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HARVARD
UNIVERSITY
LIBRAR

COPYRIGHT 1912 BY

AMERICAN SCHOOL OF CORRESPONDENCE

Entered at Stationers' Hall, London

All Rights Reserved

i

PRACTICAL MATHEMATICS

PART II

FRACTIONS

27. A fraction is an expression denoting one or more equal parts of a unit, and may be regarded as indicating division. The fraction is written, i. e., the division is indicated, by placing the dividend over the divisor with a line between.

5 Thus,

denotes that 5 of the 12 equal parts of a unit are to be 12

To say

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3 ; 4;

taken. is merely a different way of writing 5 • 12, and the

1 2 fraction is used because the dividend is less than the divisor.

1

of a 100 pound keg of bolts indicates, that the keg of

4 bolts is to be divided into 4 equal parts, and 1 of these parts taken. In this use, the unit may be considered as made up of a number of equal parts, but when used separately and without reference to any certain thing, it is convenient to consider the fraction merely

1
3

875 as an unperformed division; thus, 1 ; 4;

4
4

361 875 = 361.

The quantity below the line is called the denominator. It shows into how many equal parts the unit is divided. The quantity above the line is called the numerator. It shows

1 how many of these equal parts are taken. Thus, in the fraction

12 the 12 shows that the unit has been divided into 12 equal parts, and the 1 shows that one of the 12 parts is taken. In the fraction 7

the unit is divided into b parts and a of these parts are taken.

The numerator and denominator are called the terms of a fraction.

a

Copyright, 1911, by American School of Correspond ence.

A fractional unit is one of the equal parts into which the unit in question is divided. Thus, if the fraction is divided into fourths, 1

1 is the fractional unit; if divided into b parts, is the fractional

b unit.

Since any number divided by 1 gives a quotient equal to the dividend, any whole number may be expressed as a fraction by writing

4 4 1 for the denominator. Thus, 4 may be written

4 • 1

4.

1 1 The value of a fraction depends upon the value of the fractional units and the number of these units taken, or simply upon the division

4 of the numerator by the denominator. Thus, in the quotient

2' of 4 • 2 is 2, and the value of the fraction can be expressed

as

as 2.

If the numerator and the denominator are equal, the value of the

8 fraction is 1. Thus, may be expressed as 8 = 8 and is equal to

8 1. This shows that one unit is divided into eight parts, each part being an eighth and that 8 of these are taken, making a unit

or 1.

Strictly speaking a fraction is less than a unit; hence if the numerator is less than the denominator, the value is less than 1, and it is

8 known as a proper fraction. For example, the expression

9 that a unit has been divided into nine parts, each part forming a

1 of

means

taken. This is one less than the nine parts necessary to make a

8 unit, and therefore is less than 1.

9 If the numerator is greater than the denominator, the value of the fraction is greater than 1, and it is called an improper fraction.

8 Thus, the fraction

is an improper fraction, because 7 is contained

7 once in 8 with a remainder, or, expressing it in another way, because eight parts, each one being a seventh of a unit, have been taken, forming a unit and one seventh.

REDUCTION OF FRACTIONS

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28. To reduce a fraction is to change its form without changing its value. To reduce a fraction to higher terms multiply both numerator and

3 3 X 3 9 denominator by the same quantity. Thus,

The 4

4 X 3 12 value of the fraction has been increased three times by multiplying the numerator by 3, and then decreased just as many times by mul

b tiplying the denominator by 3. If in the fraction both the

d numerator and denominator are multiplied by c, the value of the

bc fraction is not changed but the form is changed to

Thus,

dc multiplying both numerator and denominator by the same quantity does not change the value of the fraction. To reduce a fraction to lower terms divide both numerator and

4 ; 2 2 denominator by the same number. Thus,

In this 6 • 2

3 case dividing the denominator by 2 changes the fractional parts from sixths to thirds, which are twice as large as sixths and this much of the operation has doubled the value of the fraction. Dividing the numerator by 2 decreases the number of parts to one-half of the original number. Therefore dividing both numerator and denominator by the same quantity does not change the value of the fraction. In the same way

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may be reduced to L by dividing both numerator and denominator by m.

A fraction is reduced to its lowest terms when its numerator and denominator have no common factor other than 1; that is when

1 2 15 the terms are prime to each other. Thus,

2' 3' 16'

4 reduced to their lowest terms, but is not, as 4 and 6 may both

6

2 ad be divided by 2, reducing the fraction to and is not, as ad

3

d and ac may both be divided by a, reducing the fraction to

mn

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a

are

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ac

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