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by 7, we shall have the result. 567. The reason of this is plain; because 27783 is 7 times the first quotient; and the first quotient, 3969, is 7 times the second quotient. Therefore the given dividend is 49 times the second quotient, or 567 is the 49th part of the given dividend.

7)27783

7)3969

567

The number 7, being seven times less than 49, must be contained in the dividend 7 times oftener. But 7 is contained in 27783 just 3969 times. Therefore 49 must be contained in it the 7th part of 3969 times, or the quotient sought is the 7th part of 3969.

But when this method is employed, we must carefully attend to the management of the remainders. Thus, dividing 5689 by 42, the quotient is 135, with the remainder, 19; and we employ a successive division by 7 and 6, the first quotient will be 812, with the remainder of 5, and dividing that quotient by six, we shall get the quotient 135, with the remainder of 2. But this 2 is to be considered as 2 sevens, or 14; which, added to the former remainder, gives 19 for the true remainder, as before. The reason of this will be plain from considering that by the first division we find that the dividend contains in it 812 sevens: so that any remainder on dividing that 812, must be regarded as of the denomination sevens. *

32. Any number is divided by 10, 100, 1000, &c. by cutting off as many digits from the right hand of the dividend, as there are ciphers in the divisor. The digits thus cut off express the remainder, and the remaining digits of the dividend the quotient.

Thus, dividing 234567 by 1000, the quotient is 234, with the remainder 567. This is manifest, since the dividend is equal to 1000 times 234, with 567 added to the product.

*This may be made quite clear to the youngest student by supposing that we wanted to divide 53 dollars by twelve; that is, to find how many sets of 12 dollars are contained in 53 dollars. Dividing 53 by 4, we find that it contains 13 sets of 4 dollars each, and one over. Every three of this quotient will make a parcel of 12 dollars; and now to find their number, dividing 13 by 3, the quotient is 4, (four parcels of 12 dollars) and one over. But this one is plainly one set of 4 dollars: which added to the former one dollar, gives 5 for the remainder, and 4 for the quotient. Hence appears the reason of the rule which directs us to multiply the remainder on the second division by the first divisor, and add the product to the remainder on the first division.

Hence, it is plain that if the divisor consist of any significant figures, followed by any number of ciphers, we may employ the method of division described in the last article. Thus, if we want to divide 234567 by 7000, we may divide first by 1000 and then by 7; and the quotient will be 33, with the remainder 3567. For when we divide 234 by 7, the remainder of 3 is in fact 3000, and is to be added to the first remainder, 567. And we shall have the same result, (though not so expeditiously) if we first divide by 7, and then by 1000.

When the pupil has had some practice in the methods already explained, he may be taught (as has been already observed) to omit writing the products, which will at least save much room in his operations. This method will be understood from the following example.

Ex. 3. Divide 59122 by 82.*

172

Here, the first figure put in the quotient 82) 59122 (721 is 7; then we say, 7 times 2 are 14, 14 from 21 and 7 remain; 7 times 8 are 56 and 2 (carried) are 58 from 59 and 1 remains. We then bring down 2, and place 2 in the quotient; then, twice 2 are 4, 4 from 12 and 8 remain, twice 8 are 16 and 1 (carried) are 17, 17 from 17 and nothing remains. Bring down 2, and place 1 in the quotient; then 82 from 82 and nothing remains.

82

Although the principle on which the operations in division depends has been already explained in § 29; still the demonstration alluded to may be more clearly understood by the following example.

Ex. 4. Divide 8560 by 36.

*The divisor is placed to the right of the dividend by the French, and the quotient below it, as in the margin. This mode gives the work a more compact and neat appearance, and possesses the advantage of having the figures of the quotient near the divisor, by which means the practical difficulty of multiplying the divisor by a figure placed at a distance from it, is removed. This difficulty every one must have felt, particularly in long operations; and hence this method might, with much propriety, be employed in preference to that which is employed in this country, as well as in England,

574

59122 (82

721

172

164

82

82

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Here, the first part of the quotient is 200, the product of which by 36 is 7200. This taken from the dividend leaves 1360 to be divided by 36. The next part of the quotient is 30; the product of which by 36 is 1080; which still leaves a remainder of 280 to be divided by 36. This gives 7, with the remainder 28. Hence, it appears that 36 is contained 200+30+8 times, or 238 times, with the remainder 28. By comparing this and the common process subjoined, it will be found that the latter is merely an abbreviation of this, the ciphers being omitted in the one and retained in the other.

Exercises in Division of Whole Numbers.

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Ex. 7. 81034-8

8. 41098- I

9. 10340-10 10. 3000711 11. 23456-12 12. 43078-3

Answers
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:20001

=1483843
=1432958

=501260

10041618 =1037030,7% =6395060

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33. 1000000000000000÷÷111 =9000900090001 34. 100000000000000011111=90000900009TTITY

35. Divide 74638105 by 37. 36. Divide 31086901 by 7100. 37. Divide 7380964 by 23000. 38. Divide 2304109 by 5800.

37

Ans. 2017246-3
Ans. 43788.
Ans. 320.

Ans. 3971509

5800

39. Suppose 96000 men are formed into ranks of three deep, what is the number in each rank?

Ans. 32000. Saratoga is 201 army march in Ans. 67 miles. being $38330;

40. The distance from New-York to miles; how many miles each day must an order to arrive at the latter place in 3 days? 41. The annual income of a gentleman how much per day is that equivalent to, there being 365 days in the year? Ans. $104.

42. How many lessons of ninety-five lines each are contained in Virgil's Æneid, the number of lines contained in that poem being nine thousand eight hundred and ninetytwo? Ans. 10413.

43. If it be supposed, as in common circumstances is found to be nearly true, that as many persons die in 33 years as are equal to the entire population; it is required to find how many persons die each year, at an average, in the United States, the population being ten millions two hundred and thirty thousand? Ans. 310000.

44. A prize, worth $20000 is to be divided equally among 25 men; what is each man's part? Ans. $800.

45. It is estimated that there are a thousand millions of inhabitants in the known world: if a number of persons equal to the whole population die in 33 years, how many deaths are there in a year?

Ans. 30303030, and 10 of a remainder.

46. The national debt of England, at present, cannot be less than five hundred millions sterling: how long would that be in paying at the rate of four millions and five hundred thousand pounds a year? Ans. 200 years.

47. The national debt of the United States, at present, [1827] is about seventy millions five hundred thousand and five hundred dollars: how much must the debt be reduced every year, so that it may be all paid off in fifty years? Ans. 1410010..

What is division?

Questions.

What is the divisor?

What is the dividend?

What is the quotient?

Repeat the rule of operation.

If the divisor be the product of two or more known factors, how is the operation of division performed?

If the divisor consist of unity with one or more ciphers, how is the operation performed?

CHAPTER II.

PRACTICAL APPLICATION OF MULTIPLICATION AND DIVISION TO THE REDUCTION OF MONEY, WEIGHTS, MEASURES, &c.

33. REDUCTION is the method of converting quantities from one name or denomination to another of the same value; and it is divided into Reduction Descending, and Reduction Ascending.

When quantities of a higher denomination are to be brought to a lower, it is called Reduction descending, and it is performed by multiplication.

When quantities of a lower denomination are to be brought to a higher, it is called Reduction ascending, and it is performed by division.

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From the first initiation of the youthful student into multiplication and division, he ought to be led to the practical

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