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several times together in the mind. When we say 3 times 4 are 12, our meaning is, that 4 added together 3 times, make 12; or that 4 and 4 are 8 and 4 are 12. So when we say 4 times 6 are 24, the meaning is, that 6 added together 4 times, make the sum of 24. If it were required to add 3 and 4 together, we could not perform the operation by multiplying, for neither 2 times 3, nor 2 times 4 make 7, the sum of 3 and 4; in this case, we should be obliged to add 3 and 4 to obtain their sum. But were it required to find the sum of two 3's, or two 4's, we could obtain their sum either by adding or multiplying; for we may say, 3 and 3 are 6, and 4 and 4 are 8, or two times 3 are 6, and 2 times 4 are 8; in both cases we obtain the same number, but in the former case, by adding, and in the latter by multiplying. It may at first appear singular to the young scholar, that we are able to add large numbers together in the mind, without setting down the number to be added to itself but once; but his surprise will be removed by reflecting that we have but nine figures and a cipher to represent all numbers, so that by committing to memory the sums of the nine figures, added together any number of times not exceeding 9, we can with great ease, set down the sum of any number, added together any number of times, for we multiply by one figure only at a time.

DEFINITIONS.

A Multiplicand is a number which is to be added to itself several times, or the number to be multiplied.

A Multiplier is a number by which the multiplicand is multiplied.

4 Product is a number found by multiplying the multiplicand by the multiplier, and is always equal to the number which would be obtained if the multiplicand were added to itself as many times as the multiplier contains units. [3 times 4 are 12; here

3 is the multiplier, 4 the multiplicand, and 12 the product.

4 JMultiplication Table is a table exhibiting the sum or product of any number from 1 to 12, added together, or multiplied any number of times not exceeding 12.

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MULTIPLICATION TABLE.

2 times 1 are 25 times 1 are 5R times 1 are 8 2 4 2 10 2 16 3 6 3 15 3 24 4 8 4. 20 4. 32 5 10 5 25 5 40 6 12 6 30 6 48 7 14 7 35 7 56 8 16 8 40 8 64 9 18 9 45 9 72 10 20 10 50 10 80 11 22 11 55 11 88 12 24 12 60 12 96 3 times 1 are 36 times 1 are 69 times 1 are 9 2 6 2 12 2 18 3 9 3 18 3 27 4. 12 4 24 4 36 5 15 5 30 5 45 6 18 6 36 6 54 7 21 7 42 7 63 8 24 8 48 8 72 9 27 9 54 9 81 10 30 10 60 10 90 11 33 1 1 66 11 99 12 36 12 72 12 108 4 times 1 are 417 times 1 are 710 times 1 are 10 2 8 2 14 2 20 3 12 3 21 3 30 4 16 4 28 4 40 5 20 5 35 5 50 6 24 6 42 6 60 7 28 7 49 7. 70 8 32 8 56 8 80 9 36 9 63 9-90 10 40 10 70 10 100 11 44 11 77 11-110 12 -48 12 &l 12 120

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This Table should be perfectly committed to me. mory by every scholar before he attempts to proceed any further, as little progress can be made without it.

When the multiplier consists of one figure only.

RULE.

Set the multiplier under the units’ place of the multiplicand, and multiply every figure in the muliplicand by the multiplier, beginning at the right hand.

Question 1. What is the product of 321, multiplied by 22

Operation. *
Multiplicand 3 2 1
Multiplier 2 Explanation.—In this
question, we say, twice
Product 6 4 2 1 are 2, twice 2 are 4

and twice 3 are 6. 2. What is the product of 54321, multiplied by 22 Ans. 108642. 3. What is the product of 644.312, multiplied by 22 Ans. 1288624.

4. Multiply 9321 by 3. Product. 27.963. 5. Multiply 82122 by 4 Prod. 328488, 6. Multiply 7202 by 4. Operation. Explanation.—When there are ci7 2 0 2 phers in the multiplicand, as they do 4 not represent any number of them

selves, their products cannot be any 2 8 8 0 8 thing but ciphers. The use of plao

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ing ciphers in the product, is to keep the product of the next figure in its proper place. Illustration.—The product is the same number that the sum would be, if the multiplicand were added together as many times as the multiplier contains an unit. When we add, we set down the numbers, and add one column at a time; but when we multiply, we add together in the mind, each figure of the multiplicand, as many times as the multiplier contains an unit, which always shows how many numbers, each equal to the multiplicand, are to be added together; therefore, we shall obtain the same number by multiplying, as we should by adding the multiplicand together as many times as the multiplier contains an unit. If it be required to multiply 432 by 3, we may obtain the answer by adding or multiplying.

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In the first operation, we say, 2 and 2 are 4, and 2 are 6; 3 and 3 are 6, and 3 are 9; 4 and 4 are 8, and 4 are 12. But in the second operation, when we say, 3 times 2 are 6, our meaning is the same, as when we say 2 and 2 are 4, and 2 are 6; and 3 times 3 are 9, is the same as 3 and 3 are 6, and 3 are 9; and 3 times 4 are 12, the same as 4 and 4 are 8, and 4 are 12.

As the sum of each column in the addition, is equal to the product in the same place in the multiplication, the sum must equal the product.

It has been seen, that whether we multiply tens, hundreds, &c. the operations are the same as though we multiplied units; for 2 times 3 hundred make 6 hundred, and 3 times 4 thousand make 12 thousand.

Remark.--From what has been said, it is manifest, that when several things are sold at the same price for each one, the price of the whole may be found by multiplying; for if we set down the price of one article as many times as there are articles, and add up these prices, we shall obtain the price of the whole; but by multiplying the price of one article by the whole number, we obtain the same answer. From this it appears, when any thing is sold, and the price of one unit of the quantity is given, that the quantity will be the multiplier, and the price of an unit, the multiplicand.

7. A man sells 3 bushels of corn, at 32 cents a bushel; how, many cents does he receive for the three bushels?

Operation by adding. Operation by multiplying. 3 2 3, 2

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Explanation.—in this question, the three bushels of corn is the quantity, and must be the multiplier; the 32 cents the price of one bushel, or an unit of the quantity, is the multiplicand. By setting down 32, the price of one bushel, 3 times, and adding, we obtain 96 cents, the price of 3 bushels in one number; but by multiplying 32 by 3, the number of bushels, we obtain the same answer. When the multiplier is a small number, we can readily obtain an answer by adding; but when the multiplier is a large number, it would be very tedious to set down the multiplicand as many times as there are units in the multiplier, and be obliged to add them up. This shows the great utility of multiplication. -

8. What is the price of 4 barrels of flour, at 521 cents a barrel ? Ans. 2084 cents.

9. A farmer has 5 fields of wheat. Each field produces 81 bushels; how many bushels in the 5 fields. Ams. 405.

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