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C.'s hat, and they all scrambled anew for what it contained; of which A. got \, B. Į, D. 2. and C. and E, equal shares of what was left of that stock; D. then struck of what A. and B. last acquired out of their hands : they with difficulıy recovered of it in equal shares again, but the other three carried off }a-piece of the same. Upon this they called a truce, and agreed, that the of the whole left by A. at first should be equally divided among them. How much of the prize, after this distribution, remained with each of the competitors ?

PART III.

XLVI. DECIMAL FRACTIONS.

A DECIMAL fraction is a fraction whose denominator is always unity or 1, with one or more ciphers. Thus, a unit may be imagined to be equally divided into 10 parts, and each of these into 10 more; so that, by a continual decimal subdivision, the unit may be supposed to be divided into 10, 100, 1000, and so on without end, all being equal parts, called tenth, hundredth, or thousandth parts of unit or 1.

In Decimal Fractions, the figures of the numerator only are expressed, the denominator being omitted, because it is always known to consist of a unit with so many ciphers as there are places in the numerator.

A decimal fraction is distinguished from an integer by a point or comma prefixed, thus, ,5 which stands for , or {; ,75 for 700 or ; -2752 for 47o; and 12,010 for 1276 'or 12., &c.

Ciphers at the right-hand of a decimal fraction alter not its value; for, ,5 or”,50 or ,5000 are each of them the same value, and are equal to 5 or ; but ciphers at the left hand, in a decimal fraction, crease the value in a tenfold proportion, for ,05 is táo; also, ,0005 is noiro, &c. all of which will plainly appear by the following

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By the above Table it also plainly appears, that as whole numbers increase towards the left hand by a tenfold proportion, so decimal parts decrease towards the right hand by the same proportion.

A finite decimal is that which ends at a certain number of places; but an infinite is that which no where ends.

A circulating or recurring decimal is that wherein one or more figures are continually repeated.

Thus 64,766666, &c. or 64,76, is called a single circulating or recurring decimal.

And 147,642642, &c. or 147,6+7, is called a compound recurring decimal.

Note. In all operations, if the result consist of several nines, reject them, and make the next superior place a unit

Thus, for 17,1999 write 17,2; and for 12,99 write 13, &c.

XLVII. ADDITION of DECIMALS. Case 1. ADDITION and Subtraction in Decimals are performed after the same manner as Sect. 2, 3, of whole numbers : Care being taken that like parts be placed under one another; and from their sum or difference cut off so many decimal parts as there are the most in any of the given numbers.

EXAMPLES. (1) What is the sum of ,0476, 21,476, ,0067, ,64, 17,0, and

,207647

more.

(2) Add ,427, 64,075, 27,6421, 10,3, ,0074, ,104, and

,046842, together. (3) What is the sum of ,274, ,076, ,64762, ,0706, 947, ,007,

and 968,42 ?

Case 2. To add decimals, wherein there are single repe tends.

RULE. Repeat the circulating decimals till each line has an equal number of decimal places, and ends directly under each other, annexing a cipher or ciphers to the finite terms; then add as before ; only increase the sum of the right hand row with as many units as it contains nines, and the figures in the sum under that place will be a repetend.

EXAMPLES. (4) What is the sum of 47,674, 4,02642, 32,5, 6,14, and

27,0646? (5) Add 11,4, 6,14274, 91,78, 37,671, and 146,476741. (O) What is the sum of 14, 276421, 7,4, 21,646, 9,27, and

31,1474?

Case 3. To add decimals, having compound repetends.

RULE. Make the repetends similar and conterminous ; then add as before, only increase the right-hand figure by as many units as are carried from the column of figures wherein all the repetends begin together; lastly, dash off for a repetend as many places as were so in the numbers added together.

EXAMPLES. (1) What is the sum of 14,471, 768,746, 7,064, and

26,0067? (2) Add ,248, 3,67, 27,0497694, and 9,946, together.

XLVIII. SUBTRACTION of DECIMALS.

EXAMPLES. (1) WHAT is the difference between 176, and 10,764? (2) From 647 take ,00746.

(3) What is the difference between 74,6407, and 69,5 ?

Case 2. To subtract decimals that have a single repetend.

RULE. Repeat the circulating decimals till the lines end together, as in addition ; and if the repetend of the number to be subtracted be greater than the repetend of the number it is to be taken from, then the right-hand figure of the remainder must be less by unity than it would be; or instead of borrowing ten, as in whole numbers or finites, borrow in this place 9, the rest as usual, and the rigint-hand place or figure will be a repetend.

EXAMPLES (4) What is the difference between 41,74, and 21,94648 ? (5) From 24,1 466, take 19,9. (6) What is the difference between 16,176 and 4,1942764?

XLIX. MULTIPLICATION of DECIMALS. Case 1. MULTIPLICATION of Decimals is also performed as in whole numbers, no regard being had to the decimals, as such, till the product is obtained; then observe the foHowing

RULES. 1. Strike off so many figures from the right hand of the product, as there are decimal places in the multiplier and multiplicand.

2. But if there be not so many figures in the product, supply the deficiency by prefixing ciphers to the left hand, to make them equal.

3. If the number is to be multiplied by 10, 1000, &c. remove the separating point in the multiplicand so many places towards the right hand as there are ciphers in the multiplier.

EXAMPLES. (1) Multiply ,17504, by 76. (2) Mul. 27,42, by 3,56. (3) Mul. 8,04704, by ,2575 (4) Mul, 5745, by.,0675..

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