Speed Mathematics Simplified (30 page)

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
as 7…or 7
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

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
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
is practically 5—but you start with 4.