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Thus, is read two-thirds; g, three-fifths, etc.
Read the following fractions:

65

1, †, †, 4, §, 17, 3, 27, H, HI, §4, 18, 25.

TO WRITE COMMON FRACTIONS:- Write the numerator over the denominator with a horizontal line between.

Thus, three-fourths is written ; five-sevenths, &; nine-tenths, o, etc.

Write the following fractions:

Two-thirds, three-fourths, three-fifths, four-sevenths, two-sevenths, two-ninths, six-ninths.

Two fifths, three-fifths, four-ninths, six-sixths, fiveeighths, two-thirds, eleven-twelfths, seven-seventeenths, nine-fourteenths, twelve-twenty-firsts, one hundred and seven-one hundred and eighths, ninetyfortieths, and twenty-nine-eighty-fourths.

A Proper Fraction is one whose numerator is less than its denominator; as,, 8,

1 5

An Improper Fraction is one whose numerator is equal to or greater than its denominator; as,, 27.

5

A Simple Fraction is a single fraction; as, %,

A Compound Fraction is a fraction of a fraction, or several fractions connected by the word of; as, of , 1 of 4 of 2.

A Complex Fraction has a fraction in one or both

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A Mixed Number is a fraction joined to a whole number; as, 51, 3§.

Reduction of Fractions is changing their form with out altering their value.

MENTAL EXERCISES.

1. When any thing is divided into 2 parts, what are the parts called? When divided into 3 parts? 5? 11? 16? 21? 27? 30? 32?

2. When any thing is divided into 4 equal parts, what is one part called? Two parts? Three parts? 3. How many thirds make a whole one? How many fourths? Sevenths? Nineteenths? Thirty-firsts? One hundred and firsts?

4. When any thing is divided into 12 parts, what is one part called? When divided into 14 parts?

29? 31? 47? 82?

5. How many fourths in one? How many fourths in one-half? How many eighths in one? In onehalf? In one-fourth? In three-fourths?

6. How many sixteenths in one? In one-fourth? How many twenty-fourths?

VALUE OF FRACTIONS.

90. The Value of one of the parts of a fraction depends upon the number of parts into which the unit is divided.

Thus, if an apple is divided into two equal parts, or halves, the parts are larger than if divided into three equal parts, or thirds.

The value of the fraction depends on the size of the parts, and the number of them taken.

The value of a proper fraction is always less than

one.

The value of an improper fraction is one or more than one.

The reduction, multiplication, and division of frac tions depend on the following propositions:

Prop. 1.-The value of a fraction is not altered by mul tiplying or dividing both terms by the same number.

Prop. 2.-The value of a fraction is multiplied by multiplying the numerator or dividing the denominator.

Prop. 3. The value of a fraction is divided by divid ing the numerator or multiplying the denominator.

NOTE.—The teacher should fully illustrate these propositions.

A fraction is in its lowest terms when the numerator and denominator are prime to each other; as, §. $5.

REDUCTION OF FRACTIONS.

91. TO REDUCE A FRACTION TO ITS LOWEST TERMS. EXAMPLE. Reduce to its lowest terms.

SOLUTION. Since the value of a frac

OPERATION.

2)=1.

3 ) }} =‡, Ans.

tion is not altered by dividing both terms by the same number, divide the numerator and denominator by their common factors. Dividing the terms of 4 by 2, the result is 12; dividing in like manner by 3, gives ; and 4 and 5 are prime to each other.

CONCLUSION.-Therefore, reduced to its lowest terms equals. Hence, the

Rule.-Divide the terms of the fraction by any common factor; divide the resulting fractions, in like manner, until no number greater than one will exactly divide both

terms.

NOTE.-When the numbers are large, it is better to find the greatest common divisor; and dividing by this, the fraction is reduced to its lowest terms.

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OR MIXED NUMBER.

EXAMPLE.-How many units in 291?

SOLUTION. Since equal one unit, 291 equal as many units as 8 is contained times in 291, which is 363 times.

CONCLUSION.-Therefore, 281 equal 36.

Hence, the

OPERATION.

8) 291

36%, Ans.

Rule.-Divide the numerator by the denominator; the quotient will be the whole or mixed number.

MENTAL EXERCISES.

Reduce the following to whole or mixed numbers:

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EXAMPLES FOR PRACTICE.

Reduce the following to whole or mixed numbers:

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93. TO REDUCE A WHOLE OR MIXED NUMBER TO AN IMPROPER FRACTION.

EXAMPLE.-How many fourths in 143?

SOLUTION.-Since in one unit there are 4 fourths, in 14 units there are 14 times 4

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OPERATION.

14359, Ans.

4

56

3

59

Rule.-Multiply the whole number by the given denominator; to the product add the numerator, if any, and place the sum over the denominator.

MENTAL EXERCISES.

Reduce the following to improper fractions:

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In 5? 7? 9?

17. How many fourths in 3?

18. Reduce 5 to an improper fraction whose de

nominator is 6; whose denominator is 7; 11.

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