rather than "almost 9," then whenever you need an answer digit for 8 / 70, 8 / 69, and all the intermediate possibilities down to 8 / 64, you should answer without a second thought "8."

Go through the rest of these "breaking point" combinations now:

That finishes all the possibilities. This practice section is vastly different from any other in the book, in that it calls for you to give what you know to be a very approximate answer—an answer you know full well will need adding to later. Yet this is the way we must divide, and what at first may seem very odd must become second nature.

Having done the single-digit tables, expand your practice a bit now by doing precisely the same thing with these examples. In each case, be sure to divide by only the first digit of the divider, raised in value by one. See only the first answer digit for each of these problems:

Those are the basic vocabulary elements in speed division. When you put them together with simultaneous left-to-right multiplication and subtraction, short hand division really becomes short-cut division.

See how well you remember the entire system now by working this problem out in detail on your pad:

Cover the demonstration below with your pad while you work it out. This problem comes out even, so you know without looking ahead whether or not you solved it properly.

When you are finished, compare your working with this model:

One way to speed up your working of any problem such as this is simply to jot down your working figures on whatever piece of paper you have handy. You do not always have to copy the entire problem, although this is often helpful during early stages of practice, when each move still seems rather strange.

If you have been doing your practice, you should be able to solve the following problem by the shorthand method without copying it. I do not suggest that you solve it in your head (some people can, but most of us have to lean on our pencils even with simplified techniques), but you should be able to glance from printed problem to jotted working figures and produce the answer without turning yourself into a stenographer:

 See if you can do this problem without copying it. Use your pad only for jotting down the answer and working figures. If you have never done it this way before the technique may seem difficult, but it can save you a great deal of time in your number work.

When you have finished, compare your jotted figures with these:

Try a simpler problem with this non-copying method. See if you can jot down your answer and your working figures for this example without copying the problem itself:

Cover the answer below until you have done your best. Here is the way you jot down your answer and working figures:

You have practiced all possible digit combinations. Now, before going on to other methods of speeding up your number work, do a bit more drill in actual division examples.

If you can, solve these without copying them. Use your pad for answers and working figures only.

If you did the above examples in the shorthand method with confidence, then divisions of any length at all are merely extensions of what you already know. Try the following three-digit dividers. Again, do your best to jot down only the developing answer and working figures rather than copying the problem over:

Give these your best before checking your work against the solutions following. If you were able to handle them without copying, extra good. The solutions will be given in copied form, however, so you can check your work whether or not you jotted your answer and working figures separately.

Every one of the preceding problems comes out even, as a quick check on yourself before looking at the solutions. If you have any remainders, go back and recheck now

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