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COMMON FRACTIONS.

130. If a unit, as an apple, be divided into two equal parts, one of these parts is one half.

If a unit be divided into three equal parts, one of these parts is one third; two of them two thirds, etc.

If a unit be divided into four equal parts, one of these parts

is one fourth; two of them two fourths; three of them three fourths, etc.

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In like manner, if a unit be divided into five equal parts, the parts are fifths; if into six equal parts, sixths; if into seven equal parts, sevenths; and so on.

parts of a unit are called fractions. Hence,

Such

131. A Fraction is a part of a unit, consisting of one or more of the equal parts, which compose the unit.

The UNIT of a fraction is the unit or thing divided.

A FRACTIONAL UNIT is one of the equal parts into which the unit of a fraction is divided.

132. The DENOMINATOR of a fraction is the number which shows into how many equal parts the unit is divided. Thus, Three is the denominator of two thirds.

133. The NUMERATOR of a fraction is the number which shows how many of the equal parts of the unit are taken. Thus,

Two is the numerator of two thirds.

What is a Fraction? The Unit of a fraction? A Fractional Unit? The Denominator of a Fraction? The Numerator?

NOTATION AND NUMERATION.

134. A Common Fraction is a fraction expressed by writing the numerator above, and the denominator below, a

dividing line; as,

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135. A PROPER FRACTION is one whose numerator is less

than the denominator; as, J.

136. An IMPROPER FRACTION is one whose numerator is not less than the denominator; as §, %.

137. A MIXED NUMBER is a whole number with a fraction; as 31; read three and one fourth.

138. The TERMS of a fraction are the numerator and denominator.

The numerator expresses the number of fractional units in the fraction, and the denominator their name or denomination. 139. A whole number may be expressed in a fractional form, by writing 1 under it, for a denominator. Thus,

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SOLUTION. Three fifths; the denominator 5 denoting that the name or denomination of fractional units expressed is fifths, and the numerator 3 that three of those units are taken.

How is a Common Fraction expressed? What is a Proper Fraction? An Improper Fraction? A Mixed Number? What are the Terms of a fraction? What do they express? How may a whole number be expressed in a fractional form? How may a fractional expression be explained?

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14. Express by figures seven ninths.

SOLUTION. Since the name or kind of fractional units to be expressed is ninths, we write 9 as the denominator, and since the number of these units to be taken is seven, we write 7 as the numerator, and have as the required expression.

Express by figures,

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26. Seventy-two thousand twenty-firsts.

141. A fraction may be regarded as indicated division (Art. 58), the numerator answering to the dividend, and the denominator to the divisor. Hence, the

GENERAL PRINCIPLES.

1. The value expressed by a fraction is the quotient of the numerator divided by the denominator.

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Thus,

of 2 inches =

By Art. 71-3, it follows, that

2. Multiplying the numerator or dividing the denominator multiplies the fraction.

3. Dividing the numerator or multiplying the denominator divides the fraction.

How may a fraction be regarded? What is the value expressed by a fraction? How is the value of a fraction affected by multiplying its numerator or by dividing its denominator? What effect has dividing the numerator or multiplying the denominator of a fraction?

4. Multiplying or dividing both terms of a fraction by the same number does not change its value.

Mental Exercises.

1. If a unit be divided into 4 equal parts, what will be the name or denomination of the parts?

SOLUTION. If a unit be divided into 4 equal parts, each of the equal parts will be. Therefore, the name or denomination of the parts will be fourths.

2. If a unit be divided into 5 equal parts, what will be the name or denomination of the parts? If into 9 equal parts? 13? 17? 62? 92? 150?

3. If a unit be divided into 63 equal parts, what will 1 of the equal parts be called? 3 of the equal parts? 9? 19? 25? 41? 62?

4. If a unit be divided into 19 equal parts, what is 1 of the fractional units called? 7 of the fractional units? 11? 16? 17?

of 12?of 12? ? ? 12?

5. How much is 6. How much is

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SOLUTION. Since 1 third of 18 is 6, 2 thirds of 18 are 2 times 6, or

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SOLUTION. Since of 12 is 6 and of 12 is 4, 1 of 12 is greater than of 12 by the difference between 6 and 4, or 2. Therefore, of 12 is greater than of 12 by 2.

12. Which is the greater, of 30, or of 30? of 24?

of 24, or

What effect has multiplying or dividing both terms of a fraction? Which is the greater, a fourth or a half of any thing? A third or a tenth?

13. Which is the greater, of 72 or of 72?of 100 or to of 100?

14. Which is the greater, of 15 or % of 15? of 42 or of 42?

15. How many fourths in a half?

SOLUTION. Since there are 4 fourths in 1 unit, in 1 half of 1 unit there is one half of 4 fourths, or 2 fourths. Therefore, in 1 half there are 2 fourths.

16. How many sixths in a third? How many twelfths? 17. How many eighths in a fourth? How many six

teenths?

18. How many tenths in a fifth? How many twentieths? 19. How many thirds in ?

SOLUTION. Since in 1 unit there are 6 sixths, in 1 third of 1 unit there is 1 third of 6 sixths, or 2 sixths, and in 4 sixths as many thirds as 2 sixths are contained times in 4 sixths, which are 2. Therefore, in ₫ there are 2 thirds.

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SOLUTION. Since 1 is 1 seventh of 7, 2 is 2 times 1 seventh of 7,

or 2 sevenths of 7. Therefore, 2 is 4 of 7.

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28. 8 is 1 fifth of what number?

SOLUTION. If 1 fifth of some number is 8, 5 fifths, or the number

itself, is 5 times 8, or 40.

29. 7 is of what number?

30. 10 is of what number?

of what number?

of what number?

REVIEW QUESTIONS. What is a Fraction? (131) How is a Common Fraction expressed? (134) What are the Terms of a fraction? (138) How may a fraction be explained? (140)

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