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each figure in the number is thus restored to its original place, and consequently to its original value. Thus, annexing a cipher to 12, it becomes 120, which is the same as 12x 10. On the other hand, removing the cipher from 120, it becomes 12, which is the same as 120-10.

In the same manner it may be shown that removing two ciphers from the right of a number, divides it by 100; removing three, divides it by 1000; removing four, divides it by 10000, &c. Hence,

80. To divide by 10, 100, 1000, &c.

Cut off as many figures from the right hand of the dividend as there are ciphers in the divisor. The remaining figures of the dividend will be the quotient, and those cut off the remainder. 7. How many times is 10 contained in 120 ?

Ans. 12. 8. In one dime there are ten cents : how many dimes are there in 100 cents ? In 250 cents ? In 380 cents ?

9. In one dollar there are 100 cents : how many dollars are there in 6500 cents? In 76500 cents? In 432000 cents ? 10. Divide 675000 by 10000.

Ans. 67 and 5000 Rem. 11. Divide 44360791 by 1000000. 12. Divide 82367180309 by 10000000. Case III.- When the divisor has ciphers on the right.

13. How many acres of land, at 20 dollars per acre, can you buy for 645 dollars ?

Suggestion. The divisor 20 is a composite number, the factors of which are 2 and 10. (Art. 55. Obs. 1.) We may, therefore, divide first by one factor and the quotient thence arising by the other. (Art. 78.) Now cutting off the right hand figure of the dividend, divides it by 10; (Art. 80 ;) consequently dividing the remaining

QUEST.-80. How proceed when the divisor is 10, 100, 1000, &c.

figures of the dividend by 2, the other factor of the divisor, will give the quotient. Operation. Cut off the cipher on the right of the di210)6415

visor; also cut off the right hand figure

of the dividend ; then divide the 64 by 32

2. The 5 which was cut off, is the re

mainder. Ans. 32% acres. Hence, 81. When there are ciphers on the right hand of the divisor.

Cut off the ciphers; also cut off as many figures from the right of the dividend. . Then divide the other figures of the dividend by the significant figures of the divisor, and annex the figures cut of from the dividend to the remainder.

14. How many horses, at 80 dollars apiece, can you buy for 640 dollars ?

15. How many barrels will 6800 pounds of beef make, allowing 200 pounds to a barrel ?

16. How many regiments of 4000 each, can be formed from 840000 ?

17. Divide 143900 by 2100. 18. Divide 4314670 by 24000.

81.a. The four preceding rules, viz: Addition, Subtraction, Multiplication, and Division, are usually called the Fundamental Rules of Arithmetic, because they are the foundation or basis of all arithmetical calculations.

GENERAL PRINCIPLES.

82. From the nature of division, it is evident that the value of the quotient depends both on the divisor and the dividend.

If a given divisor is contained in a given dividend a

Quest.–81. When there are ciphers on the right of the divisor, how proceed? What is to be done with figures cut off from the dividend ? 81. a. What are the four preceding rules called? Why? 82 Upon what does the vulue of the quotient depend?

certain number of times, the same divisor will obviously be contained,

In double that dividend, twice as many times ;
In three times that dividend, thrice as many times, &c.

Thus, 4 is contained in 12, 3 times; in 2 times 12 or 24, 4 is contained 6 times; (i. e. twice 3 times ;) in 3 times 12 or 36, 4 is contained 9 times; (i. e. thrice 3 times ;) &c. Hence,

83. If the divisor remains the same, multiplying the dividend by any number, is in effect multiplying the quotient by that number.

Again, if a given divisor is contained in a given dividend a certain number of times, the same divisor is contained,

In half that dividend, half as many times;
In a third of that dividend, a third as many times, &c.

Thus, 4 is contained in 24, 6 times; in 24-2 or 12, (half of 24,) 4 is contained 3 times; (i. e. half of 6 times ;) in 24 -3 or 8, (a third of 24,) 4 is contained 2 times ; (i.e. a third of 6 times ;) &c. Hence,

84. If the divisor remains the same, dividing the dividend by any number, is in effect dividing the quotient by that number.

If a given divisor is contained in a given dividend a certain number of times, then, in the same dividend,

Twice that divisor is contained only half as many times; Three times that divisor, a third as many times, &c.

Thus, 2 is contained in 12, 6 times ; 2 times 2 or 4, is contained in 12, 3 times; (i. e. half of 6 times ;) 3 times 2 or 6, is contained in 12, 2 times ; (i. e. a third of 6 times ;) &c. Hence,

85. If the dividend remains the same, multiplying the divisor by any number, is in effect dividing the quotient by that number.

QUEST.--83. If the divisor remains the same, what effect has it on the quotient to multiply the dividend ? 84. What is the effect of dividing the dividend by any given number? 85. If the dividend remains the same, what is the effect of multiplying the divisor by any given number?

If a given divisor is contained in a given dividend a certain number of times, then, in the same dividend,

Half that divisor is contained twice as many times ;
A third of that divisor, three times as many times, &c.

Thus, 6 is contained in 24, 4 times ; 6-2 or 3 (half of 6,) is contained in 24, 8 times; (i. e. twice 4 times ;) 6-3 or 2, (a third of 6,) is contained in 24, 12 times; (i. e. three times 4 times ;) &c. Hence,

86. If the dividend remains the same, dividing the divisor by any number, is in effect multiplying the quotient by that number.

87. From the preceding articles, it is evident that any given divisor is contained in any given dividend just as many times as twice that divisor is contained in twice that. dividend ; three times that divisor in three times that dividend, &c.

Conversely, any given divisor is contained in any given dividend just as many times as half that divisor is contained in half that dividend ; a third of that divisor in a third of that dividend, &c.

Thus, 4 is contained in 12, 3 times ; 2 times 4 is contained in 2 times 12, 3 times ; 3 times 4 is contained in 3 times 12, 3 times, &c.

Again, 6 is contained in 24, 4 times; 6:2 is contained in 24-2, 4 times ; 6:3 is contained in 24:3, 4 times, &c. Hence,

88. If the divisor and the dividend are both multiplied, or both divided by the same number, the quotient will not be altered.

89. If any given number is multiplied and the product divided by the sume number, its value will not be altered. Thus 12 X 5=60; and 60-5=12.

Quest.–86. What of dividing the divisor ? 88. What is the effect upon the quotient if the divisor and dividend are both multiplied or both divided by the saine number ? 89. What is the effect of multiplying and dividing any given number by the same number?'

CANCELATION.*

90. We have seen that division is finding a quotient, which, multiplied into the divisor will produce the dividend. (Art. 65.) If, therefore, the dividend is resolved into two such factors that one of them is the divisor, the other factor will, of course, be the quotient. Suppose, for example, 42 is to be divided by 6. Now the factors of 42 are 6 and 7, the first of which being the divisor, the other must be the quotient. Hence,

Canceling a factor of any number, divides the number by that factor. Obs. The term cancel means to erase or reject.

91. When the dividend is the product of two or more factors, one of which is the same as the divisor, the division may be performed by CANCELING that factor in the divisor and dividend. (Art. 89.)

21. Divide the product of 19 into 25 by 19. Common Method. 19

By Cancelation. 25

19)19x25 95

25 Ans. 38

Cancel the factor 19, which is com19)475(25 Ans. mon both to the divisor and dividend, 38

and 25, the other factor of the dividend, 95

will be the quotient. (Art. 90.) 95 22. Divide 85 x 31 by 85. Ans. 31. 23. Divide 76 x 58 by 58. 24. Divide 75 X 40 by '40. 25. Divide 63 x 28 by 7. Suggestion. 28=4x7. We may therefore contract

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Quest.-90. What is the effect of canceling a factor of any number? Obs. What is meant by the term cancel ? 91. When the divisor is a factor of the dividend, how may the division be performed ?

* Birk's Arithmetical Collections, London, 1764.

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