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16. How many times 12 in 172, and how many over? 17. How many times 15 in 630, and how many over? 18. How many times 22 in 865, and how many over? 19. 1236 is how many times 17, and how many over? 20. 7652 is how many times 13, and how many over? 21. 3061 is how many times 125, and how many over? 22. 1861 is how many times 231, and how many over? 23. 8 times 256 is how many times 9? 24. 12 times 157 is how many times 7? 25. 15 times 2251 is how many times 12? 26. 19 times 136 is how many times 75? 27. 63 times 102 is how many times 37 ? 28. 78 times 276 is how many times 136? 29. 115 times 321 is how many times 95? 30. 144 times 137 is how many times 312?

CONTRACTIONS IN DIVISION.

77. a. The operations in division, as well as in multiplication, may often be shortened by a careful attention to the application of the preceding principles.

CASE I-When the divisor is a composite number.

Ex. 1. A gentleman divided 168 oranges equally among 14 grandchildren who belonged to 2 families, each family containing 7 children: how many oranges did he give to each child?

Suggestion.-First find how many each family received, then how many each child received.

Operation.

2)168

If 2 families receive 168 oranges, 1 family will receive as many oranges, as 2 is contained times in 168, viz: 84. But there are 7 children in each family. If then 7 children receive 84 oranges, 1 child will receive as many, as 7 is contained times in 84, viz: 12. He therefore gave 12 oranges to each child

7)84

12 Ans.

NOTE. This operation is exactly the reverse of that in Ex. 1. Art. 55. The divisor 14 being a composite number, we divide first by one of its factors, and the quotient thus found by the other. The final result would have been the same, if we had divided by 7 first, then by 2. Hence,

78. To divide by a composite number.

Divide the dividend by one of the factors of the divisor, and the quotient thus obtained by the other factor. The last quotient will be the answer required.

To find the true remainder, should there be any.

Multiply the last remainder by the first divisor, and to the product add the first remainder.

OBS. 1. If the divisor can be resolved into more than two factors, we may divide by them successively, as above.

2. To find the true remainder when more than two factors are em ployed, multiply each remainder by all the preceding divisors, and to the sum of the products add the first remainder.

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3. A teacher having 36 scholars arranged in 4 equal classes, wishes to distribute 216 pears among them equally. how many can he give to each scholar?

4. How many cows, at 27 dollars a head, can be bought for 945 dollars?

5. How many times is 64 contained in 453?

6. How many times is 72 contained in 237 ?

CASE II-When the divisor is 1 with ciphers annexed to it.

79. It has been shown that annexing a cipher to a number, increases its value ten times, or multiplies it by 10. (Art. 58.) Reversing this process; that is, remo ving a cipher from the right hand of a number, will evi dently diminish its value ten times, or divide it by 10; for,

QUEST.-78. How proceed when the divisor is a composto number? How find the true remainder? Obs. How proceed when the divisor can be resolved into more than two factors? How find the remainder in this case? 79. What is the effect of annexing a cipher to a number? What is the effect of removing a cipher from the right of a number?

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 12x10. 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 10 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?

Analysis. 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. ** ୪

figures of the dividend by 2, the other factor of the divisor, will give the true quotient.

Operation. 210)6415

32-5 rem.

Cut off the cipher on the right of the divisor; also cut off the right hand figure of the dividend; then divide the 64 by 2. The 5 which we cut off, is the remainder. 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 off 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 the 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 IN DIVISION.

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 value 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?

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