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name, then divide by such number as will reduce it to the next greater, always annexing ciphers to the dividend, as occasion may require thus proceed till it be reduced to the decimal of the required integer. Or, reduce the given parts to a simple quantity, by reducing them to the lowest name mentioned; annex ciphers thereto, and divide by such numbers as will reduce them to the name required. Or, reduce the given parts to a vulgar fraction, and that fraction to a decimal.

EXAMPLE 1.-Reduce 17s. 10d. to the decimal of a pound sterling. 0.5+10d.

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10.500 .875+178.=

17.875
20

=.89375.

12

the decimal required.

EXAMPLE 2.-Reduce 2 feet 9 inches to the deci

mal of a yard.

Vulgar fraction. 3, and 33.0000 .9166 as required.

36

=

To find the value of any given decimal.

RULE.-Multiply the decimal given by the number of parts of the next inferior denomination, cutting off the decimals from the product; then multiply the remainder by the next inferior denomination; thus proceeding till you have brought the least known parts of the integer.

EXAMPLE 1.-Required the value of 89375 of a pound sterling.

.89375
20

17.87500

12

10.50000

2

1.00000 or, 17s. 10 d.

EXAMPLE 2.-Reduce .625 of a hundred weight to its proper terms.

.625 × 4 = 2.500 × 28 14.000, or 2 quarters and 14 lbs.

ADDITION OF DECIMALS.

RULE. Arrange the numbers under each other, according to their several values; find the sum, as in addition of whole numbers, and cut off for decimals as many figures to the right hand as there are decimals in any one of the given numbers.

EXAMPLE. What is the sum of 23.45, 7.849, 543.2, 8.6234, 253.004 ?

23.45 7.849

543.2

8.6234

253.004

836.1264

.75

.6666

.8888

.875

.4444

3.6250

If any of the decimals be repetends, continue them beyond the others, and make them end together; then in adding, increase the sum of the first column by as many units as there are nines therein; as,

Here the first sum 18 contains two nines; therefore 2 added to 18 20. The rest of the work is the same as usual in others; the repetend is 0, so the sum is finite.

If some of the decimals be repetends, and others circulates, continue them both beyond those that are finite, and till their periods end together; then to the sum of the first column add as many as would arise to carry to it if they were continued farther; so will you have a circulate in the sum. Thus,

2.5

7.696969

The repetend of .6, the circulate of 3.666666 69 and .372, continued till their periods end together. It may easily be observed that there would be I to carry to the first 28.286008 column if it were carried any farther.

14.372372

C

NOTE.-It is not always necessary to attend to the rule for repetends and circulates; three or four decimal figures, according to the rule, being sufficiently near the truth for common calculations.

SUBTRACTION OF DECIMALS.

RULE. Place your numbers according to their value, subtract as in whole numbers, and cut off for decimals, as in addition.

EXAMPLE.-Subtract 35.87043 from 132.005.

132.005 35.87043

If both be single repetends, make them end together; and if there be occasion to borrow at the first figure, borrow 9 96.13457 only instead of 10; thus,-.8333

.6666

.1666

If both be circulates, or one a repetend and the other a circulate, continue both till their periods end together; then if there should be occasion to borrow at the following figure, were they continued that figure farther, carry 1 to the first figure; and if the numbers be in different denominations, reduce them till they be alike.

Subtract 334 from 13; thus,

1.666666

.834834

.831831

MULTIPLICATION OF DECIMALS.

RULE. Place the factors under each other, and multiply them together, as in whole numbers; then point off as many figures from the product (counting

from right to left) as there are decimal places in both factors; observing, if there be not enough, to annex as many ciphers to the left hand of the product as will supply the deficiency.

EXAMPLE.-Multiply .4375 by .125.

.4375

Here the product of 4375 by .125 is .125 546875; but as there are three places of decimals in the multiplier, and four in the 21875 multiplicand, a cipher must be added on the left hand of the product to reduce it to its proper terms.

8750

4375

.0546875

To multiply a repetend by a single figure, add 1 to the first product for every 9 therein, so will you have a repetend in the product; and if there be several figures in the multiplier, do so with each product, and continue them till they end together; then add them as so many repetends.

If the multiplicand be a circulate, consider the increase that would arise to the first product if the multiplicand was continued farther: thus do with each product, make them end together, and add them by the rule for adding circulates.

To Contract the operation, so as to retain only as many Decimals in the Product as may be thought necessary.

RULE. Place the unit figure of the multiplier under that figure of the multiplicand whose place is the last to be retained in the product, and dispose of the rest so that they may all stand in contrary order to that in which they are usually placed.

Then, in multiplying, reject all the figures to the right hand of the multiplying digit, and set down the product, so that the right hand figures may fall in a straight line under each other; observing to increase the first figure of every line with what would arise by carrying I from 5 to 14,-2 from 15 to 24,-3 from 25

to 34, &c., from the product of the two preceding figures, when you begin to multiply; and the sum will be the product required.

EXAMPLE.-Multiply 27.14986 by 92.41035.

Common way.
27.14986
92.41035

13 574930
81 44958

2714 9860

108599 44

542997 2

24434874

2508.9280 650510

Contracted way:

27.14986

53014.29

24434874

542997

108559

2715

81

14

2508.9280

DIVISION OF DECIMALS.

RULE.-Prepare your decimals as directed for multiplication, divide as in whole numbers, cut off as many figures for decimals in the quotient as the number in the dividend exceeds the number in the divisor, namely, make the number of decimal figures in the divisor and quotient together equal to the number in the dividend.

EXAMPLE. Divide 173.5425 by 3.75. 3.75)173.5425(46.278

150.0

2354
2250

1042
750

2925

2625

3000

3000

Although you may take additional ciphers at pleasure, care must be had in reckoning the number taken in dividing for decimals in the dividend ; and if you put the decimal point in the quotient at any part of the operation, continuing the operation afterwards will not cause the point to be removed.

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