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The usual method of testing the work in subtraction is to add the subtrahend and the difference. If the sum is like the minuend, the work is supposed to be right.

CASTING OUT THE 9's*

Another method of proof is sometimes employed known as proof by casting out the 9's. (See page 18.) The excess of 9's in the minuend should exactly equal the sum of the excess of 9's in the subtrahend and the difference together.

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5+8, 9 out = 4; 4 + 3 + 1 + 2, 9 out = excess .
3+4+2, 9 out = 0; 9 out; excess, 5

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2+4+1 + 7, 9 out = excess.

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Solve, and prove by casting out the 9's:

a. From 958436 take 154018.

c. From 879514 take 252192.

b. 83.27-15.436

d. 62.573- 33.3333

e. Find the sum of the four differences.

* This may be omitted if the teacher does not care to have the pupils consider this topic. The proof by casting out the 9's is an instructive figure drill and a practical principle. Some accountants constantly cast out the 9's (or the 11's) as a means of preventing errors. A system for checking up the work by bookkeepers has been devised and sold in some parts of the country, which is nothing more than an ingenious application of this principle.

MULTIPLICATION

NOTE. For definitions of terms see page 340.

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1. Point out the multiplicand, the multiplier, and the product in each of the above examples.

Observe that in written problems in multiplication we usually write the multiplier under the multiplicand, and begin with the right-hand figure of each. In doing such work without the aid of a pencil, it is better to begin with the left-hand figure of the multiplicand. Thus, if required to multiply 4865 acres by 3 without the aid of a pencil, one would say, Three times 4 thousand acres are 12 thousand acres; 3 times 8 hundred acres are 2 thousand 4 hundred acres, which added to 12 thousand acres make 14 thousand 4 hundred acres, etc.

The multiplier is always an abstract number.

The denomination of the product is always the same as that of the multiplicand.

There are 64 primary facts of multiplication (see pages 341, 342). These make the usual multiplication table, beginning with 2 times 2, and ending with 9 times 9. There are other facts of multiplication that should be memorized; but these 64 facts must be mastered as a preparation for solving multiplication problems. The pupils must also know that once 1 is 1, once 2 is 2, etc.; and that 2 times 0 is 0, etc.; but these can hardly be said to require a memory effort, and hence are not included in the primary facts.

Remembering that the writing of one zero to the right of an integral number multiplies the number by 10, and that 20 times a number is 10 times 2 times the number, and that 30 times a number is 10 times 3 times the number, etc., solve the following. Prove by one of the methods given on page 31.

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g. Find the sum of the six products, a to f.

Remembering that the writing of two zeros to the right of an integral number multiplies the number by 100, three zeros by 1000, etc., solve the following.

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n. Find the sum of the six products, h to m.

1. Find the product of 3426 and 57.

OPERATION

3426

EXPLANATION

Seven times 3426 equals 23982. Fifty times 3426 57 equals 171300. Fifty times the number plus 7 times 23982 the number equals 57 times the number. Therefore, adding 23982 and 171300 gives 57 times 3426.

The product of 3426 and 57 is 195282.

17130

195282

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u. Find the sum of the six products, o to t.

1. Find the product of 728.37 and .6.

OPERATION

72-8.37

.6

437.022

EXPLANATION

To multiply by .6, means to take 6 times 1 tenth of the number. One tenth of 728.37 is 72.837. Six times 72.837 equals 437.022.

NOTE 1. The separatrix is used to indicate the place of the decimal point in the number denoting 1 tenth of the multiplicand.

NOTE 2. The decimal point should be written in the product when, in the process of multiplication, the place is reached where it belongs. Do not multiply all the figures and then attempt to determine the place of the point.

2. Find the product of 746.2 and .25.

OPERATION

746.2

.25

37.310 149.24 186.550

EXPLANATION

To multiply by .25, means to take 25 times 1 hundredth of the number.

One hundredth of 746.2 is 7.462. Five times 7.462 equals 37.310. Twenty times 7.462 equals 149.24. 37.310 + 149.24 = 186.550.

NOTE 3. For further direction in regard to the use of the separatrix, see note on page 73 of the Intermediate Arithmetic.

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k. Find the sum of the ten products, a to j.

7. Multiply five hundred thirty-five and six tenths by three and four tenths.

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i. Find the sum of the eight products, a to h.

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r. Find the sum of the eight products, j to q.

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aa. Find the sum of the eight products, s to z.

Multiply:

bb. Six thousand by six thousandths.

cc. Sixty-six thousand by sixty-six thousandths. dd. The sum of seventy-eight thousandths and fortyfive hundredths by their difference.

To the Teacher. It may be unnecessary for all the pupils to solve all the problems on this page. The judgment of the teacher must determine how much of such figure drill is necessary with each class.

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