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1. If a book cost 5 cents, what will 4 such books cost?

Addition. 5

4

Ans. 20 cts.

Four books will evidently cost four Multiplicationtimes as much as one book; and to answer the question by Adlition, we should write down 4 fives, and add them, as at the left hand. By Multiplication we should proIceed as at the right hand, thus, 4 times 5 are 20. Now these two operations differ only in the form of expression; Ans. 20 cts. for we can arrive at the amount of 4 times 5 only by a men tal process similar to that at the left hand. Hence, in order to derive any advantage from the use of Multiplication over that of Addition, it is necessary that the several results arising from the multiplication of the numbers below ten, should be perfectly committed to memory. They may be learned from the Multiplication Table, page 19. (16)

2. If one pound of raisins cost 9 cents, what will 7 pounds cost?

84. 3. There are 24 hours in a day; how many hours are there in 3

days?

Addition.

1st day 24 hours. 20 24 hours,

за 24 hours.

24 hours. 3 times.

Three days will evidently contain | Multiplication. 3 times as many hours as one day, or 3 times 24 hours; we may therefore write down 24 three times, and add |

them together, as at the left hand, or Ans. 72 hours. Ans. 72 hours. we may write 24 with 3, the number of times it is to be repeated, under it, as at the right, and say 3 times 4 are 12, (the same as 3 fours added together) which are 1 ten and 2 units. We therefore write down the 2 units in the place of units, and reserving the 1 ten to be joined with the tens, we say, 3 times 2 tens are 6 tens, to which we add the 1 ten reserved, making 7 tens. We therefore write 7 at the left hand of the 2, in the place of tens, and we have 72 hours, the same as by Addition. In Multiplication the two numbers which produce the result, as 24 and 3 in this example, are called factors. The factor which is repeated,as the 24, is called the multiplicand; the number which shows how many times the multiplicand is repeated, as the 3, is called the multiplier; and the result of the operation, as the 72, is called the product.

4. There are 320 rods in a mile; how many rods in 8 miles?

85. 5. A certain orchard consists of 26 rows of trees, and in each row are 26 trees; how many trees are there in the orchard?

Operation.

26

28

156

52

Here we find it impracticable to multiply by the whole 26 at once; but as 26 is made up of 2 tens and 6 units, we may separate them and multiply first by the units and then by the tens; thus, 6 times 6 are 36, of which we write down the 6 units, and reserving the 3 tens, we say 6 times 2 are 12, and 3, which was reserved, are 15, I which we write down, the 5 in the place of tens, and the 676 1 in the place of hundreds, and thus find that 6 of the rows contain 156 trees. We now proceed to the 2, and say 2 times 6 are 12; the 2 by which we multiply being 2 tens, it is evident that the 12 are so many tens; but 12 tens are 1 hundred and 2 tens; we therefore write the 2 under the place of tens, which is done by putting it directly under the 2 in the multiplier, and reserve the 1 to be united with the hundreds. We then say 2 times 2 are 4; both these 2's being in the tens' places, their product 4 is hundreds, with which we unite the 1 hundred reserved, making 5 hundreds. The 5 being written at the left hand of the

2 tens, we have 5 hundred and 2 tens, or 520 for the number of trees in 20 rows. These being added to 156, the number in 6 rows, we have 676 for the number of trees in 26 rows, or in the whole orchard.

86. 6. There are in a gentleman's garden 3 rows of trees, and 5 trees in each row; how many trees are there in the whole ?

1, 1, 1, 1. 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,

We will represent the 3 rows by 3 lines of 1s, and the 5 trees in each row by 5 1s in each line. Here it is evident that the whole number of 1s are as many times 5 as there are lines, or 3 times 5-15, and as many times 3 as there are columns, or 5 times 3-15. This proves that 5 multiplted by 3 gives the same product as 3 multiplied by 5; and the same may be shown of any other two factors. Hence either of the two factors may be made the multiplicand, or the multiplier, and the product will still be the same. We may therefore prove multiplication by changing the places of the factors, and repeating the operation.

SIMPLE MULTIPLICATION.

87. Simple Multiplication is the method of finding the amount of a given number by repeating it a proposed number of times. There must be two, or more, numbers given in order to perform the operation. The given numbers, spoken of together, are called factors. Spoken of separately, the number which is repeated, or multiplied, is called the multiplicand; the number by which the multiplicand is repeated, or multiplied, is called the multiplier; and the number produced by the operation is called the product.

RULE.

88. Write the multiplier under the multiplicand, and draw a line below them. If the multiplier consist of a single figure only, begin at the right hand and multiply each figure of the multiplicand by the multiplier, setting down the excesses and carrying the tens as in Addition. (84). If the multiplier consist of two or more figures, begin at the right hand and multiply all the figures of the multiplicand successively by each figure of the multiplier, remembering to set the first figure of each product directly under the figure by which you are multiplying, and the sum of these several products will be the total product, or answer required. (85)

PROOF.

89. Make the former multiplicand the multiplier, and the

What is Simple Multiplication?
What relation has it to Addition?
How many numbers must there be
given?

What are they called spoken of to-
gether?

What, spoken of separately?
What is the result called?

How must the numbers be written down?

How do you proceed when the mul

tiplier is a single figure? How when the multiplier consists of two or more figures? What is the method of proof?

former multiplier the multiplicand, and proceed as before; if it be right, the product will be the same as the former. (86)

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27. Multiply 848329 by 4009.

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Prod. 3400950961

28. Multiply 64+7001+103-83 by 18+6.

Prod.

170040

29. 49 x 15 x 17 x 12 x 100 how many? Ans. 14994000

CONTRACTIONS OF MULTIPLICATION.

90. 1. A man bought 17 cows for 15 dollars apiece; what did they all cost?

Operation.

17

5

85

3

If we multiply 17 by 5, we find the cost at 5 dollars apiece, and since 15 is 3 times 5, the cost at 15 dollars apiece will manifestly be 3 times as much as the cost at 5 dollars apiece. If then we multiply the cost at 5 dol. lars by 3, the product must be the cost at 15 dollars | apiece.

A number (as 15) which is produced by the multiplicaAns. 255 dolls. | tion of two, or more, other numbers, is called a composite number. The factors which produce a composite number (as 5 and 3) are called the component parts.

1. To multiply by a composite number.

RULE.-Multiply first by one component part and that product by the other, and so on, if there be more than two, the last product will be the

answer.

2. What is the weight of 82 boxes each weighing 42 pounds?

42=6x7 Ans. 3444 lbs.

3. Multiply 2478 by 36.

Product 89208,

4. Multiply 8462 by 56.

Product 473872.

91. 5. What will 16 tons of hay cost at 10 dollars a ton?

It has been shown (73) that each removal of a figure one place towards the left increases its value ten times. Hence to multiply by 10, we have only to annex a cipher to the multiplicand, because all the significant figures are thereby removed one place to the left. In the present example we add a cipher to 16, making 160 dollars for the answer.

6. A certain army is made up of 125 companies, consisting of 100 men each; how many men are there in the whole?

For the reasons given under example 5, a number is multiplied by 100 by placing two ciphers on the right of it, for the first cipher multiplies it by 10, and the second multiplies this product by 10, and thus makes it 10 times 10, or 100 times greater; and the same reasoning may be extended to 1 with any number of ciphers annexed. Hence

What is a composite number?

What is meant by the component parts of a number?

What is the rule for multiplying by

a composite number?

Prod. 3579000.

2. To multiply by 10, 100, 1000, or 1 with any number of ciphers annexed. RULE.-Annex as many ciphers to the multiplicand as there are ciphers in the multiplier, and the number thus produced will be the product. 7. Multiply 3579 by 1000. 8. Multiply 789101 by 100000. Prod. 78910100000.92. 9. What is the weight of 250 casks of sugar, each weighing 300 Here 300 may be regarded as a composite number. whose component parts are 100 and 3; hence to multiply by 300, we have only to multiply by 3 and join two ciphers to the product; and as the operation Ans. 75000 lbs. must always commence with the first significant figure, when the multiplicand is terminated by ciphers, the cipher in that may be omitted in multiplying, and be joined afterwards to the product. Hence

lbs. ?

25
3

3. When there are ciphers on the right of one or both the factors:

RULE.-Neglecting the ciphers, multiply the significant figures by the general rule, and place on the right of the product as many ciphers as were neglected in both factors.

10. Multiply 3700 by 200.

Prod. 740000.

11. Multiply 7830 by 97000. Prod. 759510000.

93. 12. Peter has 17 chesnuts, and John 9 times as many; how many has John?

170

17

Ans. 153.

Here we annex a cipher to 17, which multiplies it by 10. If now we subtract 17 from this product, we have the 17 9 times repeated, or multiplied by 9.

13. A certain cornfield contains 228 rows, which are 99 hills long, how many hills are there?

22800 Annexing two ciphers to 228, multiplies it 100; we then 228 subtract 228 from this product, which leaves 99 times 228';' and in general

Ans. 22572"

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

RULE.-Annex as many ciphers to the multiplicand as there are nines in the multiplier, and from the sum thus produced, subtract the multiplicand, the remainder will be the answer.

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