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Thus, to multiply 436 by 324 according to the rule, is the same as to multiply separately by 4 units, 2 tens, and 3 hundreds—

2. 436 Here it will be seen that 436 x 4 = 1744 324 the result of multiplying X 20 = 8720

is the same in both cases. 1744

X 300 = 130800 In No. 2, the figures have 872

141264 their proper position given 1308

to them, according to the 141264 rank of the multiplier, by

means of nothings annexed,

and in No. 1 by putting each line of figures a place further to the left, which serves the same purpose as the nothings.

PROOF.—To prove that any question in multiplication is correctly performed, we may reverse the operation, by making the multiplicand the multiplier; and if the product is the same, the account is correct.

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For example, if we multiply 36 by 27, the product is 972 ; and if we multiply 27 by 36, the product will also be 972.

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Multiply the following numbers :

Answers. 1. 18530729 by 21 42

389145309 778290618 2. 43915806 v 37 26

1624884822 1141810956 3. 70268315 51 84

3583684065 5902538460 4. 92573684 1 78 89

7220747352 8239057876 5. 39753984 147 96

1868437248 3816382464 6. 48637251 73 67

3550519323 3258695817 7. 83974695 1 89

7473747855 1931417985 8. 18370298 57 98

1047106986 1800289204 9. 95721386 1 79 89

7561989494 8519203354 10. 84397857 1 75

6329839275 51060703485 11. 39260489 38 406

1491898582 15939758534 12. 17935982 , 207 98

3712748274 1757726236 13. 31694708 78 59

2472187224 1869987772 14. 89357064 65 91

5808209160 8131492824 15. 39712584 1 456 789

18108938304 31333228776 16. 80379218 1 372 958

29901069096 77003290844 17. 92873905 " 837 325

77735458485 30184019125 18. 61938796 504 982

31217153184 60823897672 19. 54917283 1 271 659

14882583693 36190489497 20. 93650389 · 948 326

88780568772 30530026814

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Multiply

Answers. 21. 41705826 by 516 486 21520206216 20269031436 22. 18472593 i 785 827 14500985505 15276834411 23. 85149260, 6521 61594 | 555258324460 5244683520440 24. 52816937, 3298 73261 174190258226 3869421621557 25. 29583604 » 19695 4938 582649080780 146083836552 26. 96250371 1 86372 84768313337044012815818144596 27. 63927048 , 9328 60745 596311503744 3883248530760 28. 30694715, 8975 32164 275485067125 987264813260 29. Two factors are 30957 and 839; what is their product ?

Ans. 25972923 30. How many oranges are there in 47 boxes, when each box contains 279 ? .

. . Ans. 13113 31. In a desk there were 6 drawers, each drawer was divided into 8 compartments, and in each compartment were 87 pounds; how many pounds did the desk contain ?. . Ans. 4176

III. WHEN THE MULTIPLIER IS A NUMBER HAVING nothings

ANNEXED TO IT. RULE.-Extend the nothings beyond the multiplicand, then write them down as part of the answer, and multiply by the other figure or figures of the multiplier, as in Rules I. and II.

Examples.—Multiply 7348 by 70; 47683 by 70600; and 87300 by 760.

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NOTE.-WHEN THE MULTIPLIER is 10, 100, 1000, or 1 with any other number of nothings annexed, the multiplication is accomplished merely by annexing as many nothings to the multiplicand as are contained in the multiplier ; thus

386 is multiplied by 10, by annexing 1 nothing = 3860
736
, 1000,

3 nothings = 736000

IV. WHEN THE MULTIPLIER IS A FRACTION-AS ), or **

RULE.—Multiply the given number by the upper figure of the fraction, and divide † the product by the under figure.

To multiply a number by a fraction, such as 3, 4, &c., is to find what is the half or three-fourths of the number: thus-to multiply 16 by ,, is the same thing as to find how much is of 16.

† For the process of division, see page 20. Example.-Multiply 16 by 1. 16

Here 16 is multiplied by 3 and then 4148

divided by 4. 12 Answer. When the upper figure of a fraction is 1, such as }, it is unnecessary to multiply by the I, as it will obviously make no difference on the number; all that is required is to divide the given number by the under figure; thus-to multiply 24 by j, we divide 24 by 3, and the answer is 8—that is, 8 is one-third of 24.

3

V. WHEN THE MULTIPLIER IS AN INTEGER OR whole NUMBER,

WITH A FRACTION ANNEXED-AS 8%. RULE.-Multiply first by the integer, then by the fraction as in Rule IV., and add both the products for the answer. Example.-Multiply 4387 by 75. 4387 x 7

4387 x
Here 3655%, the product
30709
of multiplying by $, is

6)21935
added to the product of
36555
multiplying by 7.

36555
34364.Answer.

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* See explanation of Fractions, under that head, p. 67.

SIMPLE DIVISION.

Division is that process by which we discover how often one number is contained in another, or by which we divide a given number into any proposed number of equal parts. We can ascertain, by aid of the Multiplication Table, how many times any number is contained in another, as far as 144, or 12 times 12, without writing figures; but the calculations beyond this point are usually written down.

The number to be divided, is called the dividend; the number by which it is divided, is the divisor; and the result is termed the quotient, from a Latin word, which means, literally, How many times ?

Division is denoted by the following character • ; thus75 = 25, signifies that 75 is to be divided by 25.

RULES FOR DIVISION.
I. WHEN THE DIVISOR DOES NOT EXCEED 12.

Rule.-1. Write down the dividend; draw a curved line on the left side of it, and a straight line below it: then write the divisor on the left of the curved line.

2. Find how often the divisor is contained in the first figure, or (if the divisor is larger than it) in the two first figures of the dividend, and write the quotient below.

3. Multiply the divisor by the quotient, and deduct the product from the figure or figures just divided ; then annex to the remainder, if any, the next figure of the dividend, find how often the divisor is contained in this new sum,* and write down the quotient; and so on as before, till all the figures of the dividend have been divided, when the division is completed.

If there is a remainder after the division is finished, it is marked. down as part of the answer, with the divisor written below it, forming a fraction.

* NOTE.-If, after annexing a figure to the remainder, the number be less than the divisor, place a nothing in the quotient to express this, then annex another figure to the remainder, and proceed with the division.

The process of division here described, is termed Short Division when part of the process is carried on in the mind, and the results only written down : short division is employed when the divisor does not exceed 12. In numbers above 12, it is necessary, for convenience, to write down at length the various steps of the process; and when this is done, it is termed Long Division. The principle is the same in both cases, the sole difference being, that in the one, the operation is only partly written down, whilst in the other, all the figures of the process

19
18

are written; as will be seen in the following example, in which both methods are given :

Example.Divide 7958 by 6. Short Division.

Long Division. 6)7958

Here we find that there 6)7958(13263 Quot. 13262 Quot is one 6 in 7, the first

figure of the dividend, and
1 over : we therefore write

1 in the quotient, and multiplying the divisor, 6, by 1, the quotient, 15 subtract the product, 6, from 7 of the dividend.

12 To the remainder, 1, we bring down 9, the next figure of the dividend, making 19. As there are 3 times 6 in 19, we place 3 in the quotient,

36 and multiplying the divisor, 6, by 3, subtract

2 = å 18 from 19, which leaves 1. To this I we bring down 5, making 15; and as there are 2 times 6 in 15, we place 2 in the quotient, and multiplying 6 by 2, subtract 12 from 15, leaving 3. To this 3 we bring down 8, making 38, in which there are 6 sixes; therefore, placing 6 in the quotient, we multiply 6 by 6, and subtract 36 from 38, leaving 2 over. Here the account terminates, it being found that there are 1326 sixes in 7958, with a remainder of 2, below which the divisor is written, thus-, and the fraction is annexed to the quotient as part of the answer.

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Exercises. Divide the following numbers :

Answers. 1. 398654 by 2 3 4 199327 1328844 996632 2. 87564329 à 5 6 7 175128654 14594054 125091899 3. 876437639 • 2 4 8 4382188191 219109409 1095547041 4. .

9 11 12 973819596 79676149 730364691}

II. WHEN THE DIVISOR EXCEEDS 12.

Rule.-1. Draw a curved line on each side of the dividend, and place the divisor on the left of it.

2. Point off as many figures from the left of the dividend as make a number greater than the divisor: find how often the divisor is contained in this number, and place the result at the right side of the dividend, as the first figure of the quotient. It is to be observed, that 9 is the highest number to be placed in the quotient at any one time.

3. Multiply the divisor by the quotient, and subtract the product from the number pointed off, then bring down and annex to the remainder, if any, the next figure of the dividend.* Proceed

* NOTE.-If, after bringing down a figure to any of the remainders, during the process, the number be less than the divisor, place a nothing in the quotient to express this, then bring down another figure to the remainder, and proceed with the division.

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