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EXAMPLES.

1. Divide 45'5 by 2100.

21.00 ) .455 (0216, &c.

35
• 140

14

2. Divide 41020 by 32000. 3. Divide 953 by 21600. 4. Divide

61 by 79000.

CONTRACTION II.

Hence, if the divisor be 1 with ciphers, as 10, 100, or 1000, &c.; then the quotient will be found and by merely noving the decimal point in the dividend. so many places farther to the left, as the divisor has ciphers; prefixing ciphers if need be.

EXAMPLES.

10 s

So, 217•3 • 100 = 2:173
And 5.16 • 100 =

And 419
And 21 - 1000 =

CONTRACTION III.

When there are many figures in the divisor ; or when only a certain number of decimals are necessary to be retained in the quotient ; then take only as many figures of the divisor as will be equal to the number of figures, both integers and decimals, to be in the quotient, and find how many times they may be contained in the first figures of the dividend, as usual.

Let each remainder be a new dividend ; and for every such dividend, leave out one figure more on the right-hand side of the divisor ; remembering to carry for the increase of the figures cut off, as in the 2d contraction in Multiplication.

Note. When there are not so many figures in the divisor, as are required to be in the quotient, begin the operation with all the figures, and continue it as usual till the number of figures in the divisor be equal to those remaining to be found in the quotient: after which begin the contraction.

EXAMPLES.

1. Divide 250892806 by 92.41035, so as to have only four decimals in the quotient, in which case the quotient will colle tain six figures.

Contracted.

Contracted

Common. 92-4103,5)2508 928,06(27.1498.92 4103,5)2508 928,06 27-1498 660721

66072 06 13849

13848610 4608

46075750 912

91116100 80

79467850 6

5539570 2. Divide 4109.2351 by 230.409, so that the quotient may contain only four decimals.

Ans. 17 8345. 3. Divide 37.10438 by 57: 3.96, that the quotient may contain only five decimals.

Ans. 00649. 4. Divide 913.08 by 2137%, that the quotient may contain only three decimals.

REDUCTION OF DECIMALS.

CASE I.

To reduce a Vulgar Fraction to its equivalent Decimal,

Divide the numerator by the denominator as in Division of Decimals, annexing ciphers to the numerator as far as necessary; so shall the quotient be the decimal required.

EXAMPLES

1. Reduce s to a decimal.
24 = 4 X 6. Then 4) 7

6) 1.750000.

291666 &c.

2. Reduce y, and , and f, to decimals.

Ang. 25, and 5, and .75. 3. Reduce to a decimal.

Ans. .625. 4. Reduce a to a decimal.

Ans. •2. 5. Reduce to a decimal.

Ans. 031350. 6. Reduce 559.

Ans. •143155 &c.

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His to a decimal.

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CASE UL.

To find the Value of a Decimal in terms of the Inferior Deno

minations.

MULTIPLY the decimal by the number of parts in the next lower denomination ; and cut off as many places for a remainder to the right hand, as there are places in the given decimal

Multiply that remainder by the parts in the next lower denomination again, cutting off for another remainder as before.

Proceed in the same manner through all the parts of the integer ; then the several denominations separated on the lefthand, will make up the answer.

Note, This operation is the same as Reduction Descending in whole numbers.

EXAMPLES.

1. Required to find the value of 775 pounds sterling

775

20

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2. What is the value of .625 shil ?

Ans. 7 14. 3. What is the value of .86351 ?

Ans. 178 3.24d. 4. What is the value of .0125 lb troy ? Ans. 3 dwts. 5. What is the value of .4694 lb troy?

Ans. 5 oz 12 dwts 15.744 gr. 6. What is the value of .625 cwt ? Ans. 2 qr 14 lb. 7. What is the value of .009943 miles ?

Ans. 17 yd i ft 5.98848 inc. 8. What is the value of 6875 yd ? Ans. 2 qr 3 pls. 9. What is the value of .3375 acr? Ans. I rd 14 poles. 10. What is the value of •2083 hhd of wine ?

Ans. 13.1229 gal.

CASE

CASE IIL

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To reduce Integers or Decimals to Equivalent Decimals of

Higher Denominations. Divide by the number of parts in the next higher denomination ; continuing the operation to as many higher denominations as may be necessary, the same as in Reduction Ascending of whole numbers.

EXAMPLES.

1. Reduce I dwt to the decimal of a pound troy.

20 I dwt
12

0:05 oz
0-004166 &c. lb, Ans.

2. Reduce 9d to the decimal of a pound. Ans. 03757. 3. Reduce 7 drams to the decimal of a pound avoird.

Ans. 02734375lb. 4. Reduce 26d to the decimal of al. Ans. •0010833 &c. 1. 5. Reduce 2.15 lb to the decimal of cwt.

Ans. 019196+cwt. 6. Reduce 24 yards to the decimal of a mile.

Ans. 013636 &c, mile. 7. Reduce •056 pole to the decimal of an acre.

Ans. .00035 ac. 8. Reduce 1:2 pint of wine to the decimal of a hbd.

Ans. .00238+hhd. 9. Reduce 14 minutes to the decimal of a day.

Ans. •009722 &c. da. 10. Reduce •21 pint to the decimal of a peck.

Ans. .013125 pec. 11. Reduce 28" 12"! to the decimal of a minute. w*..,

Note, When there are several numbers, to be reduced all to the decimal of the highest :

Set the given numbers directly under each other, for dividends, proceeding orderly from the lowest denomination to the highest.

Opposite to each dividend, on the left-hand, set such a number for a divisor as will bring it to the next higher name ; drawing a perpendicular line between all the divisors and dividends.

Begin at the uppermost, and perform all the divisions : only observing to set the quotient of each division, as decimal

parts;

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parts, on the right-hand of the dividend next below it ; so shall the last quotient be the decimal required.

EXAMPLES

1. Reduce 178 91d to the decimal of a pound.

3.
12 9.75
20 17 8125

0.890625 Ans.

2. Reduce 191 178 31d to l. Ans. 19.86354166 &c. l. 3. Reduce 158 6d to the decimal of a l.

Ans. 7752. 4. Reduce 74d to the decimal of a shilling. Ans. 6258. 5. Reduce 5 oz 12 dwts 16 gr to lb. Ans. •46944, &c. Ib.

RULE OF THREE IN DECIMALS.

PREPARE the terms by reducing the vulgar fractions to decimals, and any compound numbers either to decimals of the higher denominations, or to integers of the lower. also the first and third terms to the same name : Then multiply and divide as in whole numbers.

Note, Any of the convenient Examples in the Rule of Three or Rule of Five in Integers, or Vulgar Fractions, may be taken as proper examples to the same rules in Decimals. -The following Example, which is the first in Vulgar Fractions, is wrought out here, to show the method. If f of a yard of velvet cost źl, what will {yd cost ?

yd 1 yd 2 = .375

.375 •4 : : •3125 : •333 &c. or 6 8

375) •12500 (333333 &c,
1250

20
125
8 6.66666 &c.

12

=:3125

Ans. 68 8d.

d799999 &c. = 8d.

DUODE.

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