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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 l. c. m. of 18, 56, 38, and 30.

Explanation. – Here, 2) 18, 56, 38, 30

by repeated divisions, we

take out all the factors 3) 9, 28, 19, 15

that are common to two 3, 28, 19, 5

or more of the given num

bers. The product of 1. c. m. 2 x 3 x 3 x 28 x 19 x 5 = 47880

these factors (2 and 3) and those that are not common must be the l. c. m. sought.

WRITTEN WORK.

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. 75. Of 165, 9500, and 855. 72. Of 184, 390, and 552. 76. Of 1146, 484, and 24. 73. Of 308, 616, and 77. 77. Of 880, 9680, and 176. 74. Of 84, 336, and 472. 78. Of 187, 539, and 8470.

For other examples in multiples, see page 123.

SECTION IX.

COMMON FRACTIONS.

189. If a unit, as 1 inch, is divided into two equal parts +

one of the parts is called one half. If the unit is divided into three equal parts, one of the +

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.

All

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

Writing Common Fractions. 194. The terms of a fraction are written, the numera

tor above and the denominator below a Numerator,

inch. Denominator, 3 line. Thus, two thirds of an inch is written as in the margin.

195. Exercises. Write in figures the following: a. One half of a mile.

d. Twenty twenty-fifths. b. One third of a day.

e. Twelve thirds. 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 ? or ?

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. The fact that of 1

+ equals 1 of 2 may be illustrated as in the margin.

of 1 equals ß of 2.

ILLUSTRATION.

197. Exercises.

a. What is meant by the expression ß?

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? ?

Illustrate the fact that i of 1 equals of 3; that ; of 1 equals } of 2.

c.

REDUCTION.

WRITTEN WORK.

ILLUSTRATION.

CHOD

To change a Fraction to smaller or larger terms. 198. ILLUSTRATIVE EXAMPLE I. Change 1 to equivalent fractions of smaller terms.

Explanation. — By dividing both terms of tif by 1% = f = }

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 1by 6, we should have made the terms one sixth as large, and obtained at once the fraction f.

The illustration shows that the same part of the unit is expressed by 14, $, and . In

obtaining & and f from 1', the 8

number of parts taken has 3

been diminisned as the size of

the parts has been increased. If both terms of n fraction are divided by the same number, the value of the fraction will not be changed.

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

Explanation. -- By multiply= $; f = 1 %

ing both terms of s 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 1

fraction it. Here the num

ber 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.

WRITTEN WORK.

ILLUSTRATION.

201. Oral Exercises.

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

a. Change to their smallest terms: $; $; $; ; $; 1; 12; ਨੂੰ ਲੈ ; ; ; ; ; ਲੈ ; ਹੈ ; ; ; ; ; fਲ ; •

b. Change to their smallest terms: 14; 12; si isi ; ਗੁ; ; ; ; ਫ਼ੈਲ ; ਨ ; ਲ ; ; ; ਨੂੰ; ਤੌਲ ; ; $ 5

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c. Change to their smallest terms: 13; 14; 18; $; $; 14; * ; 18; 8; 9; 14; ; }$M; }$; 8; 1979%

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

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

f. Change ș, 4, }, 7o, 11, H4, each to an equivalent fraction having 120 for a denominator.

g. How many thirtieths in 1? in }? in } ? in ? in ? in ? in ?

h. How many 24ths in ? in ? in ? in ? in 3? 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. Change to equivalent fractions of smallest terms: (1.) 36 (4.) 388 (7.) 14% (10.) 1784. (2.) 4%

(5.) 38 (8.) 46.28 (11.) 3948 (3.) 93 (6.) 780 (9.) 1789 (12.) 3399. For other examples, see page 123.

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