CHAP. IX.

DECIMAL FRACTIONS.

INTRODUCTION.

THE practice of collecting units into tens, has univerfally pre

vailed; it has therefore been found convenient to follow a like method in fractions, by dividing an unit into ten equal parts, and each of thefe into ten fimilar parts, which are called DECIMAL

FRACTIONS.

A decimal fraction is expreffed by writing only the numerator with a POINT before it; the denominator being an unit with as many cyphers as the numerator has places. The figure next the point fignifies tenth parts, the next, hundredth parts, the next, thousandth parts, and so on; as in the following examples. .5 fignifies Five tenth parts.

.05

.005

723

6.500

Five hundredth parts.

Five thoufandth parts.

723. thousandth parts.

Six and five tenth parts.

From these examples it appears, that the value of decimals depends upon their diftance from the POINT, and therefore each cypher placed on the left of decimals diminishes their value ten times; but when cyphers are placed on the right, they have no fignification, as the distance of the figures from the point is not thereby

altered.

When a vulgar fraction can be reduced to a decimal without a remainder, the decimal is called FINITE, or TERMINATE; if it cannot, the decimal is called INFINITE, OF INTERMINATE.

Sect. I. REDUCTION OF DECIMALS.

PROBLEMI. To reduce vulgar fractions to decimals; annex cyphers to the numerator, and divide by the denominator.

If to form a dividual, it be neceffary to annex more cyphers than one to the numerator, prefix as many to the figures of the quotient. This occurs in example 2d.

Ex. 1ft, Red. to a decimal. 50)17.00(.34 finite

150

200

200

Ex. 2d. Reduce

16)1.0000

.0625 finite alfo.

The numerator of a vulgar fraction-is understood to be divided by the denominator, and this divifion is actually performed when it is reduced to a decimal. Hence the reason of the rule is evident. Thus in example ift, 1700 hundredth parts by 50, as above.

or 17 divided by 50, is equal to that is 34 hundredth parts, or .34,

Interminate decimals may be diftinguished into, It, REPEATING decimals, when the quotient repeats the fame figure; 2d, CIRCULATING decimals, when two or more figures return in their order. If the figures repeat or circulate from beginning as in example, 3d and 5th, they are called pure repeaters, or circulates; but if there be figures before them, as in examples 4th and 6th, they are called mixt repeaters, or circulates; and the figures before the repeating ones are called the Finite part.

Repeaters are generally marked either with a point, or a dash, and circulates, with a point above the first and last figures of the circle.

PROBLEM II. To reduce lower denominations to decimals of higher. Annex cyphers to the lower denomination, and divide it by the value of the higher. If there be feveral denominations, begin at the lowest, and reduce them in their order to the highest.

In example ift, To reduce d. to a decimal, we annex a cypher, and divide by 4 the value of a penny, which gives 25. We next prefix the 8d. to it, and divide by 12 the value of a fhilling, and it comes to .6875, and then prefix 125. and divide by 20, the fhillings in L.1, the quotient is the decimal required. In example 2d, we divide by the component parts of 24, the decimal we obtain is 3 a repeater; and in dividing by 28 and 12, the repeating figure is annexed mentally.

The following examples for practice are reverfed by those in the next problem.

PROBLEM III. To value a decimal fraction. Multiply it by the value of the different denominations, fucceffively pointing off from the right, as many places as in the given decimal. The figures pointed off give the answer.

When the given decimal ends with a repeater, carry by 9 when you multiply the repeating figure, and when a cypher occurs on the right of the multiplier, annex the repetend for it: This is done in example 2d.

To value the decimal of a pound fterling by Inspection. Double the first decimal place for fillings, if the fecond place be 5 or above, reckon a fhilling more, then the remainder of the fecond with the third is farthings, abating 1 for 25, and 2 for 40.

In example ift, we proceed thus: twice 4 is 8s. and 23 'far. is 51d. In example 2d, twice 6 is 128. and becaufe it is above 25 we abate 1, which makes 84d. In example 3d, because (8) the fecond decimal place exceeds 5, we add to twice 7, which gives 155. then we take 5 from 8, the remainder is farthings, abating as before, and it comes to 8d. In example 4th, the third place is fuppofed to be filled with a cypher, which makes 30. and abating 1, is 29 far. or 74d. In example 5th, the fourth decimal place is rejected, being lefs than farthings.

Because 960 farthings make a pound, and the rule fuppofes 1000, we therefore abate 1 for 25. The answer obtained by it, unlefs the decimal terminate in 25, 50 or 75, a little more than the true value; for there should be abatement not only of 1 for 25. but also of on the remaining figures. To correct this we abate 2 when the farthings amount to 40 or upwards.

PROBLEM IV. To reduce finite decimals to vulgar fractions. Place the given decimal for the numerator, and a unit with as many cyphers annexed as there are figures in the decimal, for the denominator.

Ex. .125

125 and when reduced to its lowest terms is .

1000

As the arithmetic of circulating decimals is of little or no use in bufinefs, we shall only give rules by which they may be reduced to vulgar fractions, by which any queftion where they occur may be wrought more eafily than by circulating decimals.

RULE I. If the decimal be a pure repeater or circulate, place 9 for the denominator of repeater, and 9 for every figure in the circle. II. If the decimal be mixt (whether repeating or circulating) subtract the finite part from the whole decimal. The remainder is the nume rator; place 9 below each figure of the circle, with a cypher for every finite place, for the denominator.

Ex. Ift, 6= or when reduced . 2d. 8i a pure circ, or r

z.

=

If be reduced to a decimal it will produce a pure repeater 44 and fince is the decimal equivalent to, 2; 3 and fo on till 9; hence every repeating figure is the numerator of a fraction whofe denominator is 9.

The reafon of this will appear by dividing the decimal into its finite and circulating parts. Thus in example 4th, the finite decimal is 9, and the circle .027; but 9, and .027, fhould be provided the circulation began immediately after the place of units; but as it begins after the place of tenths it is

27

27

of or The fum of the two vulgar fractions fhould be the value of the decimal fought. To add which, we reduce them to a common denominator. Thus,, and add to it, the fum is or, which is the fame as by the rule.

27

918

9989

RULE I. Write down the numbers, placing the decimal points under each other.

CASE I. When the decimals are finite, find their fum as in integers.