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VULGAR FRACTIONS. DEFINITION.-Vulgar Fractions are parts of whole numbers, and are expressed thus, %; &c. The top figure is a remainder left after division, and is called the numerator ; the bottom figure is the divisor used in division, and is called the denominator. Fractions are read thus, to, is read one tenth ; and 4, is read six sevenths, &c.

Fractions are proper, improper, compound, or mixed ; a Proper Fraction has its numerator the smallest ; an Improper Fraction has its numerator, equal, or the largest ; a Compound Fraction is the fraction of a fraction, and is coupled by the word of ; a mixed number is a whole number and fraction. A proper fraction is written thus, į, , , , &c. An improper fraction is written thus, i, j, or, 7,&c. A compound fraction is written thus, i of of 4 of 4. A mixed number is written thus, 124, 68, 57 &c.

CASE I. To find the greatest common measure. RULE.--Divide the denominator by the numerator, and the last divisor by the last remainder; the last divisor used is the common measure ; if 1 is the last divisor, the fraction is in its lowest terms.

Examples 1. The greatest common measure of is required.

26)252(9

234

.

18)26(1

18

8)18(2
16
2)8(4

8 The last divisor being two, it is the greatest com mon measure of the fraction.

2. What is the greatest common measure of $?

Ans. 8. 3. What is the greatest common measure of me?

Ans.-4.

CASE II. To reduce fractions to their lowest terms. Rule.--Find a common measure by case first; di- ! vide the numerator and denominator of the given fraction, by the common measure, the quotient is the fraction in its lowest terms.

Examples. 1. Reduce to its lowest terms.

4. + 2 Comm. Ans. 2. It is required to reduce to its lowest terms. 3. Required to reduce * to its lowest terms.

Ans. 1

Ans. 13

108

16

CASE III.

To reduce a mixed number to an improper fraction.

RULE.-Multiply the whole number by the denominator of the fraction, to the product add the -numerator of the fraction; this sum placed over the given denominator will form the fraction required.

Erumples. 1. Reduce 101 to an improper fraction.

10X4=40+3=43 numerator, Ans. 2. Reduce 1121 to an improper fraction.

112x3+1=3.97 Ans. 3. Reduce 281% to on improper fraction.

Ans.

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CASE IV.

To reduce an improper fraction to its proper terms.

RULE.-Divide the numerator by the denominator, the quotient is the whole number ; the remainder is the numerator of the fraction.

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Examples.
1. Reduce to its proper terms.

43-4-10%. Ans.
2. Reduce 317 to its

proper terms.

Ans. 1124 3. Reduce to its proper terms.

Ans. 280

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CASE V.. To find the least common denominator. RULE.—Divide the given denominators by any number that will divide two, or more of them without a remainder; then divide the undivided numbers and last quotients, by any number that will divide two or more of them, without a remainder; and thus continue dividing till no two numbers can be divided without a remainder ;. then multiply the divisors and remainders together, the product is the common denominator required.

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Eramples.

1. What is the least common denominator of เอ อ:

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346 8 7 10. 18 given denominators.

22 8 7 10 6

com. denom: 1X 4 X 7 X 5 X 3 X 3 X 23520 Ans.

6

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2. What is the common denominator of the 1, and 1?

Ans. 1008.
CASE VI.
To reduce fractions to a common denominator.

Rule. Find the common denominator by case 5th, and use it as a dividend; use each given denominator as a divisor; divide, multiply the quotients by the respective numerators, and the product placed over the common denominator will form the fractions required.

1
Eramples.
Reduce , , j and is to common denominators.
Com. denom, 407 4310X3530 1st numerator.

40 = 58X25 16 2d numerator.
40 - 8= 5x7535 38 numerator.

40-105 4X936 4th numerator. The new numerators being written over the common denominator, the fractions appear thus is, 18, 16,6 Answers.

2. Reduce this, and to common dem nominators.

Ans. fon ***, , os ito. 3. Reduce $0, 127

12, and it to common denominators.

Ans.

280, 336

1

330

315
420) 420) 420) 420

CASE VII. To reduce compound fractions to simple ones. Rule.-Multiply all the numerators together for a new numerator ; and all the denominators together for a new denominator, and it is done.

Examples.
1. Reduce of of of of to a simple frac-
tion.
Num. 1 X 3 X 5 X 4 X

Ans.
Den. 2 X 4 X 6 X 7 X 8 -2688

- 360

2. Reduce io of į of g of is of % to a simple fraction, and to its lowest terms.

Ans.

27

1600

CASE VIII.

RE fract

To reduce a fraction of a lower denomination to the

fraction of a higher, retaining the same value.

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3

1 10

3 10000

1

RULE.-Reduce the fraction to a compound one, by comparing it with all the denominations between itself, and the denomination, to which you would reduce it, then reduce it to a simple fraction by case, seventh.

Eramples.
1. Reduce to of a mill, to the fraction of a doll.
By comparing it, it becomes to of to of ý of

Ans.
2. Reduce of a grain to the fraction of a pound,
Froy weihgt.

of
of

of 12 of 11520 Ans.
3. Reduce of a day the fraction of a year, con-
taining 365 days.

Ans. 1167 RULE 2.-Reduce the Integer to the same denomination with the denominator of the fraction, for a denominator ; over which place the numerator of the given fraction, and it will form the fraction required.

Examples
I. Required to reduce je of a mill, to the fraction of
a dollar.
dolls. d.

m, tenths.
1x10x10x10x10=10000 denominator.

Ans. Τσόσο 2. Reduce of a day to the fraction of a year.

Ans, ico

1 20

24

c.

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