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174. A fraction may be expressed by two numbers, one of them being written above and the other below a short horizontal line ; thus, $, 14, 149.
175. The number above the line is the numerator of the fraction; the number below the line, the denominator of the fraction.
176. KINDS OF FRACTIONS.
1. A fraction whose numerator is less than its denominator is a proper fraction.
3, %, 1%, are proper fractions.
2. A fraction whose numerator is equal to or greater than its denominator is an improper fraction.
$, %, 4, are improper fractions.
NOTE.—The fraction . 7 is a proper fraction. 2.7 may be regarded as an improper fraction or as a mixed number. If it is to be considered an improper fraction it should be read, 27 tenths; if a mixed number, 2 and 7 tenths.
3. Such expressions as the following are compound fractions:
of 4 of 1, of t.
4. A fraction whose numerator or denominator is itself a fraction or a mixed number, is a complex fraction.
are complex fractions.
Fractions. 5. Any fraction that is neither compound nor complex is a simple fraction.
, 11, 4, are simple fractions.
6. A fraction whose denominator is 1 with one or more zeros annexed to it, is a decimal fraction.
, .7, .25, 36, are decimal fractions.
Note 1.—The denominator of a decimal fraction may be expressed by figures or it may be indicated by the position of the right-hand figure of its numerator with reference to the decimal point. When the denominator is thus indicated, the fraction is called a decimal and is said to be written decimally.
NOTE 2.-A11 fractions that are not decimal are called common fractions. A decimal fraction when not “written decimally” (or thought of as written decimally) is usually classed as a common fraction.
7. A complex decimal is a decimal and a common fraction combined in one number.
.73, .254, .056}, are complex decimals.
177. There are three aspects in which fractions should now be considered.
1. THE FRACTIONAL UNIT ASPECT.
The numerator tells the number of things and the denominator indicates their name. In the fraction & there are 5 things (magnitudes) called sevenths. In the fraction ☆ there are five fractional units each of which is one eighth of some other unit called the unit of the fraction.
NOTE:--The function of the denominator is to show the number of parts into which the unit of the fraction is divided ; the function of the numerator, to show the number of parts (fractional units) taken. Fractions.
II. THE DIVISION ASPECT. The numerator of a fraction is a dividend; the denominator, a divisor, and the fraction itself, a quotient : thus, in the fraction , the dividend is 5; the divisor, 8, and the quotient, .
NOTE.—In the case of an improper fraction, as q, it may be more readily seen by the pupil that the numerator is the dividend, the denominator the divisor, and the fraction (352) the quotient; but the division relation is in every fraction, whether proper or improper, common or decimal, simple or complex,
III. THE RATIO ASPECT.
The numerator of a fraction is an antecedent; the denominator, a consequent, and the fraction itself, a' ratio : thus, in the fraction 18, 7 is the antecedent, 10 the consequent, and
NOTE 1.—This relation may be more readily seen by the pupil, in the case of an improper fraction. In the fraction 42, 12 is the antecedent; 4, the consequent; 4, or 3, the ratio.
NOTE 2.-Every integral number as well as every fraction is a ratio. The number 8 is the ratio of a magnitude that is 8 times some unit of measurement to a magnitude that is 1 time the same unit of measurement. 8 (units of measurement) is the antecedent; 1 (unit of measurement) is the consequent, and the pure number 8 is the ratio.
178. REDUCTION OF FRACTIONS.
1. The numerator and the denominator of a fraction are its terms.
2. A fraction is said to be in its lowest terms when its numerator and denominator are integral numbers that are prime to each other.
3. Reduce 16 ó to its lowest terms. Operation.
Explanation. 160 16
Dividing each term of 188 by 10 we have 10)
20 1 tenth as many parts, which are 10 times as 16 4 large. Dividing each term of jg by 4 we have 4)
1 fourth as many parts, which are 4 times as 20 5
large. Hence, 18% = 4. But 4 and 5 are prime to each other, and the fraction is in its lowest terms.
RULE. - Divide each term of the fraction by any common divisor except 1, and divide each term of the fraction thus obtained by any common divisor except 1, and so continue until the terms are prime to each other.
(a) Find the sum of the ten results. I 4. Reduce to higher terms — to 120ths.
* Divide each term by 12). This involves the reduction of a complex to a simple fraction; but it will lead to thoughtful work for the pupil to solve such problems in this manner.
+ Divide each term by 1.
# If the pupil has not had sufficient practice in addition of fractions to do this, the finding of the sum may be omitted until the book is reviewed.
Reduce to higher terms to 160ths.
(3) ਝੂਠ (4) 3
5. Two or more fractions whose denominators are the same, are said to have a common denominator.
6. Two or more fractions that do not have a common denominator may be changed to equivalent fractions having a common denominator.
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7. Two or more fractions that do not have a common denominator may be changed to equivalent fractions having their least common denominator. The 1. c. d. of two or more fractions is the 1. c. m. of the given denominators.
Change it, do, and 27 to equivalent fractions having their least common denominator.
11 X 4
44 (2) 120 = 30 = 4
30 x 4
9 X 3 27 (3) 120 – 40 3
40 x 3 120
37 x 2 74 (4) 120 60 = 2
60 X 2 120 jf =
7 = 1