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ADDITION OF DECIMALS. ROLE. In setting down the proposed numbers to be added,

great care must be taken in placing every figure directly underneath those of the same value, whether they be mixed rumbers, or pure decimal parts; and to perform which there must be a due regard had to the commas or separating points which ought always to stand in a direct line, one under another, and to the right hand of them carefully place the decimal parts, according to their respective values; then add them as in whole numbers.

EXAMPLES.

1. Add 72,5+32,071+2,1574+371,4+2,75. Facit 480,8784. 2. Add 30,07 +2,0071 +59,432 +07,1. 3. Add 3,5+47,25+927,01 +2,0073+1,5. 4. Add 52,75+47,21 +724 +31,452+,3075. 5. Add 3275+27,511 +1,005+725+7,32. 6. Add 27,5+52 +3,2675+,5741 + 2720.

SUBTRACTION OF DECIMALS. RULE

ULE. Subtraction of Decimals differs but little from whole

numbers, only in placing the numbers, which must be carefully observed, as in Addition.

EXAMPLES.

1. From ,2754 take ,2371
2. From 2,37 take 1,76
3. From 271 take 215,7
4. From 270,2 take 75,4075

5. From 571 take 54,72 6. From 625 take 76,91 7. From 23,415 take ,3742 8. From ,107 take ,0007

MULTIPLICATION OF DECIMALS. RULE. Place the factors, and multiply them, as in whole

numbers, and from the product towards the right-hanıl, cut off as many places for decimals as there are in both factors together ; but if there should not be so many places in the product, supply the defect with cyphers to the left hand.

EXAMPLES.

1. Multiply ,2365 hy ,2435 Facit ,05758775. 2. Multiply 2,071 by 2,27 7. Multiply 27,35 by 7,70071 3. Multiply 27,15 by 25,3 8. Multiply 57,21 by ,0075 4. Multiply 72347 by 23,15 9. Multiply ,007 by ,007 5. Multiply 17105 by ,3257 10. Multiply 20,15 by ,2705 6. Multiplý 17105 by 0237 | 11. Multiplý ,907 by ,0025

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When any number of decimals is to be multiplied by 10, 100, 1000, &c. it is only removing the separating point in the multiplicand so many places towards the right-hand as there are cyphers in the multiplier ; thus, ,578X10=5,78. ,578 X 100=57,8. ,578 x 1000=578. 578 X 10000=5730.

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CONTRACTED MULTIPLICATION OF DECI.

MALS.

under place of the multiplicand that is intended to be kepė in the product, then invert the order of all the other figures, i. e. write them all the contrary way; then in multiplying, begin at the figure in the multiplicand, which stands over the figure you are then multiplying with, and set down the first figure of each particular product directly one under the other, and have a due regard to the increase arising from the figures on the right-hand of that figure you begin to multiply at in the multiplicand

Nole, "That in multiplying the figure left out every time next the righishund in the multiplicand, if the product be 5, or upwards, to 15 corry 1; if 13, or upwards, to 25, carry 2 ; and if 25, or upwards, to 35, carry 3, &c.

EXAMPLES 12. Multiply 384,672158 . by 36,8345, and let there be only four places of decimals in the product. Facit 14169,2065. Contracted Way.

Common Way. 384,672150

284,672158 5438,63

36,8345

115401647
23080329
3077377
115402
15387
1923

1923|360790 15386 88632 11510116174 307737726+ 23080329 48 115401647,4

14169,2065

14169,2065 038510

Pacit ,1166. 13. Multiply 3,141592 by 52,7438, and leave only 4 places of deciinals.

Facit 165,6994. 14. Multiply 2,38645 by 8,2175, and leave only 4 places of decimals.

Facit 19,6107. 15. Multiply 375,13758 by 16,7324, and let there be only ! place of decimals.

Facit 6276,9 16. Multiply 375,13758 by 16,7324, and leave only 4 places of decimals,

Fucit 6276,9520. 17. Multiply 395,3756 by ,75642, and let there be only 4 places of decimals.

Fucit 899,0700.

DIVISION OF DECIMALS.

difficulty is in valuing the quotient, which is done by any of the following rules :

RULE 1. The first figure in the quotient is always of the same value with that figure of the dividend, which answers or stands over the place of units in the divisor.

2. The quotient must always have so many decimal places, as the dividend has more than the divisor.

Note 1. If the divisor and dividend have both the same number of decimal parts, the quotient will be a whole number.

2. If the dividend hus not so many places of decimals as are in the livisor, then so many cyphers must be annexed to the dividend as will nuke them equul, und the quotient will then be a whole number.

3. But if, when the division is done, the quotient has not so many figures as it should huve places of deciinals, then so muny cyphers must te prefixed us there are places wanting.

EXAMPLES.

1. Divide 85643,825 by 6,321. Facit 13549,09428 +. 2. Divide 48 by 144.

7 Divide 7382,54 by 6,4252 3. Divide 217,75 by 65. 8 Divide ,0851618 by 423. 4. Divide 125 by ,1045. 9 Divide 267,15975 by 13,25. 5. Divide ,709 by. 2,574. 10 Divide 72,1564 by 1347. 6. Divide 5,714 by 8275. 11 Divide 715 by 30,75.

When 'umbers are to be divided by 10, 100, 1000, 10,000, &c. it is pt. formed by placing the separating point in the dividend so many places towards the left hand, as there are eyphers in the divisor. Thus, 6784; 10=578,4 5784 1000=-5,784.

5784---100=57,84 5784--10000=,5784 X.

CONTRACTED DIVISION OF DECIMALS. RULE. By the first rule find what is the value of the first

figure in the quotient ; then by knowing the first figure's denomination, the decimal places may be reduced to any number, by taking as many of the left-hand figures of the dividend as will answer them; and in dividing omit one figure of the divisor at cach following operation. Note.

That in multiplying every figure left out in the divisur, you must carry 1, if it be 5, or upwards, to 15; if 15 or upwards, to 25, carry 2 ; if 25, or upwards, to 35, carry 3, &c.

EXAMPLES.

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12. Divide 721,17562 by 2,257432, and let there he only places of decingals in the quotient.

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Contracted.

Common Way. 2,257432)721,17562(319,467 2,257432)721,17582(319,407 6772296

6772296

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RULE. Add cyphers to the numerator, and divide by the

denominator, the quotient is the decimal fraction required

EXAMPLES.

1. Reduce

.to a decimal. 4)1,00(25 Facit. 2. Reduce

.to a clecimal. Facit ,5. 3. Reduce ......to a decima!. Facit ,75. 4. Reduce .......tv a decimal. Facit ,375. 5. Reduce

....to a decimal. Faeit 1923076+. 6: Reduce of 10 ......to a decimal. Facit ,6013956+.

Note. If the given parts be of several' denominations, they may be radused either by so many distinct operations as there are diferent pars, op by the first reducing ihem into their loreest venuminatić!, and !567 Javide us before ; or,

2dly, Bring the lowest into decimals of the nest superior nomine tion, ard on the right hand of the decimal found, place the pasts given of the next superior denomination ; so proceeding till you bring out the decimal parts of the highest integer required, by still dividing the product by the next superior denominator ; or,

3dly, so ronder pence, shillings, and farthings. If the number of shillings be cven, take half for the first place of decimals, and let the second and third pl«ccs 'be filled up with the farthings contained in the remaining pence and farthings, always remembe: ing to add 1, when it is or exceeds 25. But if the number of shillings be odd, the second place of decimals must be increased by 5. 7. Reduce 5s. to the decimal of a £.

Facit , 25. 8. Reduce 9s. to the decimal of a £.

Facit ,45. 9. Reduce 16s. to the decimal of a £.

Facit ,j. 10. Reduce 8s. 4d, to the decimal of a £. Facit ,4166. 11. Rexluce 16s. 73d. to the decimal of a £. Facit ,8322916. first.

second.

third. 165. 74.

4)3,00

2)16 7용 12

12)7,75

,832 199

39 210)16,64583

960)799(,8322916 ,8322916

12. Reduce 19s. 5dı. to a decimal of a £. Facit ,972916. 13. Reduce 12 grains to the decimal of a ib. troy.

Facit ,002083. 14. Reduce 12 drams to the decimal of a lb. avoirdupoise.

Facit ,046875. 15. Reduce 2 qrs. 14 lb. to the decimal of an cwt. Facit ,625. 16. Reduce two furlongs to the decimal of a league.

Facit ,0833. 17. Reduce 2 quarts, 1 pint, to the decimal of a gallon.

Facit ,625. 18. Reduce 4 gallons, 2 quarts of wine, to the decimal of an hogshead.

Facit ,071428+. 19. Reduce gallons, 1 quart of beer, to the decimal of a barrel.

Facit ,0625. 30. Reduce 52 days to the decimal of a year. Facit ,142465+. To find the vulue of any Decimal Fraction in the known parte

of an Integer. Rule. Multiply the decimal given by the number of parts of the next interior denomination, cutting off the decimals from the product; then multiply the remainder by the next inferior denomination; thus proceeding, till you have brought in the least known parts of an Iuteger,

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