CASE II.

When there are ciphers on the right hand of either the multiplicand or multiplier, or both.

RULE.-Neglect those ciphers; then place the significant figures under one another, and multiply by them only; add them together, as before directed, and place to the right hand as many ciphers as there are in both the factors.

4. What is the value of a farm of 600 acres, at 20 dollars an acre?

5. Multiply 50.750.000 by 75.000.

Ans. 12.000 dollars.

Ans. 3.806.250.000.000.

39. How do you proceed when any of the right hand figures of the multiplicand or multiplier are ciphers?

CASE III.

To multiply by 10, 100, 1000, &c.

RULE.-Set down the multiplicand underneath, and join the ciphers in the multiplier to the right hand of them.*

When the multiplier is a composite number, (or exactly equal to the product of any two figures in the multiplication table)—

RULE Multiply first by one of those figures, and that product by the other, the last product will be the answer.t

* This is evident from the nature of numbers, since every cipher annexed to the right of a number increases the value of that number in a tenfold proportion.

The reason of this rule is obvious; for if a given number be multiplied by any other, and that product again by another, the last product must be the same as if the given number were multiplied by the product of the two multipliers.

40. What is the rule when the multiplier is 10, 100, 1000, &c. ?- -41. When the multiplier is a composite number, what is your rule?

D

CASE V.

To multiply by 9, 99, 999.

RULE.-Annex as many ciphers to the right of the multiplicand as there are figures in your multiplier, and from the number thus produced, subtract the given multiplicand, and the remainder wice be the product.

EXAMPLES.

1. Multiply 5384976 by 9999.

53849760000

There being four 9s in the given multi5384976 plier, add four ciphers (0000) at the right hand; then write the multiplicand underneath, and sub

53644375024 tract according to the rule.

2. Multiply 371967 by 999.

Ans. 371595033.

PROMISCUOUS EXERCISES.

1. What is the product of 4. Multiply 876956 by 990000. 237856, multiplied by 3729?

Ans. 887.099.968.

Ans. 868.186.440.000

2. If 4 bushels of wheat make

5. If a man rise an hour earlier 1 barrel of flour, and the price of every day, how much useful time wheat be one dollar a bushel, what will he gain for study or labour in will 225 barrels of flour cost? 20 years, there being 365 days in

a year ?

3. Multiply 308879 by twenty thousand five hundred and three. Ans. 6.332.946.137.

6. What will be the total product of ninety-eight millions seven hundred sixty-three thousand five hundred and forty-two, multiplied by the same sum?

Ans. 9.754.237.228.385.764,

42. What is the rule for multiplying by 9, 99, &c.?

DIVISION.

DIVISION is the method of finding how many times a less number is contained in a greater; or dividing a quantity given, into any number of parts assigned; and is a concise way of performing several subtractions.

There are four parts to be noted in Division:

1. The Dividend, or number given to be divided.

2.

3.

The Divisor, or number given to divide by.

The Quotient, or answer to the question, which shows how often the divisor is contained in the dividend.

4. The Remainder (which is always less than the divisor, and of the same name with the dividend) is very uncertain, as there is sometimes a remainder, and sometimes none.

SIMPLE DIVISION,

Is the dividing of one number by another, without regard to their values; as 56, divided by 8, produces 7 in the quotient : That is, 8 is contained 7 times in 56.

GENERAL RULE.

1. Draw a curve line on each side of the dividend, and write the divisor at the left hand.

2. Take the same number of the first left hand figures in the dividend that there are in the divisor, if they be equal to, or greater than the divisor; but if they be less than the divisor, take one more; find the number of times the divisor is contained in them, and write a figure representing the number at the right hand of the dividend, which will be the first figure of the quotient.

3. Multiply the divisor by this quotient figure, and write the product under that part of the dividend taken.

4. Subtract this product from that part of the dividend taken, and bring down the next figure of the dividend, and place it at the right hand of the remainder; then find a quotient figure, multiply and subtract as before directed; proceed in the same manner until all the figures in the dividend are brought down and divided.

5. When a figure has been brought down and placed at the right hand of the remainder, if the number be less than the divisor write a cipher in the quotient, and bring down another figure.

43.

What is Division?- -44. How many parts are there in Division, and what -45. What is Simple Division?46. What is your rule?

are they?

PROOF.-Multiply the divisor and quotient together, and add the remainder, if there be any, to the product: If the work be right, that sum will be equal to the dividend.

DIVISION TABLE.

1 2 3 4 5 6 7 8 9 10 11 19

TO USE THE TABLE.

2 4 6 81012141618 20 22 24Look for the divisor or

number by which you wish

91215 18 21 2427 30 33 36 to divide, in the left hand

31

4

162024 28 32 36 40 44 48

5

6

7

8

9

10

11

112

perpendicular column

Then trace the horizontal column in which the

25 30 35 40 45 50 55 60divisor stands, until you find

36 42 48 54 60 66

49 5663 70

77

the dividend or number 72into which you wish to di

vide, then trace that col84umn to the top and you will find the quotient or

6472 80 88 96mumber of times the divi

sor is contained in the div

81 90 99108idend.-If you cannot find the exact number into

100 110 120 which you wish to divide in the table, look for the

121 132 next less one, and the difference between them 144 will be what is over.

EXAMPLES.

1. How many times is 3 contained in 175817? Divisor. Dividend. Quotient.

3) 1 7 5 8 17 (58605

1 5

2 5

24

1 8

1 8

Here we first write down the dividend, and making a curve on each side, place the divisor (3) at the left hand. In this example, we see, that 3, the divisor, cannot be contained in 1, the first figure of the Idividend; therefore we take two figures, (17) and inquire how often 3 is contained therein, which finding to be 5 times, we place the 5 in the quotient, and multiply the divisor by ît, setting the first figure of the mul2 Remainder. tiplication under the 7 in the divi

17

1 5

dend,

47. How do you prove your work to be right?