ARITHMETIC.

PART II.

FRACTIONS.

The term fraction is derived from the Latin word frango, which signifies to break; from the idea that a number or thing is broken or separated into parts.

134. A Fraction, therefore, is an expression denoting one or more equal parts of a unit, and may be regarded as expressing unexecuted division, usually by writing two numbers one above the other with a line between them. The line means divided by.

135. The number below the line is called the denominator. It shows into how many equal parts the unit is divided.

136. The number above the line is called the numerator. It shows how many of these equal parts are taken. Thus, in the fraction, the 12 shows that the unit has been divided into 12 parts, and the 1 shows that one of the 12 parts is taken.

137. The terms of a fraction are its numerator and denominator.

138. A fractional unit is one of the equal parts into which the unit in question is divided.

Any whole number may be expressed as a fraction by writing

4 1 for the denominator. Thus, 4 may be written and read four 1 ones, etc.

4

2

139. The value of a fraction is the quotient obtained from the division of the numerator by the denominator, or from the expression of this division. Thus, in the quotient of 4÷2 is 2, and the value of the fraction can be expressed as 2; but the quotient of 3-4 cannot be otherwise expressed, and the value is written 3.

Thus the line between the terms of a fraction indicates division, viz.: the division of the numerator by the denominator.

If the numerator and denominator are equal, the value of the fraction is 1, because any quantity is contained in itself once.

If the numerator is greater than the denominator, the value of the fraction is greater than 1; if the numerator is less than the denominator, the value is less than 1.

140. A proper fraction is one whose numerator is less than its denominator, as,

141. An improper fraction is one whose numerator is equal to or greater than its denominator. Hence the value of every improper fraction must be 1 or more than 1, and that of every proper fraction less than 1.

A fraction, strictly speaking, is less than a unit; hence, if the numerator is equal to or greater than the denominator, the fraction expresses a unit or more than a unit and is therefore called an improper fraction.

The following explanation shows the actual value of a proper fraction:

4

142. Take for example the fraction 5 is the denominator,

is the

and shows that the unit is divided into 5 equal parts, fractional unit, since it is one of the 5 equal parts into which the original unit is divided; 4 is the numerator, and shows that four of these equal parts are taken; 4 and 5 are the terms of the fraction. It is a proper fraction, since the numerator is less than the denominator, and since its value is less than 1; it is read four-fifths.

143. Fractions indicate division, and all changes in either term of a fraction affect the value of the fraction. They may be reduced, added, subtracted, multiplied, and divided. For example, in the fraction, if we divide the numerator by 2 we divide the value of the fraction and obtain the fraction; or if we multiply the denominator we divide the value, thus:

Also in multiplying the numerator of a fraction we multiply

144. The following rules should be remembered:

Multiplying the numerator, or ) multiplies the value

Dividing the denominator

Dividing the numerator, or
Multiplying the denominator

Multiplying or dividing both
numerator and denominator by
the same number

of

the fraction.
divides the value
of
the fraction.

does not change the
value of
the fraction.

The following general rule will help to fix the above prin. ciples firmly in mind:

A change in the numerator produces a like change in the value of the fraction, but a change in the denominator produces an opposite change in the value of the fraction.

145. Reducing a fraction is changing its form without changing its value. A fraction is reduced to higher terms by multiplying both terms of the fraction by the same number.

We readily see that we have changed the form of the fraction

4

but not the value, because is equal to 6

2

146. This is seen in Fig. 2, where we have a unit divided into two, three, four, six, and twelve parts.

multiply both terms by 2. This gives us

2

Take one-half and

which we see is equal

change is made in both terms of a fraction, the value of the frac

tion is not changed.

147. To reduce a fraction to lower terms, divide both numerator and denominator by the same number. Thus:

In this case also we have not changed the value, since dividing both terms of the fraction by the same number is equivalent to dividing the fraction by unity, and does not change the value. A fraction is reduced to its lowest terms when its numerator and denominator have no common factor other than 1; thus, 1, 3, 15 26 are reduced to their lowest terms.

169

148. A mixed number is a whole number and a fraction. To reduce a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction, add the product to the numerator, and place their sum, over the denominator.

149. To reduce an improper fraction to a mixed number, divide the numerator by the denominator, write the quotient as the integer of the mixed number, and the remainder placed over the divisor (denominator of improper fraction) as the fraction of the mixed number.

47
6

For example, if we wish to change to a mixed number,

we divide the numerator (47) by the denominator (6), and place the remainder (5) over the divisor (6) as the fraction of the mixed number. Thus:

6)47
78

Now, applying the rule for changing a mixed number to an improper fraction, we multiply the whole number (7) by