COMMON FRACTIONS.

129. A FRACTION is an expression representing one of more of the equal parts of a unit.

NOTE. A unit, or any other whole number, is often called an Integer, it is also called an Integral or Entire Number.

130. A COMMON or VULGAR FRACTION is expressed by two numbers, one above and the other below a line; thus (one half), (two fifths), etc.

(a) The number below the line shows into how many parts the unit is divided, and is called the DENOMINATOR, because it denominates or gives name to the parts; thus, if a unit is divided into 3 equal parts, each part is one third; if into 8, each part is one eighth; etc?

(b) The number above the line is called the NUMERATOR, because it numerates or numbers the parts taken.

(c) The numerator and the denominator are the TERMS of the fraction.

131. A fraction is nothing more nor less than unexcuted division, i. e. division indicated but not performed, the numerator being the dividend, and the denominator the divisor. Hence,

(a) The value of a fraction is the quotient of the numerator, divided by the denominator; thus, 2=12÷4 = 3; and, .'.,

(b) Any change in the NUMERATOR causes a LIKE change in the value of the fraction, and any change in the DENOMINATOR causes an OPPOSITE change in the value of the fraction (Art. 84).

These principles are developed in the following Problems.

129 What is a Fraction? Other names for a whole number? 130. A Common Fraction, how expressed? Number below the line, what called? Why? Number above, what called? Why? Terms of a fraction, what? 131. A fraction, what is it? Value of a fraction? What follows?

132. A PROPER FRACTION is one whose numerator is less than the denominator; as, }, 71, 24.

133. An IMPROPER FRACTION is one whose numerator equals or exceeds its denominator; as, 4, 7, 8, 1§. An improper fraction equals or exceeds a unit; hence its name, IMPROPER fraction.

134. A SIMPLE FRACTION has but one numerator and one denominator, and is either proper or improper; as, 3, §, 14.

135. A COMPOUND FRACTION is a fraction of a fraction; as, 3 of 11, of 3 of §.

136. A MIXED NUMBER is a whole number and a fraction united; as, 8, 20.

137. A COMPLEX FRACTION is one that has a fraction or a

313 mixed number for one or for each of its terms; as, 72

138. The RECIPROCAL of a number is a fraction whose numerator is 1, and whose denominator is the number itself; thus, the reciprocals of 4, 9, and 4 are ‡, †, and

1

PROBLEM 1.

139. To reduce a mixed number to an improper fraction.

Ex. 1. In 3 how many fourths?

OPERATION.

31

4

Ans. 13.

Since 4 fourths make a unit, there will be 4 times as many fourths as units, therefore, in three units there will be 4 times 3 fourths 12 fourths, and the 1 fourth in the example added to the 12 fourths, gives 13 fourths, i. e. 13. Hence,

13,

Ans.

132. A Proper Fraction, what? 133. An Improper Fraction? 134. A Simple Fraction? 135. A Compound Fraction? 136. A Mixed Number? 137. A Complex Fraction? 138. The Reciprocal of a Number? 139. Explain the Operation in Ex. 1.

RULE. Multiply the whole number by the denominator of the fraction; to the product add the numerator, and under the sum

(a) To reduce an integer to a fraction having any given denominator:

Multiply the integer by the proposed denominator, and under the product write the denominator (Art. 84, c).

27. Reduce 12 to a fraction whose denominator is 7.

Ans. .

28. Reduce 9 to a fraction whose denominator is 8. 29. Reduce 9 to a fraction whose denominator is 5. 30. Reduce 7 to a fraction whose denominator is 1.

Ans. 7.

31. Reduce 87 to a fraction whose denominator is 87. 32. Reduce 16 to a fraction whose denominator is 1. 33. Reduce 16 to a fraction whose denominator is 4. 34. Reduce 20 to a fraction whose denominator is 4. 35. Reduce 14 to five different fractional forms.

139. Rule for reducing a mixed number to an improper fraction? Reason? An integer, how reduced to a fractional form?

PROBLEM 2.

140. To reduce an improper fraction to a whole or

mixed number.

Ex. 1. How many units in 13?

1343, Ans.

Ans. 31.

Since the numerator is a dividend and the denominator a divisor (Art. 131), the

\fraction is reduced to an equivalent whole or mixed number by the following

RULE. Divide the numerator by the denominator; if there is any remainder, place it over the divisor, and annex the fraction so formed to the quotient.

2. Reduce to a whole or mixed number. 3. Reduce §§ to a whole or mixed number. 4. Reduce to a whole or mixed number. 5. Reduce 56 to a whole or mixed number. 6. Reduce $34.

7. Reduce 34.

8. Reduce 783.

9. Reduce 4.

10. Reduce §.

Ans. 3.

Ans. 3.

Ans. 214.

Ans. 26.

11. Reduce 128.

PROBLEM 3.

141. To reduce a fraction to its lowest terms.

Ex. 1. Reduce to its lowest terms.

FIRST OPERATION.

Ans. .

Dividing both terms of a frac tion by any number does not alter the value of the fraction (Art. 84, b, and 131); .. dividing each

term of 2 by 3 gives the equal fraction ; then dividing each term of this result by 4 gives 2, and as 3 and 4 are mutually prime (Art. 112), §, in its lowest terms, equals 2.

SECOND OPERATION.

12), Ans.

In this operation both terms of the fraction 3 are divided by their greatest common divisor, 12 (Art. 119), and thus the fraction is re

duced at once to its lowest terms. Hence,

143. Rule for reducing an improper fraction to a whole or mixed number? Reason?

RULE 1. Divide each term by any factor common to them, then divide these quotients by any factor common to THEM, and so proceed till the quotients are mutually prime. Or,

RULE 2. Divide each term by their greatest common divisor.

142. To multiply a fraction by a whole number.

Ex. 1. Multiply by 3.

FIRST OPERATION.

X3, Ans.

Ans.org.

It is just as evident that 3 times are as that 3 times 2 cents are 6 cents, or that 3 times 2 are 6; i. e. when the numerator

is multiplied by 3 the fraction represents 3 times as many parts as before, and each part continues of the same size; .. the fraction is multiplied by 3.

SECOND OPERATION.

X3, Ans.

If the denominator is divided by 3, the fraction represents just as many parts as before, but each part is three times as great, and

.. the whole fraction is three times as great. Hence,

RULE 1. Multiply the numerator by the whole number. Or, RULE 2. Divide the denominator by the whole number.

NOTE 1. The correctness of Rule 1 is also evident from Art. 83 (a), and Art. 131. Rule 2 also depends on Art. 83 (d).

141. First rule for reducing a fraction to its lowest terms? Second rule? Reason? 142. First rule for multiplying a fraction by a whole number? Why? Second rule? Why? Another reason?