SECTION IX.

FRACTIONS.

Art. 118. When any thing is divided into two equal parts, each part is called one half.

There are two pints in one quart; one pint is therefore one half of one quart. One half of two apples is one apple. One half of 100 apples is 50 apples.

When any thing is divided into three equal parts, each part is called one third. Two of the parts are called two thirds.

There are three feet in one yard. One foot is therefore one third of one yard; two feet are two thirds of one yard.

There are two halves in one thing; three thirds; four fourths; five fifths; etc.

119. A Fraction is one or more of the equal parts of a unit, or of any number regarded as a unit.

If an orange is divided equally among six boys, the part which each boy receives is called a fraction. Two of the parts are also called a fraction; so are three of them, or more.

120. The number or object which is divided into equal parts is called the Unit of the Fraction.

Each of the equal parts into which the unit of the fraction is divided is called the Fractional Unit.

In the illustration given above (Art. 119), the orange is the unit of the fraction; one sixth of the orange is the fractional unit.

121. Fractions are represented by figures, as follows:

122. The numbers, or Terms, which represent fractions, are called Numerator and Denominator. 123. The numerator is written above a short horizontal or oblique line, and the denominator below it., §, 23. 124. The Denominator shows the number of equal parts into which the unit of the fraction is divided. 125. The Numerator shows either the unit of the fraction, or the number of fractional units in the fraction.

When one orange is divided into six equal parts, each part is represented by the fraction, in which 6 is the denominator, and shows into how many equal parts the orange has been divided. The numerator. 1, shows that one orange has been divided.

The fraction represents two of the six equal parts into which one orange has been divided; or, it represents one sixth part of two oranges. The pupil can readily see that one sixth of two oranges is equal to two sixths of one orange.

126. A Mixed Number is composed of an integer and a fraction. 1, 254.

127. If three oranges are each divided into six equal parts, or sixths, there are eighteen sixths, 18. The number of oranges in these eighteen sixths, 18, can be found by dividing the numerator by the denominator, 18÷6=3. So, 24-24÷6-4; 2=24÷8=3; etc.

The quotient resulting from the division of the numerator by the denominator is the Value of the Fraction. 128. Whenever the numerator equals the denominator, the value of the fraction is one. =1.

12=3.

Whenever the numerator exceeds the denominator, the value of the fraction is greater than one. Whenever the numerator is less than the value of the fraction is less than one. is less than 1.

the denominator, is less than 1.

129. A Proper Fraction is a fraction whose value is less than one., 1.

130. An Improper Fraction is a fraction whose value is equal to, or greater than, one., .

REVIEW QUESTIONS.

What is a fraction? What are the terms of a fraction? Define each term. What is a mixed number? How is the value of a fraction found? What is the difference between a proper fraction and an improper fraction? Define each.

REDUCTION OF FRACTIONS.

CASE I.-To reduce an integer or a mixed number to an improper fraction.

ORAL.

131. 1. How many half oranges in 3 oranges? SOLUTION.-In 3 oranges there are 3 times as many half oranges as in 1 orange. In one orange there are 2 half oranges; in 3 oranges there are three times 2 half oranges, or 6 half oranges.

2. How many thirds of an orange in 3 oranges?
3. How many fourths of an apple in 4 apples.
4. How many fifths of a dollar in ten dollars?
5. In 5 how many halves? Thirds? Fourths?
6. In 8 how many fifths? Sixths? Ninths?
7. Change 6 to 10ths. To 12ths.

SOLUTION. 3=3+4. In 3 units there are 3 times as many fourths as in 1 unit. In one unit there are 4 fourths; in 3 units there are 3 times 4 fourths, or 12 fourths. 12 fourths+3 fourths=15.

Rule. Find the product of the integer and the denominator. To this product add the numerator of the fraction, if there be any. The result is the numerator of

the required fraction.

NOTE.-It will be observed that the rule directs the reduction of the integer to the same denominator as the fraction, previous to their addition. This is in accordance with the principle (Art. 33) that like numbers only can be added.

CASE II.-To reduce an improper fraction to an integer or a mixed

number.

ORAL.

132. 1. In 24 sixths of an orange, how many oranges? SOLUTION. There are six sixths of an orange in one orange. In 24 sixths of an orange there are as many oranges as 6 sixths are con

tained times in 24 sixths, which are 4 times. Hence, in 24 sixths of an orange there are 4 oranges.

SOLUTION. There are 3 thirds in one unit. In 13 there are as

many units as are contained times in 13, which are 4 times. Hence, 13-41.

Rule. Divide the numerator of the fraction by its denominator

Reduce to integers or mixed numbers:

To reduce a fraction to higher or lower terms.

133. In one orange there are four fourths; therefore, in one half of an orange there is one half of four fourths, or two fourths; that is,=. In one orange there are ten tenths; therefore, in one half of an orange there is one half of ten tenths, or five tenths; that is,. These fractions, 1, 2, and have the same value; each is equal to one half of the same unit. They are, therefore, called equivalent fractions.

134. Equivalent Fractions are fractions of different expression, but of the same value.

135. The value of a fraction is the quotient resulting from the division of the numerator by the denominator. (Art. 127.) That is,