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Complement. The difference between a number and 10, or some higher power of 10 (as 100, 1000, 10,000, and so on), is called the complement of that number.

For example, the complement of 3 is 7, of 8 is 2, of 70 is 30, of 48 is 52, of 175 is 825, of 4000 is 6000.

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Complement method of subtraction. If instead of subtracting a number from a larger number, its complement is added to the larger number, the result is too large by some power of 10. If two complements are added, the result is too large by some multiple of 10 (as 20, 30, 200, 300, and so on).

Thus, if instead of subtracting 7 from 9, its complement, 3, is added to 9, the sum is 12, which is 10 too large. If instead of subtracting first 3 and then 2 from 9, the complements 7 and 8 are added to 9, the sum is 24, which is 20 too large.

Subtraction can thus be made a process of adding complements, provided that the proper power or multiple of 10 is discarded from the result.

EXAMPLES. 1. Use the complement method in subtracting 8 from 15.

Solution. The complement of 8 is 2. Then 15+2 = 17, which is 10 too large. Hence the correct difference is 7.

2. Use the complement method in finding the value of the

expression

21-583+6—8—2.

127

Solution. Here there are several numbers to be added and several to be subtracted. Wherever a number is to be subtracted set down its complement with the proper power of 10 as shown at the right. Then adding the left-hand column, the result is 57. But four complements have been added and hence the result is 40 too large. The correct value of the expression is therefore 17.

In business the need for finding the value of such an expression as the one in the second example above is rare. The principle of the complement method is used, however, a great deal where arithmetical operations are performed on adding machines.

EXERCISES

5-10

8

7-10

6

2-10

8-10

57-40

Using the complement method, find the value of each of the following expressions :

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Multiplication. The process of taking (adding together) one of two numbers as many times as there are units in the other is called multiplication; that is, multiplication is, in effect, a short method of addition.

For example, 6 × 4 = 24, but the same result is obtained by adding 6 + 6 + 6 + 6.

The number to be multiplied is called the multiplicand, and the number by which to multiply is called the multiplier. The result obtained by multiplying two numbers is called the product.

The term multiplicand is rarely used in business practice.

ORAL EXERCISES

1. Multiply each of the following numbers: (a) by 2; (b) by

(e) by 3. State only the products:

5; (c) by 4; (d) by 6;

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2. Multiply each of the following numbers: (a) by 2; (b) by 5; (c) by 6; (d) by 8. State only the products:

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Short methods. Only those short methods of multiplication which have a practical value merit consideration here. In addition to the following methods, the student should always be on the alert for short and simple ways to save time in getting products:

1. To multiply by 10, 100, 1000, and so on, annex to the number to be multiplied as many ciphers as there are at the right of the multiplier.

Thus, 654 x 10 = 6540, and 4034 × 1000 = 4,034,000.

2. To multiply by 11, multiply by 10 and add the number multiplied to the product.

Thus, 451 x 11 = 4510 + 451: = 4961.

In multiplying a number of two figures by 11, the following rule is helpful: For the product write the tens' figure of the number multiplied at the left, the units' figure at the right, and the sum of the two between them.

Thus, in multiplying 36 by 11, write 396, the 3 and 6 being written first and 9, the sum of 3 and 6, being inserted between them.

In multiplying 75 by 11, the sum of 7 and 5 is 12, and hence the product should be written 825, adding the 1 (ten) to the 7 before setting it down.

3. To multiply by 101, multiply by 100 and add the number multiplied to the product.

Thus, 635 × 101 = 63,500 + 635 = 64,135.

This rule may be extended to cover multiplying by 102, 103, and so on by adding twice the number multiplied to the product, three times the number multiplied, and so on.

4. To multiply by 99, multiply by 100 and subtract the number multiplied from the product.

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This rule also may be extended to multiplying by 98, 97, and so on by subtracting twice the number multiplied, three times the number multiplied, and so on.

ORAL EXERCISES

Multiply each of the following numbers by 10; by 100; by 1000:

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Checks in multiplication. The most practical check in multiplication is the check of 9's (see page 7). In applying this check, the principle is that the excess of 9's in the product to be checked should equal the excess of 9's in the product of the excesses of 9's in the numbers multiplied.

EXAMPLE. Multiply 458 by 38 and check by casting out the 9's.

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Since the excess of 9's in 17,404 is 7, and that in 16 is also 7, the work is probably correct.

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The chief purpose of Exs. 1-12 above is to give practice in the use of the check of 9's. In the following exercises, in which only the results should be stated, emphasis should be placed upon rapid work and upon the application of short methods wherever possible.

ORAL EXERCISES

1. (a) Multiply each of the following numbers by 7, and to each product add 5:

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(b) Multiply each number in (a) by 4, and to each product

add 8.

2. (a) Multiply each of the following numbers by 5, and from each product subtract 7:

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(6) Multiply each number in (a) by 3, and from each product

subtract 11.

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