Speed Mathematics Simplified (35 page)

Check: 4 x 8 is 32, which reduces to 5. This is the digit sum of the number divided.

You can easily see the trouble you would have trying to divide the digit sum of the divider (8) into the digit sum of the number divided (5) and produce any rational whole-digit result. The reason for this lies in the special reduction of digit sums, which pretends (for digit-sum purposes) that 10 is 1, that 14 is 5, and that 9 is 0. The system works perfectly if you multiply as outlined above, but cannot possibly work if you try to divide.

If there is a remainder in the answer to the division, add one more step. First multiply the digit sums of the divider and the answer, as before. Now, however,
add
the digit sum of the remainder. This total, reduced, should equal the digit sum of the number divided.

Here is how it works:

Check: 2 x 5 is 10, which reduces to 1. 1 plus 2 is 3, which is the digit sum of the number divided. Right.

Now try these problems, using your pad to cover the correct solutions as you always do. Caution: Any digit multiplied by 0 must give 0.

The illustrations below will, as always, be in shorthand division. Look at them after you have finished your practice.

Check: 1 x 8 is 8. Right. Problem

Check: 0 x 8 is 0. 0 plus 3 is 3. Right.

This is all there is to know about digit-sum checking. The back-up check in the next chapter works the same way, but the check figures will be quite different.

11

ACCURACY: THE BACK-UP CHECK

T
HE digit-sum, or “casting out nines,” method is the quickest and easiest way to check any problem. Once you become fully accustomed to it, you will find yourself checking a problem about as quickly as you could read it over.

It is not, however, completely foolproof. The last chapter explained the types of errors to which it is quite blind. As someone once pointed out, the digit-sum method will tell you that a problem is
wrong
, but it will not tell you for sure that it is right.

This chapter explains how to “cast out elevens.” This is a little slower but inherently more accurate than casting out nines. In cases of critical accuracy, some experts advise using both methods. You can easily do one right after the other in much less time than it would take to check by conventional methods, and if both your digit-sum and your “elevens” results check out, you can be quite sure you have a perfect answer.

Casting out elevens, or simply “elevens” as we will call it, works on precisely the same check-figure method as does casting out nines. In fact, adding up the digits is really only casting out nines because the proof of a number's divisibility by nine is the addition of its digits. If the sum is nine (or 0), the number is exactly divisible by nine. Any other result is the
remainder
you will have after dividing by nine.

Both casting out nines and casting out elevens are merely special (and convenient) applications of a general rule. You could check a problem by “casting out” any number at all. You could find the remainder of each number after dividing it by four, say, and use these remainders as check figures. Nines and elevens are merely the easiest numbers to cast out that also depend for their divisibility on every digit in the number.

This use of a division-remainder is not as odd as it might sound at first. If you add a series of numbers exactly divisible by four, then their total must obviously be divisible by four. If one of those numbers has a remainder of two after a division by four, then the answer must also have a remainder of two after a division by four. If you multiply two numbers each of which is exactly divisible by seven, then their product must also be exactly divisible by seven.

When the numbers are not exactly divisible by whatever number you use for your check figure, then the remainders of each number get carried along through the arithmetic too, and once you do to these remainders whatever you did to the numbers themselves, they must come out in exactly the same relationship to the remainder of the answer.

In order to get a clearer understanding of what is behind this general method of checking, try “casting out” the fives in the following example. That is, use as a check figure the remainder of each number after dividing it by five:

Five-remainder of answer: 1. Right.

Note that in none of these check figures do we count the answer to any division by the “base” of our check figure. It is only the remainder we watch—because the remainders must stay in order through the calculations. If the remainders do not check out, we know the answer is wrong.?

As a general exercise in number sense, try “casting out” the sevens in the next example. Your check figure in each case is now the
remainder
after dividing by
seven
, and you use the check figures just as you would use digit sums:

Cover up the explanation below with your pad while you do this problem (from left to right, canceling in the answer) and then check your results by dividing each number by seven and using only the remainder as your check figure. Handle the check figures just as you would digit sums.

Here is the working:

Check figure of answer: 4

The one weak point of casting out any single-digit number for checking purposes is that any one digit in your answer that happens to wrong by the exact size of the digit you are casting out will not be caught. “Casting out” (or dividing by) a two-digit number is by nature more accurate. The easiest two-digit number to cast out—which also depends on every digit in the number when casting it out, unlike ten for example—is 11. There are three different ways to test divisibility by 11, or to determine the remainder after a division by 11 to use as a check figure. None of them is quite as simple as adding up the digits (which casts out nines), but with a little practice it goes quite fast.

Dividing By Eleven

In your work with numbers in the past, you may have learned to recognize numbers exactly divisible by eleven because of the pattern they form.?

All two-digit numbers divisible by eleven, for instance, are paired digits—from 11 through 99.

For two-digit numbers, then, you can quickly get the elevens-remainder by subtracting from the number (mentally) the next lower number with paired digits. Here are some examples: