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Tut. Before I give any examples I will direct you how to proceed when the divisor is any number not over 12.

5. When the divisor does not exceed 12, the division may be performed without writing the product of the quotient and divisor. First find how often the divisor is contained in the left hand figures as before : then subtract in your mind the product of the quotient figure and the divisor, from those figures of the dividend used, and consider the remainder as prefixed to the next figure which will form another dividend, with which proceed as before. Proceed in like manner with all the figures of the dividend, setting the quatient under the dividend.

4) 4 2 6 8 5 6

1 0 6 1 4

Here I begin with 4 in the dividend, and find 4 is contained in it once, which I set underneath; then I seek how many times 4 are contained in the next figure, which being smaller than the divisor, I set down 0, and seek how often 4 is contained in 26, and find it is 6 times and 2 over; then after setting down the 6, I consider the 2 placed at the left of 8, which makes 28 ; then I seek how often 4 is in 28, and proceed in this manner with all the figures. Divide 8649 by 6.

Ans. 14414 Divide 6246963 by 8.

Ans. 7808703 Divide 84656 by 12.

Ans. 70544 Divide 3211473 by 27.

Ans. 118943. Divide 1406373 hy 108.

Ans. 13021. 6. When there are ciphers at the right of the divisor, cut them off, and cut off as many figures from the right of the dividend. Then proceed as when there are no ciphers, and the quotient will be the same as if the ciphers had not been cut oft. The figures cut off at the right of the dividend must be considered as a remainder, and placed at the right of the other remainder. Divide 14286 by 2400. 24, 0 0) 1 4 2, 8 6 ( 5

1 2 0

2 2 8 6 remainder.

From what you have been taught you will readily perceive, that Multiplication and Division prote each other. If you wish to prove Multiplication, divide the product by one of the factors, and the .quotient will be the other factor. If you wish to prove Division, multiply the quotient and divisor together, and if there is a remainder, add it to the product, and you will have the dividend.

Divide the following numbers, and prove the work.
18) 6 8 5 7 3 ( 3 8 0 9

3 8 0 9
5 4

18

1 4 5
1 4 4

3 O 4 7 2
3 8 0 9

1 1

1 2 3
1 6 2

* 6 8 5 7 3

1 1

Questions. What is Simple Multiplication ? How many numbers are required to perform an operation

in this rule ? What are these numbers called ? What is the answer called ? How must the factors be placed for Multiplication ? Why do you place the right hand figure of each successive product one place further to the left than the

preceding? When there are ciphers on the right of the significant fig

ures, how do you proceed? How is Multiplication proved ? What is Simple Division ? How many numbers are given to perform an operation ? What are these numbers called ? How do you place them for the operation ? How does it appear that, by multiplying the divisor by

the quotient figures, and subtracting their products from each successive dividend, you get the right answer

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When there are ciphers on the right of the divisor, why

does it not give a wrong answer to cut them off, if you

cut off the same number of figures from the dividend ? How is Division proved?

Examples for Practice. 1. Multiply 684389 by 156.

Ans. 106764684. 2. Multiply 9465287 by 48964. Ans. 463458312668. 3. Multiply 82164973 by 3027. Ans. 248713373271. 4. Multiply 8496427 by 874359.

Ans. 7428927415293. 5. Divide 49561776 by 5137.

Ans. 9648. 6. Divide 4637064283 by 57606. Ans. 80496.

Divide 293839455936 by 8405. Ans. 34960078. 8. Four men enter into partnership; each man puts in $4060; What is the whole amount of stock ?

Ans. $16240. 9. If one acre of land will produce 88 bushels of

corn, how much will 168 acres produce ?

Ans. 14784. 10. What sum of money must be divided among 18 men so that each man may receive $112 ? Ans. 2016.

11. A vessel which had a crew of 875 men captured another vessel, which gave each man $245, prize money; what was the value of the ship?

Ans. $214375. 12. If a man travel 35 miles in one day, how far will he travel in six weeks and three days, allowing six days to a week?

Ans. 1365. 13. If a man spend $4, per day, how much will he spend in one year, there being 365 days in a year?

Ans. $1460 14. A general bas an army consisting of 16 regiments ; each regiment consists of 12 companies, and each company, of 3 officers, 8 non-commissioned officers, 2 musicians, and 64 privates; he has an order to detach 6 noncommissioned officers and privates from each company; after the detachment he has to meet an enemy; how many men has he?

Ans. 13824.

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15. A man raised 64562 bushels of corn on 1565 acres; how many bushels was that per acre ? Ans. 41.

16. What number multiplied by 246 will give 2429629578 ?

Ans. 9876543. 17. There is an estate of $16896 to be divided among 8 heirs.; how much is each share ?

Ans. $2112. 18. if a man's income be $12060 per year, how much is it per day?

Ans. $33. 19. What number must I multiply by 80496 that the product may be 4637064283?

Ans. 57606. 20. A man bought a farm of 425 acres, for which he gave $6840; what was that per acre ? Ans. $165

21. Four men fitted a fleet of 6 vessels for sea; the first vessel and cargo were worth $1648; the second double the first; the third double the first and second ; the fourth was worth as much as the first and third ; the fifth was worth as much as the fourth and half the third ; the sixth was worth as much as the first and fifth ; the fourth vessel and cargo were lost at sea ; the others re." turned, and were sold for three times as much as the whole cost; what was each man's share of the proceeds from the sale ?

Ans. $45732

CONVERSATION III.

VULGAR AND DECIMAL FRACTIONS.

Tut. What do you think Vulgar Fractions are ? Pup. I cannot tell unless they are what the name implies, broken parts of a whole.

Tut. You have formed a very correct idea of them; they are truly parts of a whole, or integer.

1. In division you must have observed that frequently after you had got all the quotient figures, there would be a remainder. This remainder is a fraction. If you divide 17 by 5 you will have 3 for a quotient, and 2 remainder. This 2 shows the number of units there are in 17 over and above 3 times 5. Had 20 been used instead of

17, there would have been no remainder, because 4 times 5 are 20; and had 15 been used instead of 17, there would have been no remainder.

The divisor, in division, expresses the number of units, which one of those parts contains, which a unit of the quotient expresses. In the above example, 5 shows the number of units which a unit or one in the quotient expresses, for 3 in the quotient expresses 3 times 5, because 5 is the divisor, and is contained 3 times in 17. Now in dividing 17 by 5, there is a remainder of 2, which shows how many units there are left, after taking the divisor from the dividend a given number of times. In the example above, the 2 which remains shows that after taking 5 from 17, 3 times, there are 2 fifths of another 5. This remainder or fraction is expressed thus, š, so that the quotient will read 3, three and two fifths. Every fraction, then, consists of two numbers; the first, which shows how many parts of a whole the fraction expresses, is called the numerator ; the second, which shows how many of these parts it takes to make a whole, is called the denominator ; because the denomination of the fraction is deduced from it.

Pup. Then I suppose there are a great number of denominators.

Tut. There are. There is no number which may not be used both for a numerator and denominator.

Pup. If the terms are very large, is it not many times difficult to obtain their value ?

Tut. It would be sometimes if we could not reduce them to less numbers ; for the terms may be reduced to other numbers and retain the same value.

Pup. I cannot see how a number can be reduced, and not alter its value.

2. Tut. Suppose you have it, and you wish to reduce it to lower terms. If you divide each term by 4, you will retain the same value, although the figures are altered, for me are just equal to ž, 4 being one third of 12. Thus the fractions , , , , , &c. are equal

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