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135. When the divisor is 15, 35, 45, or 55.

Double the dividend, and divide the product by 30, 70, 90, or 110, as the case may be. (Art. 132.)

Note. This method is simply doubling both the divisor and dividend. We must therefore divide the remainder, if any, by 2, for the true remainder. 27. Divide 1256 by 15.

28. Divide 2673 by 35. 29. Divide 3507 by 45.

30. Divide 7853 by 55. 136. When the divisor is 25.

Multiply the dividend by 4, and divide the product by 100. (Art. 131.)

Note. This is obviously the same as multiplying both the dividend and divisor by 4. (Art. 134. Note 2.) Hence, we must divide the remainder, if any thus found, by 4, for the true remainder. 31. Divide 2350 by 25.

32. Divide 4860 by 25. 33. Divide 42340 by 25.

34. Divide 94880 by 25. 137. To divide by 125.

Multiply the dividend by 8, and divide the product by 1000. (Art. 131.)

Note.—This contraction is multiplying both the dividend and divisor by 8. For the true remainder, therefore, we must divide the remainder, if any, by 8.

35. Divide 8375 by 125. 36. Divide 25426 by 125. 138. To divide by 75, 175, 225, or 275.

Multiply the dividend by 4, and divide the product by 300, 700, 900, or 1100, as the case may be. (Art. 132.)

Note. For the true remainder, divide the remainder, if any thus found, by 4. 37. Divide 1125 by 75.

38. Divide 2876 by 175. 39. Divide 3825 by 225.

40. Divide 8250 by 275. 139. The preceding are among the most frequent and useful modes of contracting operations in division. Various other methods might be added, but they will naturally suggest themselves to the inventive student, as opportunities occur for their application.

41. How long would it take a vessel sailing 100 miles per day to circumnavigate the earth, whose circumference is 25000 miles ? 42. The distance of the Earth from the Sun is 95,000,000 of miles : how long would it take a balloon going at the rate of 100,000 miles a year, to reach the sun ?

43. The debts of the several States of the Union, in 1840, amounted to 171,000,000 of dollars, and the number of inhabitants was 17,000,000 : how much must each individual have been taxed to pay the debt ?

44. The national debt of Holland is 800,000,000 of dollars, and the number of inhabitants 2,800,000 : what is the amount of indebtedness of each individual ?

45. The national debt of Spain is 467,000,000 of dollars, and the number of inhabitants 11,900,000 : what is the amount of indebtedness of each individual ?

46. The national debt of Russia is 150,000,000 of dollars, and the number of inhabitants 51,100,000 : what the amount of indebtedness of each individual ?

47. The national debt of Austria is 380,000,000 of dollars, and the number of inhabitants 34,100,000 : what is the amount of indebtedness of each individual ?

48. The national debt of France is 1,800,000,000 of dollars, and the number of inhabitants 33,300,000: what is the amount of indebtedness of each individual ?

49. The national debt of Great Britain is 5,556,000,000 of dollars, and the number of inhabitants 25,300,000 : what is the amount of indebtedness of each individual ?

50. Divide 467000000000 by 25000000000. 51. 568240:42.

62. 462156; 75. 52. 785372-63.

63. 3562189:225. 53. 896736:- 72.

64. 685726-32000 54. 67234568-5.

65. 723564:175. 55. 34256726;15.

66. 892565--225. 56. 42367581-45.

67. 456212;-275. 57. 16753672-35.

68. 925673: 125. 58. 3256385:55.

69. 763421-175. 59. 45672400:- 25.

70. 8762407-275. 60. 6245634:45.

71. 7825600-80000. 61. 8245623-125.

72. 92004578;100000.

GENERAL PRINCIPLES IN DIVISION.

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

141. If a given divisor is contained in a given dividend a 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. Hence,

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

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

142. 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. Hence,

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

Thus, 8 is contained in 48, 6 times ; in 48:-2 or 24, (half of 48,) 8 is contained 3 times ; (i. e. half of 6 times ;) in 48-3 or 16, (a third of 48,) 8 is contained 2 times; (i. e. a third of 6 times ;) &c.

143. 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. Hence,

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

Thus, 4 is contained in 24, 6 times ; 2 times 4 or 8 is con

Quest.–140. Upon what does the value of the quotient depend? 141. If the divisor re. mains the same, what effect has it on the quotient to multiply the dividend ? 142. What is the effect of dividing the dividend by any given number? 143. If the dividend remains the same, what is the effect of multiplying the divisor by any given number ?

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

144. 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. Hence,

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

Thus, 6 is contained in 36, 6 times ; 6--2 or 3, (half of 6,) is contained in 36, 12 times; (i. e. twice 6 times ;) 6-3 or 2, (a third of 6,) is contained in 36, 18 times; (i. e. thrice 6 times ;) &c.

145. 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. Hence,

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

Thus, 6 is contained in 12, 2 times ; 2 times 6 is contained in 2 times 12, 2 times ; 3 times 6 is contained in 3 times 12, 2 times, &c.

Again, 12 is contained in 48, 4 times; 12;-2 is contained in 48:-2, 4 times; 12+ 3 is contained in 48=-3, 4 times, &c.

147. If the sum of two or more numbers is divided by any number, the quotient will be equal to the sum of the quotients which will arise from dividing the given numbers separately.

Thus, the sum of 12+18=30; and 30:6=5.
Now, 12:6=2; and 18=6=3; but the sum of 2+3=5.

Again, the sum of 32+24+40=96; and 96-8=12.
Now, 32+8=4; 24:-8=3; and 40:8=5; but 4+3+5=12. -

QUEST.-144. What of dividing the divisor? 146. What is the effect upon the quotien If the divisor and dividend are both multiplied, or both divided by the same number?

CANCELATION.* 148. We have seen that division is finding a quotient, which, multiplied into the divisor, will produce the dividend. (Art. 112.) 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. Therefore,

Canceling a factor of any number, divides the number by that factor. Hence,

149. When the dividend is the product of two factors, one of which is the same as the divisor.

Cancel the factor common to the dividend and divisor ; the other factor of the dividend will be the answer. (Ax. 9.) Note.—The term cancel, signifies to erase or reject. 1. Divide the product of 34 into 28 by 34. Common Method.

By Cancelation. 34

34)34 X 28 28

28 Ans. 272 68

Canceling the factor 34, which is com34)952(28 Ans. mon both to the divisor and di

nd, we 68

have 28 for the quotient, the same as be272

fore. 272 150. The method of contracting arithmetical operations, by rejecting equal factors, is called CANCELATION.

Obs. It applies with great advantage to that class of examples and problems, which involve both multiplication and division; that is, which require the product of two or more numbers to be divided by another number, or by the product of two or more numbers. 2. Divide 76 X 45 by 76.

3. Divide 63 X 81 by 81. 4. Divide 65 X 82 by 82.

5. Divide 95 X 73 by 95. 6. Divide the product of 45 times 84 by 9.

* Birk's Arithmetical Collections: London, 1764.

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