Speed Mathematics Simplified (20 page)

Go through the mental processes in your mind, making sure that you too would put down each new digit as it appears in the step-by-step unfolding of this answer. Note especially how the underlines are handled.

Now try a couple of three-digit multiplier problems on your own. Work from left to right, in the no-carry method, and remember to put down a left-hand digit for your first product in each line even if it is a zero:

You will find the detailed working of these two examples at the end of this chapter. Get your pad now, though, and go through them to the end before reading on. Save your working for a check against the solutions to come later.

Automatic Estimating

One of the beautiful features of left-to-right, no-carry multiplication is the way it produces quick estimates. It is as fully automatic in this respect as is left-to-right addition or subtraction.

There is no easy and accurate way of doing this with traditional multiplication. Yet it is built right in, at no extra cost, to any left-to-right system.

You can get a two-digit estimate in a twinkling. You can get a three-digit estimate (which equals the accuracy of almost any slide rule) while the man with the slide rule is still getting out his “slip stick” and setting it.

This is not a criticism of the slide rule. If you must do a great deal of multiplying and dividing and are satisfied with rounded-off answers—which the slide rule provides by its very nature—then it is well worthwhile getting one and learning how to use it. It is not hard. But do not pass up this estimating short cut even if you have a slide rule, because the system is both useful and impressive. It also works when your slide rule is somewhere else.

The technique for estimating with no-carry multiplication to any required degree of accuracy is simply this: Multiply as far as you have to and stop. Raise the last digit by one for each two digits in the multiplier.

Suppose you face a really formidable multiplication such as the cost of 53,926 items at $48.75 each. You must give a rapid approximation to three digits.

All you need to do is quickly scrawl each part of your new no-carry multiplication as far as three digits from the left. Here is how you do it:

So far, you have 260. There are four digits in the multiplier—count 4, 8, 7, 5—so raise the 0 by 2. Now you have 262.

A slide rule would not do any better. Carry the multiplication further, if you wish, and see how close we are.

Does this mean $262,000 or $2,620,000? One simple rule gives you an unfailing answer to this question. Your answer has exactly as many digits as the total of the two numbers multiplied. Just add the digits in these two numbers, and figure on that many in the answer. Special note: If the first digit of the answer is a zero (the first digit of the first line of partial answers), this must be counted too.

In the above estimate, your answer is $2,620,000. Try working it out and see—keeping in mind that two of the digits in the multiplier are behind the decimal point and therefore are a fraction.

An estimate of

would have five digits in the final answer, or 68,900. The total number of digits in number multiplied and multiplier is six, but in the answer one of that total is lost in the initial zero.

Two other special points are interesting in this matter of estimating. First, note that you work your answer out to only as many places as you need and, in order to do it, you work out each partial answer to this same number of places
starting at the top left.
For a three-digit estimate, you will have three digits in the top partial answer, two in the second, and one in the third. If the first line begins with a zero (as in the one above) then you will go to four digits in the first line. Should your multiplier have twenty digits in it, you would ignore all but the first few.

Perhaps you wonder why you raise the last digit by one for each two digits in the multiplier. Check back to the section on estimating in the chapter on subtraction, and you will find a very similar rule. The reason is this: The average of any random number of digits including 0 is 4½. The average for each two lines in addition is therefore 9
—plus
the likelihood of tens recorded (or carried back to this column) from the column to the right, at the rate of about one for each 2½ lines. The best average for estimating, then, is to increase your final digit by one for each two lines in the addition. And the number of lines in the final addition of a multiplication problem is determined by the number of digits in the multiplier: one line of partial answer for each digit.

So raise the final digit of your estimate by one for each two digits in the multiplier. Forget any extra digits, and count five as two, seven as three.

Practice estimating these two problems accurate to three digits. Use pencil and paper. Remember to raise the last digit of your estimate in the way described above, and to count the digits in both numbers and use this total as the number of digits in your answer—including an initial zero if it appears in the top line of partial answers.

Do these now:

The estimates of these appear at the end of the chapter. Do them yourself, though, before you look.

Here are the answers to the two three-digit problems you were asked to work out on page 78. Compare them with your solutions:

And here is the way we estimate to three-digit accuracy the two examples at the top of this page:

The next chapter will help you to develop greater familiarity and speed with these techniques. If you feel that everything in this chapter is completely clear, go on ahead. If not—review.

7

BUILDING SPEED IN MULTIPLICATION

Y
OU recall that we stated three basic secrets for speed in multiplication:

First, work from left to right (possible only with this system).

Second, “see” the result of each multiplication of two digits, rather than the problem.

Third, use the no-carry method.

The second point is the one that obviously requires the most practice. The foundation of all your speed is the easy, natural, painless use of the no-carry system—but the way to make it easy and painless is to make as automatic and unthinking as possible the process of “seeing” 8 x 7 as “50's” and “ends in 6.”

Your job now is to go over these half-products enough times to make the automatic response a habit. Since you un doubtedly got far more drill in the multiplication tables than you did in addition and subtraction tables, learning to see each product as only the left-hand or right-hand digit is not really all that much more work. Once you become fully used to it, you will find it quicker and simpler than the old way.

Let's review for a moment what we mean by left-hand and right-hand digits in multiplication. Try to “see” the
left-hand
(tens) digit of