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186. The above is a good method for finding the least common multiple when the numbers are easily separated into their prime factors. For larger numbers observe the following method:

ILLUSTRATIVE EXAMPLE II. Find the 1. c. m. of 18, 56, 38, and 30.

WRITTEN WORK.

2) 18, 56, 38, 30

3) 9, 28, 19, 15

3, 28, 19, 5

1. c. m. = 2 × 3 × 3 × 28 × 19 × 5=47880

X

Explanation.

Here,

by repeated divisions, we take out all the factors that are common to two or more of the given numbers. The product of these factors (2 and 3)

and those that are not common must be the 1. c. m. sought..

Rule II.

187. To find the least common multiple of two or more numbers:

1. Write the given numbers in a line as dividends. Make any prime number which is a factor of two or more

of the given numbers a divisor of those numbers.

2. Write the quotients and undivided numbers beneath as new dividends, and so continue dividing till the last quotients and undivided numbers are prime to each other.

3. The product of all the divisors, last quotients, and undivided numbers is the least common multiple required.

188. Examples for the Slate.

Find the least common multiple
71. Of 338, 364, and 448.
72. Of 184, 390, and 552.
73. Of 308, 616, and 77.
74. Of 84, 336, and 472.

For other examples in multiples, see page 123.

75. Of 165, 9500, and 855. 76. Of 1146, 484, and 24. 77. Of 880, 9680, and 176. 78. Of 187, 539, and 8470.

SECTION IX.

COMMON FRACTIONS.

189. If a unit, as 1 inch, is divided into two equal parts, Hone of the parts is called one half.

If the unit is divided into three equal parts, one of the ᅡ + + parts is called one third; two of the parts are called two thirds.

One of the equal parts of a unit is a fraction, or fractional unit. A collection of fractional units is a fractional number.

NOTE I. For the sake of brevity, fractional units and fractional numbers are both called fractions.

NOTE II. A number whose units are entire things is an integral number, or an integer.

Name a fractional unit; a fractional number; an integer.

190. The unit of which the fraction is a part is the

unit of the fraction.

191. The number of equal parts into which the unit of the fraction is divided is the denominator of the fraction.

Thus, in the fraction two thirds the denominator is three. 192. The number of equal parts taken is the numerator of the fraction.

Thus, in the fraction two thirds the numerator is two.

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

NOTE. Decimal fractions have been treated of in previous articles. All fractions except decimal fractions are called common fractions.

Numerator,
Denominator, 3

Writing Common Fractions.

194. The terms of a fraction are written, the numerator above and the denominator below a line. Thus, two thirds of an inch is written as in the margin.

inch.

195. Exercises.

The fact that of 1 equals of 2 may be illustrated as in the margin.

Write in figures the following:

a. One half of a mile.

d. Twenty twenty-fifths. e. Twelve thirds.

b. One third of a day.

c. Seven tenths of a dollar. f. Seven sevenths. g. Write any fraction you please, having for a denominator five; seven; ten; seventeen; one hundred.

h. Write any fraction you please, having for a numerator six; eight; sixty; one hundred.

i. Where is the denominator of a fraction written? Where is the numerator written?

j. Which is the greater part of a thing, or ? for?

196. The form of writing fractions as shown above is the same as the fractional form used to indicate division. (Art. 94.) Thus the expression may mean two thirds of one or one third of two.

ILLUSTRATION.

of 1 equals of 2.

197. Exercises.

a. What is meant by the expression?

Ans. It means 5 of the 9 equal parts into which a unit is divided, or it means 1 ninth of 5 units.

b. What is meant by the expression ? ? ? 13? ft? c. Illustrate the fact that of 1 equals 4 of 3; that of 1 equals of 2.

13323141

REDUCTION.

To change a Fraction to smaller or larger terms. 198. ILLUSTRATIVE EXAMPLE I. Change to equivalent fractions of smaller terms.

1} = § = }

WRITTEN WORK. Explanation. By dividing both terms of 1 by 2, we make the terms half as large, and have the fraction. Now dividing both terms of the fraction by 3, we make its terms one third as large, and have the fraction . If we had divided both terms of 1 by 6, we should have made the terms one sixth as large, and obtained at once the fraction .

The illustration shows that the same part of the unit is expressed by 1, §, and . In obtaining and from 1, the 83 number of parts taken has been diminished as the size of the parts has been increased.

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3

If both terms of a fraction are divided by the same number, the value of the fraction will not be changed.

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199. ILLUSTRATIVE EXAMPLE II. Change to equivalent fractions of larger terms.

WRITTEN WORK.

3=1; 3=13

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ILLUSTRATION.

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ILLUSTRATION.

Explanation.-By multiplying both terms of by 2, we make the terms twice as large, and have the fraction . By multiplying both terms of by 6, we make the terms six times as large, and have the fraction. Here the number of parts in each case is

increased as the size of the parts is diminished.

If both terms of a fraction are multiplied by the same number, the value of the fraction will not be changed.

200. When the terms of a fraction have no common factor, the fraction is said to be expressed in its smallest terms.

201. Oral Exercises.

Perform mentally the examples given below, naming results merely; thus, "; ; ; 1," and so on.

66

a. Change to their smallest terms: ; ; ; † ; † ; fk; P2; t; ; ; ; ; ; ; ; ; ; f8; 2°r.

b. Change to their smallest terms: 14; ; fe; 12; 20; 32; 84; 72; 4; *; *; &; fa; f; fr; Po; ; 84; 63; 81.

c. Change to their smallest terms: 1; 14; 38; 11; 11; 11; 18; 18; 18; 15; 14; 13; 13; 2; 18; 28; 17%.

00

d. Change to equivalent fractions, having 12, 16, 28, 44, 100, and 120 for denominators.

e. Change to equivalent fractions, having 27, 54, 99, and 900 for denominators.

f. Change,,, fo, 11, 13, each to an equivalent fraction having 120 for a denominator.

g. How many thirtieths in ? in ? in ? in ? in ? in ? in ?

h. How many 24ths in ? in ? in ? in ? in ? in ?

To change a Fraction to its smallest terms. 202. From previous illustrations we may derive the following

Rule.

To change a fraction to an equivalent fraction of the smallest terms: Strike out all the factors which are common to the numerator and denominator; or divide both terms by their greatest common factor.

203. Examples for the Slate.

75

900.

Change to equivalent fractions of smallest terms:
(1.) 30%. (4.) 758. (7.) 342.
(2.) 448. (5.) $35.
(3.) 3. (6.) 7,2%.
For other examples, see page 123.

1710
562

(8.) 5.
(9.) 1974.

39

(10.) 1734.

(11.) 3348.

(12.) 3394.

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