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CHAP. IV.

OF DECIMAL ARITHMETIC.

SECT. I.

REDUCTION OF DECIMAL FRACTIONS,

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A DECIMAL fraction is that fraction whofe denominator has an unit in the first place on the left hand, with as many cyphers annexed as neceffary: thus, Too, T, TOOOO! &c. are decimal fractions. But these fractions are usually expreffed in writing without a denominator, by writing the numerator, and prefixing as many points or cyphers before it on the left hand, as there are more places of figures in the denominator than in the numerator. Thus the foregoing fractions are written .5, .25, .048, .0572, and expreffed five tenths, twenty-five hundredths, forty-eight thoufandths, &c.

Cyphers placed on the right hand of decimal fractions make no alteration in their value: thus, if a cypher be annexed to the foregoing fraction .5, it will be then .50 fifty hundredths or half an integer, as before; if two cyphers .500 it will be five hundred thousandths, or, and the fame of the others.

But cyphers placed on the left hand of a decimal fraction decrease their value, every cypher decreafing it in a tenfoid proportion thus .5, .05, .005, are five tenths, five hun dredths, five thoufandths parts refpectively.

The

The value of every figure in a decimal fraction increases in a tenfold proportion from the first place on the right hand,

as in whole numbers.

The figures in the first place of a decimal fraction on the left hand are called primes, thofe in the fecond place feconds, thofe in the third place thirds, &c.

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TABLE VTun the Integer.

Gallons Decimals. Gallons Decimals.

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Hogfhead the

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Integer.

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Avoirdup. Wt. 100 ,396825,30

pts. Dec. Pk|

,476190

09375 3

1lb. the Integer.

90 5357141 20

,317460

80

317460 10

Ounces. Decimals.

70 27772

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,158730 Γ ,03125 1

142857

,126984 Decimals.pecks

21111110234375

095238015625
0793650078125

,063492 Decimals. Pints

,031746,003906

280625 2

3

2

I

2

I

015873,001953

Cafe 1. To reduce a vulgar fraction to a decimal one of the fame value.

Rule. Divide the numerator by the denominator, and the quotient will be the decimal fraction required.

If the numerator be too fmall to be divided by the deno minator, add one or more cyphers thereto, in which cafe as many places must be feparated from the quotient for decimals as the difference of the number of places of decimals in the dividend and the divifor, and the remaining figures in the quotients (if any) are integers.

But if the number of places in the quotient be less than the required number of decimal places, as many cyphers muft be prefixed on the left hand thereof as are necessary. Example 1. What is the decimal fraction of a pound equal to ?

Here

4)300

75

Here two cyphers must be added to the nume rator, which divided by 4 quotes 75, which I mark for decimals, as there are two decimals added to the dividend, and the quotient is the feventy-five hundredth part of a pound, or , equal to 155.

Example 2. What is the decimal fraction of a pound equal to gd.?-Anf. .0375.

As there are but three places of figures 24,0)90000(.0375 in the quotient, I place a cypher on the

left thereof, for there should be four figures pointed off for decimals, being fo many cyphers annexed to 9; thus the quotient is of a pound.

Example 3. What is the decimal fraction of a pound for 3 farthings? Anf. .003125.

Here the vulgar fraction for 3 96,0)300000,0(.003 125 farthings is; I therefore place

as many cyphers as are neceffary to the numerator 3, which in this cafe is 6 cyphers, and the quotient is 3125, but there must be 6 decimals in the quotient, as there are 6 in the dividend and none in the divifor; I therefore prefix two cyphers to the left hand thereof as before directed.

Example 4. What is the decimal fraction of a year for 73 days?-Anfwer.2, equal to 35.

365)730(.2

Cafe 2. To find the value of a decimal fraction in money, weight, or measure.

Rule. Multiply the decimal fraction by the number of parts of the next inferior denomination, and from the product cut off as many figures from the right hand side as there are decimal places in the fraction.

Then multiply these figures fo feparated on the right frand by the number of parts of the next inferior denomination, and from the product cut off as many figures as this laft multiplicand has decimal places, as before.

Proceed in the fame manner through all the denominations, then the separated figures on the left hand fide will be the answer.

Example

Example 1. What is the value of .564 of a pound sterling?

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20 fhillings in a pound

11,280

12. pence in a fhilling

3,300

4 farthings in a penny

1,440

Answer 115. 3 d. 440 of a farthing.

Here I multiply the given decimal by 20, the number of parts of the next lefs denomination, viz. fhillings, and from the product I separate the three - figures on the right hand, 280, as there are three places of decimals in the given number. I then multiply these separated figures 280 by 12, and from the product 3360 I feparate the three first figures 360, as before, to be multiplied by the parts of the next denomination, viz. 4; then the figures ftanding on the left hand of thefe feparated figures will be the answer.-11s. 34d.

The fame rule ferves to discover the value of any other integers, having regard to the number of parts contained in each inferior denomination.

Thus to find the value of a decimal fraction of a pound troy, I first multiply the fraction by 12 to find the ounces, then the separated figures by 20 to find the pennyweights, and lastly by 24 to discover the grains.

Qu. 2. What is the value of .6725 of a cwt. ?—Answer 2qrs. 19lb. 5oz.

Qu. 3.

What is the value of .61 of a ton of wine?Anf. 2hhds. 27galls. 2qts. 1pt.

Cafe 3. To find the value of a decimal of a pound sterling, by inspection.

Rule. Double the first figure of the decimal on the left hand for fhillings; and if the next figure be 5, or above 5, add 1 fhilling more to it: then call the figures in the second and third places farthings (after deducting 5 from the second figure if necessary), and if the number of farthings be above 12 abate one, and if above 37 abate 2. The other figures in the fraction, if more, are neglected.

Example

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