Speed Mathematics Simplified (37 page)

Here is how it goes: 4 from 6 is 2, from 12 (1 plus 11) is 10, from 18 (7 plus 11) is 8, from 9 is 1, from 8 is 7. 11's remainder, 7.

Take special care to go through any zeros at the end of the number. Zeros after a decimal point do not count (unless followed by another digit), but zeros before a decimal must be included in your calculation. For instance:

You can “feel” what the 11's remainder of this is by mentally subtracting the next-lower two-digit number with paired digits: 99 from 100 is 1. Continuous subtraction, for demonstration, would go like this: 1 from 11 (0 plus 11) is 10, from 11 (0 plus 11) is 1.

Use of Complements

If you have learned your complements thoroughly, you will find that they can speed up this process. You subtract, of course, by adding the complement of the number to be subtracted to the number from which you are subtracting—if the number to be subtracted is larger than the other.

You can make a routine of this for continuous subtraction, with the extra little kicker that you add one
extra 1
each time you use a complement. This gives the same result as adding 11.

Try this technique on this number:

Complement-kicker subtraction goes like this: Complement
of 8 (2) plus 4
plus 1
is 7; complement (3) plus 2
plus 1
is 6; complement (4) plus 5
plus 1
is 10; (no complement) plus 3
plus 1
is 4. 11's remainder, 4.

Checking Addition

Except that you extract your check figures in a different fashion, proving your answers with 11's works precisely the same way as checking with digit sums. Find your 11's remainders, do to them whatever you did to the numbers, and the result must equal the 11's remainder of the correct answer.

When adding, you add the check figures, reduce if need be by casting out the 11's of your total (you can no longer reduce by adding the digits, remember; that is for digit sums only) until you have a final check figure of 10 or less. This is equal to the 11's remainder of the answer.

Follow the checking of this problem step by step:

11's remainder of answer: 2 from 4 is 2 from 8 is 6 from 11 (0 plus 11) is 5;
or
10 (8 plus 2) from 15 (4 plus 0 plus 11 to adjust) is 5.

Try this one on your pad:

Work out the answer and check it with 11's before comparing your results with this explanation:

The check figure of 638 is 0; of 147 is 4; of 269 is 5.
The total of these is 9. The correct answer is 1054, which has a check figure of 9: 1 from 11 (for the 0) is 10, from 16 is 6, from 15 is 9. Or the even-placed digits 5 and 1 total 6, from 4 plus 0 plus 11 (to adjust) is 6 from 15, or 9.

Checking Subtraction

In subtraction, just as in using digit sums, you subtract your check figures to see if the result equals the check figure of your answer. If the check figure of the larger number is smaller than the check figure of the smaller number, add 11 to it before subtracting. If you prefer, add the check figures of the answer and smaller number; this must equal the check figure of the larger number.

Check figure of answer: 9.

Try this one on your pad before looking at the answer and its proof:

Remember to work from left to right and cancel in the answer.

The 11's remainder of the larger number is 5, of the smaller number is 3. 3 from 5 is 2. The check figure of the correct answer, 108352, is 2. Right.

Checking Multiplication

You prove your multiplication answer by multiplying the check figures of the numbers you multiplied to see if the
result—reduced by casting out 11's—equals the check figure of your answer.

Check figure of answer: 3. Try it yourself.

Now carry one through on your own:

Cover the answer and its proof with your pad until you have finished.

The 11's remainder of 735 is 9. The check figure of 48 is 4.9 × 4 is 36, which reduces (3 from 6) to 3. The correct answer is 35280, and has a check figure of 3.

Checking Division

You recall that in checking division with digit sums, you could not divide the digit sums even though you had divided the numbers. This is inherent in all check figures because (with the two remainders we use as check figures) either 9 or 11 is “0.”

Just as in checking with digit sums, you check with 11's by multiplying the check figure of the answer by the check figure of the divider—adding the check figure of the remainder, if any—and seeing if this equals the check figure of the number divided.

Here is an example:

The check figures work like this: 11's remainder of answer (0) times remainder of divider (3) is 0, plus check figure of remainder (2) is 2. The 11's remainder of the number divided is 2. Everything checks out.

Try this one, working out the solution in shorthand division and checking it by casting out 11's: