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3. Simple Multiplication. SIMPLE MULTIPLICArion is the method of finding the amount of a given number, by repeating it any proposed number of times; as, 6 repeated 4 times, or 4 tinies 6 is 24.*

In Multiplication there must be at least two numbers given to find a third.

The two given numbers spoken of together are called factors. Spoken of separately, the number to be repeated, or multiplied, is called the multiplicand, the number by which it is repeated, or multiplied, is called the multiplier, and the number found by the operation is called the product.

Before the scholar can proceed in this rule, the following table must be thoroughly committed to memory.

Multiplication and Division Table.

11

**************

96

1 2 3 1

6 | 7 | 8 | 9 | 10 12 2 4 6 | 8 | 10 | 12 | 14 16 18 | 20 | 22 | 24 3 61 9 : 2 19 18 | 21 | 24 27 | 30

36 4 1 8 11 21 10 20 2438 32 35 40 44 | 48 *5 10 15 20 | 25 i 3.) | 35 #0

43 | 50)

55 60 6 / 12 18 | 24 30 | 36 | 42 | 48 5+60 66 | 72 7 '14 | 21 25 .;5 42

49 | 50 63 711 | 77 84 8 | 16. | 24 | 32 | 41) 48 | 56 | 64 | 72 | 80 88 9 | 18 1.27 | 36 | 45 | 54 | 6.3 1 72 81 90 99 1108 10 | 20 30 | 10 50 60 70) 80 90 1100 1110 1120 11). 26 | 33 | 44 1 66 | 77 | 88 | 99 1110 1121 |132 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 1108 1120 1132 (144

*** Use of the preceding Table for Multiplication.--Find the multiplier in the left hand column and the multiplicand in the upper line; the product will be found in the line with the multiplier, directly under the multiplicand. Thus 48 the product of 6 and 8, is found in the line with 6 under 8.

For Division. Find the divisor in the left hand column, run your eye along to the right hand till you find the dividend, and right over it in the upper line is the quotient. Thus 48 divided by 6 the quotient is 8.

55

* Multiplication is only an abridged method of addition, and all the questions in Multiplication may be solved by that rule. Thus 6 multiplied by 4, is the same as 4 sixes added together, or 6 x4=6+6+6+6=24 But the solution by addition would be extremely tedious, particularly when the multiplier is a large number.

Rule. * 1. Write the multiplier under the multiplicand, with units under units, tens under tens, and so on, and draw a line under them.

2. Begin at the right hard and multiply all the figures of the maltiplicand se; arately by each figure of the multiplier, setting the first figure of the product directly under the figure of the multiplier, which is employed, and carrying for the tens as in Addition.

3. Add these several products together, and their sum is the total product, or answer required.

Prooft Make the former multiplicand the multiplier, and the former multiplier the multiplicand, and proceed as before; if the product be equal to the former, the product is right.

Examples. 1. Multiply 376 by 4.

2. Multiply 43 by 25. 3 7 6 Here 4 times 6 is 24, 4 which being 2 téns and 4

4 3

2 5 over, write down the 4 and

2 5

4 3 1 5 0 4 say 4 times 7 is 28 and 2 carried, is 30; write a ci

2 1 5

75 pher and say again 4 times 3 is 12,

8 6

100 and 3 carried is 15, which write down, and the work is done.

Ans. 1075

1 0 5

OPERATION,

PROOF.

* When the multiplier is a single digit, multiplying every figure of the multiplicand by that digit, is evidently multiplying the whole by it; and carrying for the tens is only assigning the several parts of the product to their proper places. This must be obvious from the following analysis of the first examples. Multiplicand. 37.6 or 376 Here 4 times 6 is 24, 7 being in the Multiplier.

4

4

place of tens, is 70, and 4 times 70 is 280,

and 3 being in the place of hundreds, is Product. 1 5 0 4

24 300, and 4 times 300 is 1200. Here the 2 80 multiplicand is divided into parts, and 1 2 0 0 each of the parts multiplied by 3. Their

product added together amounts to 1504,

1 50 4 the same as by the rule. Where the multiplier consists of more than 1 digit, it is considered to be divided into as many parts as there are digits, and the whole multiplicand being multiplied by each of these parts, is evidently multiplied by the whole multiplier. The product arising from multiplying by the second figure in the multiplier, or the figure in the place of tens, is ten times as great as it would be if that figure occupied the unit's place, and that arising from the third figure one hundred times as great, and so on, and these values are truly expressed by writing the first figure of each product directly under the figure by which we multiply, as will be evident by inspecting the operation ; hence the sum of the several products is the product of the whole multiplicand into the whole multiplier.

+ This method of proof depends upon the proposition that two numbers being multiplied together, either of them may be made the multiplier, or the multiplicand, and the product will still be the same, which may be thus proved.-Suppose the two factors to be 6 aud 3. 1, 1, 1, 1, 1, 1,

Now if we write three lines of 1s with six is in a line, it is 1, 1, 1, 1, 1, 1, evident that the whole number of ls will make as many 1, 1, 1, 1, 1, 1, times 6 as there are lines, that is, 3 times 6, and as many

times 3 as there are columns, that is, 6 times 3. Hence it

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3. Multiply 37934 by 2. Product.

75868 4. Multiply 357 by 56.

Prod.

19992 5. Multiply 46891 by 325. Prod.

15239575 6. Multiply 653246 by 408. Prod.

2665 24368 7. Multiply 5452176 by 1234. Prod.

670330584 8. Multiply 848329 by 4009. Prod.

3400950961 9. Multiply 99886 by 98. 10. Multiply 6842 by 2486.

CONTRACTIONS. I. When there are ciphers on the right hand of one or both the factors,

RULE. Neglecting the ciphers, proceed as before, and place on the right hand of the product, as many ciphers as were neglected in both the factors.

EXAMPLES. 1. Multiply 3700 by 200. 3 7

Here neglecting the ciphers, I multi2

ply 37 by 2, and annex four ciphers, the

number neglected in the two factors. 7 4 0 0 0 0 2. Multiply 461200 by 72000. 3. Multiply 5036000 by 70300. 4 6 1 2

5 5 3 6 72

7 0 3

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Prod. 3 3 2 0 6 4 0 0000 Prod. 3 5 4 0 3 0 8 0 0 0 0 0

cess

3

is plain that 3 times 6 are the same number of units, or give the same product, as 6 times 3, and the same may be shown of any other two factors.

There are several other methods of proof. The following by division will be found very convenient after becoming acquainted with that rule.

I. Divide the product by the multiplier, and if it be right the quotient will be equal to the multiplicand. Another method much practised, is by casting out the nines.

RULE. II. Cast the pines out of the multiplicand and multiplier ; multiply the two excesses together, cast the nines out of their product and write down the ex

then cast the nines out of the product of the sum, and if the excess be equal to the former, the work is supposed to be right.

I first cast the pines out of the mulMultiplicand. 3 5 7

6

tiplicand and find the remainder Multiplier.

5 6
2

to be 6; then out of the multiplier

and find the remainder 2; these 2 1 4 2 12 3 being multiplied together, the pro1785

duct is 12, and the excess 3. I

then cast the nines out of the proProduct. 1 9 9 9 2

3 duct, and find the excess there to

be 3 also. Hence I conclude the work is right. This method may generally be depended upon, but it is liable to the same inconvenience as in Addition, that a wrong operation sometimes appears to be right, and for the reason mentioued under that rule. There are other methods of proving Multiplication, but these are deemed sufficient.

EXAMPLE.

II. When the multiplier is a composite number. A composite number is one which is produced by the multiplication of other numbers, and the component parts are the nutnbers employed in producing the conposite number. Thus 4 times 6 is 24. Here 24 is a composite number, and 4 and 6 are its component parts.

Rule. *
Multiply by one of the component parts, and that product by the
other.

EXAMPLES.
1. Multiply 2478 by 36. 2. Multiply 8462 by 56.
2 4 7 8 Because 6 times

Ans. 473872. 6 6 is 36

3. Multiply 59375 by 35. ] 4 8 6 8

Ans. 2078125. 6

Pro. 8 9 2 0 8

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III. When the multiplier is 9, 99, or any number of nines.

Rule.t
Annex as many ciphers to the multiplicand as there are nines in
the multiplier, and from the sum thus produced, subtract the multi-
plicand, the remainder will be the product.

EXAMPLES.
1. Multiply 478 by 99. 2. Multiply 6473 by 999.
4 8 0 0

Ans. 6466527.
4 7 8

3. Multiply 99 by 9. Product. 4 7 3 2 2

Ans. 891.

*The season of this rule is obvious ; for in the first example, the product of 6 tis.es 2478 multiplied by 6, is as evidently 36 times 2478, as 6 times 6 is 36. A composite number may have 2, 3, or more component parts. Thus 30 is a composite number whose component parts may be 6 and 5, or 3 and 10, or 5, 3 and 2, &c.

† The reason of this rule will appear by considering that annexing a cipher to. any som, is the same as multiplying it by 10, annexiug two ciphers the same as multiplying by 100, &c. Now when the muitiplier is 9, annexing a cipher to the multiplicand, multiplies it by 10, which repeats it once more than is proposed hy the niultiplier; therefore if we take the multiplicand from this sum, we have the amount of the multiplicand nine times repeated, or the product arising from multiplying by 9. When there are two nines in the multiplier, annesing two ciphers to the multiplicand caultiplies it by 100, which repeats it once more than proposed by the multiplier. Hence, taking the multiplicand once from this sumi, we have the true product arising from multiplying it by 99, and the same reasoning is applicable to any number of nines.

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Application. 1. If a man earn 3 dollars per 8. If a person count 180 in a week, how much will he earn in minute, how many will he count a year, which is 52 weeks ? in an hour?

Ans. 10600. Ans, 156 dollars.

9. A inan had 2 farms, on one 2. If a man thrash 9 bushels be raised 360 bushels of wheat, of wheat per day, how much will and on the other 5 times as much; he thrash in 29 days ?

how much did he raise on both ? Ans. 261 bushels.

Aris, 2160. 3. In a certain orchard there 10. I sold 742 thousand of are 26 rows of trees, and 26 trees i boards at 24 dolls.

per

thousand, in a row; how many trees are what did they come to ? there in the orchard ?

Ans. 17808 dollars. Ans. 676.

11. Says Tom to Dick, You 4. In dividing a certain sum have only 77 chesnuts, but I have of money among 352 men, each seven times as many; how inany received 17 dollars; what was have I ?

Ans. 539. the sum divided ? Ans. 85984.

12. If 4 bushels make a barrel 5. A certain city is divided of flour, and the price of wheat into 12 wards, each ward con be one dollar a bushel, what will sists of 2000 families, and each | 225 barrels of flour cost? family of 5 persons; what is the

Ans. 900 dollars. number of inhabitants in the city? 13. Forty-seven men shared

Ans, 120,000. equally in a prize, and received 6. If a man's income be 1 dol. 25 dollars each ; how much was lar per day, what will be the a

the prize? Ans. 1175 dolls. mount of his income in 45 years, 14. Multiply 308879 by twenallowing 365 days to a year? ty thousand five hundred and Ans. 16425 dollars. three.

Ans. 6332946137. 7. A certain brigade consists 15. An army is drawn up in a of 32 coinpanies, and each com: solid body, and the number of pany of 86 soldiers ; how many rank and file is equal, being 69 soldiers are there in the brigade ? each ; what is the whole number Ans. 2752. of them?

Ans. 4761. QUESTIONS 1. What is Simple Multiplication? 9. What is the method of proof? 2. How many numbers must there 10. When there are ciphers at the

be given to perform the opera right hand of one or both the tion ?

factors, how do you proceed? 3. What are the given numbers call 11. What is a composite number? ed, spoken of together?

12. What are its component parts ? 4. What are they called, spoken of | 13. How do you proceed when the separately?

multiplier is a composite num5. What is the number found by the

ber? operation called ?

14. How do you proceed when the 6. What is the first step in the rule? multiplier is 9, 99, or any num7. What the second step?

ber of pines ? 8. What the third ?.

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