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296.14364
48517.21

29614364
5922873
2073005

29614
14807
2369
118

3765.7150 2. Multiply 743.56815 by 52.647, and let there be only 3 places of decimals in the product. Ans. 39146.632.

3. Multiply 17.14 by 62.197, reserving only the integers in the product.

Ans. 1066. 4. Multiply .7164 by 12.1, reserving 3 places of deciinals in the product.

Ans. 8.668. 5. Multiply 1.0034 by 799.99, reserving only 2 places of decimals in the product.

Ans. 802.72.

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CONTRACTED DIVISION OF DECIMALS.

Rule. Find what place of integers or decimals the first figure of the quotient will possess, and consider how many quotient figures will serve the present purpose; then take the same number of the left hand of the divisor, and as many of the dividend as will contain them, (less than ten times, rejecting the rest; then, instead of bringing figures down from the dividend, separate one from the right of the divisor as often as necessary, till the whole be exhausted, remembering to carry from the right hand figures of the divisor, as in contracted multiplication.

When there are not so many figures in the divisor, divide as usual, till there be as many of the quotient figures found as the divisor is short of the intended quotient; then use the contraction.

Examples 1. Divide 642.27541 by 3.671265, and let there be only four places of decimals in the quotient.

3.671265)642.27541(174.9466 Ans.

3671265

2751489
2569886

181603
146851

34752
33041

1711
1468

243 220 23 22

1 2. Divide 2508.928065051 by 184.8207, so as to have 4 places of decimals in the quotient.

Ans. 13.5749. 3. Divide 43.538163 by 4.6827035, and let there be 7 places of decimals in the quotient. Ans. 9.2976552.

4. Divide 1254.46403 by 46.205175, and let there be 4 places of decimals in the quotient.

Ans. 27.1498. 5. Divide 3765.715 by 296.14364, and let there be 5 places of decimals in the quotient.

Ans. 12.71584. 6. Divide 39146.632 by 743.56815, and let there be three places of decimals in the quotient.

Ans. 52.647.

CONTRACTED DIVISION. The following contracted method of dividing, being taken notice of in few works that I have seen, I have chosen to deliver it by itself.

Rule. Set down the sum after the usual manner; find the first quotient figure, and multiply the divisor as usual, but instead of setting down the product, subtract the product of each re

spective figure, from the figure above, borrowing as many as necessary, which must be carried to the product of the next figure ; bring down the figures as necessary for a dividual, and thus proceed to the end.

Examples. 1. Divide 59143684 by 627812.

627812)59143684 (94 quotient.

2640604

129356 remainder. Explanation.-I find the first quotient figure is 9, and say 9 times 2 are 18, take 18 from 18 and O is left; I set this down under the 8, and carry 1; then 9 times 1 are 9, and 1 that I carry make 10; take 10 from 16 and 6 are left; I set this down, and carry 1; then I say 9 times 8 are 72, and 1 that I carry make 73; take 73 from 73, and 0 is left; then 9 times 7 are 63, and 7 that I carry make 70; take 70 from 74, and 4 are left; then 9 times 2 are 18, and in that I

carry are 25; take 25 from 31, and 6 are left; then 9 times 6 are 54, and 3 that I carry are 57; take 57 from 59 and 2 are left; I then bring down the next figure (4) for a new dividual, and proceed in every respect as before. 2. Divide 7854 by 67.

67)7854(117 quotient.

115
484

15 remainder. 3. Divide 61427 by 121.

121)61427(507 quotient.

927

80 remainder. 4. Divide 7157264 by 23144.

23144)7157264(309 quotient.

214064

5768 remainder. 5. Divide 96215496 by 514217.

514217)96215496(187 quotient.

4479379
3656436

56917 remainder.

6. Divide 623917842 by 7219543.
7219543)623917842(86 quotient.

46354402

3037144 remainder. 7. Divide 543 by 17.

Quotient 31 ; rem. 16. 8. Divide 7259 by 72.

Quotient 100; rem. 59,

EQUATION OF PAYMENTS.

CASE I. To find the equated time for the payınent of a sum of money due at several different times.

Rule 1. Find the present worth of each payment, for its respective time; add all the present worths together, and deduct the sum from the sum of the payments; then divide this remainder by the product of the sum of the present worths and the ratio: the quotient will be the true equated time.

RULE 2. Multiply each several payment by the time it has to run: then divide the sum of the products by the sum of the payments; the quotient will be the equated time, nearly.

Examples. 1. A. owes B. $1800, whereof $200 is to be paid at 6 months, $400 at 9 months, and $1200 at 20 months—At what time may the whole debt be paid together, rebate being made at 6 per

cent. ? $

S $ As 103 200 :: 100 : 194.174 present worth of the

first payment. 104.50 : 400 :: 100 : 382.775

2d paym't. 110 : 1200 :: 100 : 1090.909

3d paym't.

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$

1667.858 sum of present worths. 1800.000 sum of the payments. 1667.858 sum of the present worths.

132.142 discount.

1667.858 sum of the present worths.

.06 ratio. 100.07148 divisor.

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10800

10800 Note.--Rule 2 is more compendious than rule 1, but cannot be depended upon as sufficiently accurate,

2. D. owes E. $1200, which is to be paid as follows, viz. $200 down; $500 at the end of 10 months, and the rest at the end of 20 months; but they agreeing to have an equal payment of the whole, the true equated time is required, rebate at 6 per cent.

Ans. By rule 1, 12 mo. 7 days.

By rule 2, 124 months. 3. A merchant has owing to him $200 to be paid as follows, viz. $100 at 3 months, $100 at 4 months, and the rest at 8 months; but it is agreed to make one payment of the wholeWhen will that time be, rebate at 6 per cent. ?

Ans. By rule 1, 5 mo. 21 da.+

By rule 2, 59 months. 4. G. owes H. $800, of which $500 are to be paid present, and the rest at 8 months; but they agree to make one payment of the whole, and wish to know the time, rebate at 6 per cent. ?

Ans. By rule 1, 2 mo. 28 da.

By rule 2, 3 months.

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