Speed Mathematics Simplified (32 page)

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 coyping 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.

10

ACCURACY: THE QUICK CHECK

P
RODUCING a quick answer is not always the end of a problem in arithmetic. A wrong answer can be worse than none at all.

When you balance your checkbook, you care whether every stub was done perfectly because otherwise you face an unpleasant hour or so finding out why your balance does not agree with the bank statement. When you make out your income-tax return, you check and double-check every operation to make sure you are not paying too much, or else getting yourself into trouble with wrong arithmetic. And in business, where so many decisions are based on numbers, the wrong numbers can lead to wrong decisions.

There are two fast ways of checking your answers. The faster of the two is the subject of this chapter. A slightly more complex “back-up” check is discussed in the next chapter. Both of them are infinitely quicker than the standard technique of doing the problem over.

The standard way of checking an answer is effective, but very slow. It takes just as much time to tell whether an answer is right as it does to produce it in the first place. The standard way, of course, is to do the problem over again in the opposite way. If we got the answer by adding down, we check by adding up. This (to some extent) keeps us from repeating some habitual error that we might commit twice if we handled the figures in the same order both times. If we subtracted, we check by adding the answer and the smaller number to see if the total equals the larger number. If we multiplied 897 by 123, we check by multiplying 123 by 897. And if we divided, we check by multiplying the answer by the divider, adding the remainder, and seeing if the result equals the number divided.

There are two serious weaknesses to this “backward” method of checking.

First, it is slow and rather boring. Our object is speed and accuracy, with as little boredom as possible.

Second, it is not really a proof at all. If we get the same answer both times, we
assume
the first solution to be correct. Yet if we habitually think of 4 x 7 as 32 (and such habitual mistakes are not uncommon), then we might indeed get the same wrong answer twice. Even if our second try produces a different answer, we still do not know if one of them is right—or which one it is. We must do the problem still a third time.

The techniques of simplified mathematics you have learned are inherently more accurate (because they are simpler) than traditional methods, but it is still unwise to assume an answer is correct unless you
know
it is correct because you have checked it.

The two methods for checking answers you are about to learn are very similar. Neither one is new, although some of the short cuts in applying them are. The first method is known in mathematical circles as “casting out nines” or “the digit sum” method. The second is “casting out elevens.” Both work on a system of check figures completely divorced from your calculations in solving the problem, so habitual errors are unlikely to be repeated, but the methods of deriving the check figures are quite different. Actually, each of them is a way of testing whether the remainders of nine or eleven remain properly constant through your calculations. This will be discussed at greater length in the next chapter. First, learn the technique of handling what is known as the digit sum.

The Digit Sum

The digit sum, as the phrase suggests, is simply the sum of all the digits in a number. This sum will be your “check figure” for each number.

Learn first how to find a digit sum. Then we will go on to the ways of using it. After you have found a few digit sums, you will be able to derive one almost as fast as you can read the number itself. It is really that quick.

If the digit sum is merely the sum of the digits in a number, then the digit sum of 23 should be 2 plus 3, or 5. Odd as this may seem at first, that is precisely right. The digit sum of 23 is 5.

The digit sum of 341 is 3 plus 4 plus 1. The digit sum of 341, then, is 8.

Just add the digits. The digit sum of 42 is—

Did you get 6?

Now, however, it becomes a little trickier. For quick utility, the check figure must always be a single digit. But the sum of the digits in longer numbers goes over ten.

In this case, we use the digit sum of the digit sum. This is the digit sum of the number itself.

This is how it works. The digit sum of 587, for instance, goes into two digits by the time we add 5 and 8, which make 13. When we add the final 7, we have a digit sum of 20.

You can reduce this to a single digit at the end, by adding 2 plus 0 and getting 2. Or you can
reduce as you go along
, like this: 5 plus 8 is 13. Reduce this by adding 1 plus 3 to get 4. 4 plus the final 7 is 11. Reduce this by adding 1 plus 1 and get 2.

This peculiarity of the digit sum is only a foretaste of those to come. Let us finish this thought before getting to that, however. Try one digit sum now. Add all the digits of the number 6934 and then add the digits of the answer until you come out with a single digit. Then reduce as you go along through the same number 6934 and see if you come out with the same final digit sum.

Done the first way, you add 6 plus 9 plus 3 plus 4 and get 22. Reduce this by adding 2 plus 2 to get 4. The digit sum of 6934 is 4.

Done the second way, you add 6 plus 9 to get 15. The digits of this total 6. 6 plus 3 is 9, plus 4 is 13. 1 plus 3 is 4. The digit sum is still 4.

There is, however, a third way. This third way is called casting out nines. The reason for the name is inherent in the digit sum, and is a fascinating byway in the mysteries of numbers.

The odd fact boils down to this: If you divide any number by nine, the remainder is the same as the sum of all the digits of that same number—reduced to one digit by continually adding the digits of the sum of the digits until you wind up with one digit.

In other words, the digit sum of any number divisible by nine will be nine. The algebraic proof of this is a little complicated for this book, but you can demonstrate it for yourself.

Take one of our examples of a minute ago. We found that the digit sum of 587 is 2. If you divide 587 by 9, you will get an answer of 65—and a remainder of 2.

The last example we tried was 6934. Our digit sum was 4. Try dividing 6934 by 9. The answer is 770—and a remainder of 4.

The digit sum is, in essence, the same as the “nines remainder”—the amount left over after an even division by nine. This is important not only to digit sums but in understanding how the entire check-figure system works, A more complete explanation comes in the next chapter.

The fact that the digit sum is the same as the remainder after dividing by nine brings up two more useful oddities. First, nine (for digit-sum purposes) becomes zero. Second, a digit 9 counts for nothing in the number itself.

This brings up a great short cut in deriving digit sums. As you add the digits, simply ignore any nines. They do not count.