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The figure below the line shows into how many parts the apple is divided, and is called the denominator. The figure above the line, shows how many parts are contained in the fraction, and is called the numerator.

From the above, we derive the following definition :

A fraction is a part of any thing. It is derived from the Latin word frango, which signifies to break. When, therefore, any thing is broken into parts, those parts are called fractions. If a stick be broken into pieces, each piece becomes a fraction of the whole. Fractions are of two kinds, decimal, and vulgar, or common. Vulgar, when the unit is divided by any of the nine digits; that is, a vulgar fraction may express any assignable part of a unit, thus, 2, 3, &c., are vulgar fractions. But when the unit is divided by 10, or into parts expressed by 1 with any number of ciphers, as 10, 100, 1000, &c., it is called a decimal fraction.

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DECIMAL FRACTIONS.

The term decimal signifies tenths. It is derived from the Latin word decem, which signifies ten. It is, therefore, applied to all fractions whose denominator is 10, or 1, with any number of ciphers. If a dollar be divided into ten parts, one of these parts, being worth ten cents, is one tenth of a dollar. If the dollar be divided into one hundred parts, one of these parts is the one hundredth part of a dollar. It is, nevertheless, a decimal fraction, because 100 is the product of 10's. The same may be said of a thousand, or ten thousand. A fraction is always known to be decimal, if its denominator be ten, a hundred, or a thousand. The denominator of a decimal fraction is not always expressed, but it can always be ascertained by the numerator. If it contains but one figure, the denominator is ten; if two, it is a hundred, &c. It is always 1, with as many ciphers annexed as the numerator has places.

When the denominator is not expressed, the fraction is distinguished from a whole number by a comma placed at the left of it. [The comma is called the separatrix.] Example: 5,50 is read five tenths, fifty hundredths, as though they were written, If the numerator have not so many places as the denominator has ciphers, supply the defect by prefixing ciphers, thus: for T, write,05; Too write ,005. Ciphers placed at the right hand of a deci

mal, do not affect its value, as, are the same in value; for while the addition of the cipher indicates a division into parts ten times smaller than the preceding, it makes the decimal express ten times as many parts. Thus, 5 tenths denotes 5 parts of a unit, which is divided into 10 parts; and 50 hundredths denotes 50 parts of a unit, which is divided into 100 parts. It is, therefore, plain, that the value is not altered, since 5 is half of 10, and 50 is half of 100.

The value of a decimal depends upon its distance from the unit's place; as whole numbers increase from the unit's place towards the left in a tenfold proportion, so decimals, in the same ratio, decrease from the unit's place towards the right hand; as will appear from the following TABLE.

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From the above Table it is evident that each figure, whether a whole number or decimal, takes its value from the unit's place. If it be in the first place after units, it is tenths; if in the second, it is hundredths, &c. Consequently, every decimal will have for its denominator 1, with as many ciphers as the decimal is distant from the unit's place; thus-2 in the Table is ; 3 is 80; 4 is Too, &c.

Although ciphers placed at the right hand of a decimal. fraction do not affect its value, yet, placed at the left, they

QUESTIONS. On what does the value of a decimal depend? In what proportion do decimals decrease from the units' place towards the right? From what does each figure in the table take its value? What is the value of the first figure at the right ?

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05

diminish it in a tenfold proportion, by removing the significant figure so much farther from the unit's place. As,5,05,005 express different values, viz.-,5 is,,05 is %, ,005 is 100%. As whole numbers are written units under units, tens under tens, from right to left, so decimals are written tenths under tenths, from left to right.

Example 1. Write 2 tenths; 3 hundredths; 4 thousandths; 6 ten thousandths.

,2

,03

,004

,0006

2. Write twenty-nine thousandths; three hundred and fourteen thousandths; five ten thousandths,

and sixty-seven millionths.

,029

,314

,0005

,000067

3. Write five tenths; five hundredths; fifty thousandths, and forty-nine; one hundred thousandths, and sixteen thousandths.

ADDITION OF DECIMALS.

RULE.

Place the numbers so that the decimal points may stand directly under each other. Add as in whole numbers; observing to point off as many places for decimals in the amount, as will be equal to the greatest number of decimals in any of the given numbers.

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QUESTIONS. What is the rule for addition of decimals?

4. Add thirty-five and four tenths; five hundred twentynine and seven millionths; sixty-nine, four hundred and sixty-three thousandths; two hundred, sixteen and two hundredths; seventy-seven, nine hundred and two tenths. Answer, 1827,083007.

5. Add forty-nine and sixty-seven hundredths; six hundred seventy-nine, two hundred seventy-five thousandths; one thousand four hundred, fifty-five thousandths, nine hundred and ninety-nine millionths.

6. Add 249,39-6712,9123-6,3219-2739,235-5,671

-723,2674-926,679-72,601.

7. Add ,7+9, 2+, 321+279,+4, 67+349, 2+3, 956. 8. Purchased of one man 325,5 lbs. of beef-of another, 175,75-of another 178,028; what was the amount?

9. I receive of A $183,25-of B $138,89——of C $372,218—of D $88,99—of E $139,29; what is the amount of the whole?

10. Add $59,67-8158,355-$375,752-8167,375$567,756

SUBTRACTION OF DECIMALS.

RULE.

Write the numbers and point the result, as in Addition of Decimals, and subtract as in whole numbers.

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5. From two hundred and sixty-nine and three tenths, take fifty-seven and thirty-nine hundredths.

Answer, 211,91. 6. Take twenty-four thousandths from nine hundredths.

Ans. ,066.

7. Take sixty-five millionths from five tenths. 8. From three hundred seventy-five thousand and three tenths, take two hundred forty-nine and thirty-nine one hundred thousandths. Ans. 374751,29961.

9. From 361,2 take 276,75.

QUESTIONS. Rule for the subtraction of decimals?

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Multiply as in whole numbers, and point off as many places for decimals in the product as there are decimal places in both factors.

If there are not so many places, supply the defect by prefixing ciphers.

EXAMPLES.

1. Multiply 49,5 by 3,2. 2. Multiply 569,39 by 27,05. 3,2

99,0 1485,

158,40

3. Multiply 6,791 by 2,67.

4. Multiply 549,05 by 35,257.

5. Multiply six hundred and seventy-five by twentyseven and three tenths.

6. Multiply sixty-seven thousand by three hundredths. 7. Multiply 34,56 by 1,3. 8. Multiply 674,49 by 37,16. 9. Multiply 5648 by 6,78.

10. Multiply 7864 by 467.

11. Multiply fifty-seven and three tenths by twenty-nine. 12. Multiply thirty-seven thousand by three hundredths. 13. Multiply fifty thousand and seven tenths by four hundredths.

14. Multiply sixty-nine and five tenths by three thousandths.

DIVISION OF DECIMALS.

RULE.

Divide as in whole numbers, and point off so many places in the quotient for decimals as the decimal places in the dividend exceed those of the divisor; or, so many that

QUESTIONS. What is the rule for the multiplication of decimals? Rule for the division of decimals?

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