plier under it, we may say, 5 times 7 are 35, writing down
the 5, and reserving the 3 (tens) as in addition. Again, 5

Multiplicand, 27 trees in each row.
Multiplier, 5 rows.

times 2 (tens) are

10, (tens,) and 3,
(tens,) which we
reserved, make 13,

(tens,) as before.

10. 12. There are on a board 3 rows of spots, and 4 spots in each row; how many spots on the board?

A slight inspection of the figure will show, that the number of spots may be found either by taking 4 three times, (3 times 4 are 12,) or by taking 3 four times, (4 times 3 are 12;) for we may say there are 3 rows of 4 spots each, or 4 rows of 3 spots each; therefore, we may use either of the given numbers for a multiplier, as best suits our convenience. We generally write the numbers as in subtraction, the larger uppermost, with units under units, tens under tens, &c. Thus,

Multiplicand, 4 spots."

Multiplier, 3 rows.

Product,

12 Ans.

Note. 4 and 3 are the factors, which produce the product 12.

Hence, Multiplication is a short way of performing many additions; in other words,-It is the method of repeating any number any given number of times.

SIGN. Two short lines, crossing each other in the form of the letter X, are the sign of multiplication. Thus, 3 x 4 = 12, signifies that 3 times 4 are equal to 12, or 4 times 3 are 12.

Note. Before any progress can be made in this rule, the following table must be committed perfectly to memory.

II. When the multiplier is 10, 100, 1000, &c.

47768.

78192.

T 12. It will be recollected, ( 3.) that any figure, on being removed one place towards the left hand, has its value increased tenfold; hence, to multiply any number by 10, it is only necessary to write a cipher on the right hand of it. Thus, 10 times 25 are 250; for the 5, which was units before, is now made tens, and the 2, which was tens before, is now made hundreds. So, also, if any figure be removed two places towards the left hand, its value is increased 100 times, &c. Hence,

When the multiplier is 10, 100, 1000, or 1 with any number of ciphers annexed, annex as many ciphers to the multiplicand as there are ciphers in the multiplier, and the multiplicand, so increased, will be the product required. Thus, Multiply 46 by 10, the product is 460.

83 100,

...

95 1000,

8300. 95000.

EXAMPLES FOR PRACTICE.

1. What will 76 barrels of flour cost, at 10 dollars a barrel? 2. If 100 men receive 126 dollars each, how many dollars will they all receive?

3. What will 1000 pieces of broadcloth cost, estimating each piece at 312 dollars?

4. Multiply 5682 by 10000.

¶ 13. On the principle suggested in the last T, it follows, When there are ciphers on the right hand of the multiplicand, multiplier, either or both, we may, at first, neglect these ciphers, multiplying by the significant figures only; after which we must annex as many ciphers to the product as there are ciphers on the right hand of the multiplicand and multiplier, counted together.

OPERATION.

Dividend. Divisor, 5) 13,462,725

2,692,545

Quotient,

PROOF.

Quotient. 2,692,545

5 divisor.

In this example, as we cannot have 5 in the first figure, (1,) we take two figures, and say, 5 in 13 will go 2 times, and there are 3 over, which, joined to 4, the next figure, makes 34; and 5 in 34 will go 6 times, &c.

In proof of this example, we multiply the quotient by the divisor, and, as the product is the same as the dividend, we conclude that the work is right. From a bare inspection of the above example and its proof, it is plain, as before stated, that division is the reverse of multiplication, and that the two rules mutually prove each other.

13,462,725

25. How many yards of cloth can be bought for 4,354,560 dollars, at 2 dollars a yard?

at 5 dollars ?
-at 9?

at 3 dollars?

at 6 dollars?

at 10?

at

at

Note. Let the pupil be required to prove the foregoing, and all following examples.

26. Divide 1005903360 by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,

and 12.

27. If 2 pints make a quart, how many quarts in 8 pints?

in 12 pints?

in 248 pints?

in 24 pints in 47632 pints?

in 20 pints?

in 3764 pints?

28. Four quarts make a gallon; how many gallons in 8

29. A man gave 86 apples to 5 boys; how many apples

would each boy receive?

Dividend.

Divisor, 5) 86

Quotient, 17-1 Remainder.

Here, dividing the number of the apples (86) by the number of boys, (5,) we find, that

each boy's share would be 17 apples; but there is one apple

left.

117. 5)86

17

In order to divide all the apples equally among the boys, it is plain, we must divide this one remaining apple into 5 equa.

Then

parts, and give one of these parts to each of the boys. each boy's share would be 17 apples, and one fifth part of another apple; which is written thus, 17 apples.

Ans. 17 apples each. The 17, expressing whole apples, are called integers, (that is, whole numbers.) The (one fifth) of an apple, expressing part of a broken or divided apple, is called a fraction, (that is, a broken number.)

Fractions, as we here see, are written with two numbers, one directly over the other, with a short line between them, showing that the upper number is to be divided by the lower. The upper number, or dividend, is, in fractions, called the numerator, and the lower number, or divisor, is called the denominator.

Note. A number like 17, composed of integers (17) and a fraction, (,) is called a mixed number.

In the preceding example, the one apple, which was left after carrying the division as far as could be by whole numbers, is called the remainder, and is evidently a part of the dividend yet undivided. In order to complete the division, this remainder, as we before remarked, must be divided into 5 equal parts; but the divisor itself expresses the number of parts. If, now, we examine the fraction, we shall see, that it consists of the remainder (1) for its numerator, and the divisor (5) for its denominator.

Therefore, if there be a remainder, set it down at the right hand of the quotient for the numerator of a fraction, under which write the divisor for its denominator.

Proof of the last example.

171
5

86

In proving this example, we find it necessary to multiply our fraction by 5; but this is easily done, if we consider, that the fraction expresses one part of an apple divided into 5 equal parts; hence, 5 times is=1, that is, one whole apple, which we reserve to be added to the units, saying, 5 times 7 are 35, and one we reserved makes 36, &c.

30. Eight men drew a prize of 453 dollars in a lottery; how many dollars did each receive?

Dividend. Divisor, 8) 453 Quotient, 56 answer 56 dollars to each man.

Here, after carrying the division as far as possible by whole numbers, we have a remainder of 5 dollars, which, written as above directed, gives for the and § (five eighths) of another dollar,

¶ 18. Here we may notice, that the eighth part of 5 dollars is the same as 5 times the eighth part of 1 dollar, that is, the eighth part of 5 dollars is of a dollar. Hence, expresses the quotient of 5 divided by 8.

Proof. 56 §

8

453

is 5 parts, and 8 times 5 is 40, that is, 405, which, reserved and added to the product of 8 times 6, makes 53, &c. Hence, to multiply a fraction, we may multiply the numerator, and divide the product by the denominator.

Or, in proving division, we may multiply the whole number in the quotient only, and to the product add the remainder; and this, till the pupil shall be more particularly taught in fractions, will be more easy in practice. Thus, 56 × 8= 448, and 448 +5, the remainder, 453, as before.

*

31. There are 7 days in a week; how many weeks in 365 days? Ans. 524 weeks. 32. When flour is worth 6 dollars a barrel, how many barrels may be bought for 25 dollars? how many for 50 dol lars?

for 487 dollars?

for 7631 dollars?

33. Divide 640 dollars among 4 men.

T19. 41. Divide 4370 dollars equally among 21 men. When, as in this example, the divisor exceeds 12, it is evident that the computation cannot be readily carried on in the mind, as in the foregoing examples. Wherefore, it is more convenient to write down the computation at length, in the following manner :