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48

Examples. 1. Reduce to its lowest 2. Reduce as to its lowest terms.

terms.

Ans.
48)272(5
240

3. Reduce 144 to its lowest Thus 16),42=An. terms.

Ans. . 32)48(1 32

4. Reduce me to its lowest terms.

Ans. ż. Gr.com.mea. 16)32(2

32

Case IV. To reduce a mixed number to its equivalent improper fraction.

Rule. * Multiply the whole number by the denominator of the fraction, and add the numerator to the product; this sum written over the denominator will be the fraction required.

Examples. 1. Reduce 8 to an improper 2. Reduce 27} to an improper fraction.

fraction.

Ans. 245 8

3. Reduce 354 to an improper 3

fraction.

4. Reduce 1001% to an impro24 then Ans.

Ans. 5919 2

5. Reduce 364 to an improper 26

fraction.

Ans. 23

9

Ans. 179

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per fraction.

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Case V. To reduce an improper fraction to its equivalent whole or mixed number.

Rule.t Divide the numerator by the denominator and the quotient will be the whole number, and the remainder, if any, will be the numerator to the given denominator.

*All fractions represent a division of the numerator by the denominator, which taken together are proper and adequate expressions for the quotient. Thus 2 divided by 3, is 2.3; whence the reason of the rule is manifest ; for if a quantity be multiplied and divided by the same number, it evidently remains the

A whole number may be changed into an equivalent fraction with a given denominator, by multiplying the whole number by the denominator and writing the product over said denominator.

† This rule is evidently the reverse of the preceding, and is the same as Simple Division,

same.

Examples. 1. Reduce to its equivalent 2. Reduce to its equivalent whole or mixed number.

whole number.

Ans. 7. 3) 76 ( 254 Ans. 6

3. Reduce 993 to its equiva16

lent mixed number. 15

Ans. 6116

1

Case VI. To reduce a compound fraction to an equivalent single one. Rulf.* Multiply all the numerators together for a new numerator, and all the denominators together for a new denominator ; then reduce this new fraction to its lowest terms.

Examples. 1. Reduce } of off to a sin 2. Reduce 3 of } off of ir to gle fraction.

a single fraction.
1x3x5
3x5x8=14=;Ans,

Ans. It

Case VII. To reduce fractions of different denominators to equivalent fractions having a common denominator.

Rule. Multiply each numerator into all the denominators except its own for a new numerator, and all the denominators together for a common denominator.

Examples. 1. Reduce ļ, and I to a com 2. Reduce ļ, and 4 to a common denominator.

mon denominator. 1 X3 X4=12 new num. for 1

Aps. + 48 49 2x2x4=16

3. Reduce it, of 1^, 's and 3x2x3=18

to a common denominator. 2x3 x4=24 common denom.

Ans. 1'shž Thus 13, ji and Ans.

4. Reduce ì fand {to arom| mon denominator. Ans. 48 48 70

ISOIS 13194 11446

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* If part of the compound fraction be a whole or mixed pumber, it must be reduced to an improper fraction. If any denominator of a compound fraction be equal to a numerator of the same, both may be expunged, and the other numbers, multiplied as by the rule, will produce the fraction required in lower terms.

+ By examining the operation it will be seen that the numerator and denominator of every fraction are multiplied by the very same numbers, and consequently their values are not altered.

Case VIII.

To reduce fractions of different denominators to equivalent fractions having the least common denominator.

Rule 1*_Find the least common multiple of all the denominators of the given fractions, and it will be the common denominator required.

2. Divide the common denominator by the denominator of each fraction, and multiply the quotient by the numerator, the products will be the numerators required.

Examples
1. Reduce 3, 5, and to the least common denominator.

6;=3 and 3x1=3
6:3=2 2x1=2 New numerators.
6:6=1 1 x 5=5

2)2 3 6

3)1 3 3

111 2x3=6 least com. mult. then ž, ž; & Ans. 2. Reduce y I and

to the 3. Reduce è, , and to the least common denominator. least common denominator.

II
18

3

Ans. 14

11
18

Ans. g go

18

Case IX. To find the value of a fraction in known parts of an integer. RULE.--Multiply the numerator by the parts of the next inferior denomination, and divide the product by the denominator; if any thing remain, multiply it by the next inferior denomination, and divide by the denominator as before, and so on as far as necessary; the quotients will be the answer required.

Examples. 1. What is the value of of a 2. What is the value of i, of pound?

a pound ?

Ans. 5s. 3 20

3. What is the value of } of a day?

Ans. 10h. 17m. 84s. 8)60(7s. Aps. 7s. 6d. 56

4. What is the value of of a

mile ? 4

Ans. 6fur. 26rds. 3yds. 2ft. 12

8)48(60.

* The common denominator is a multiple of all the denominators, and consequently will divide by any of them: therefore, proper parts may betaken for all the numerators as required.

Case X. To reduce a fraction of one denomination to that of another, retain

ing the same value. Rule.-Make a compound fraction of it, and reduce it to a single

one.

Examples. 1. Reduce of a penny to the

fraction of a penny.

5. Reduce I's of a pound to the fraction of a pound. of tofzo, compound fraction. It's of 29 of comp. fraction. Then & Tax=100=o ! There's ***Y=42=4 Ans.

Ans.

6. Reduce is of a month to the 2. Reduce & of a pound to the fraction of a day. Ans. 14. fraction of a cwt.

Ans. 2 S. Reduce 3s. 6d. to the frac. 7. Reduce tło cwt. to the frac

tion of a pound.

Ans. tion of a pound.

Ans. Jo 4. Reduce of a pound to the 8. Reduce 120 lb. Troy, to fractiou of a guinea. Ans. 4.

the fraction of a pwt.

Ans. j

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2. ADDITION OF VULGAR FRACTION'S.

Rule.-Reduce compound fractions to single ones ; mixed numbers to improper fractions, fractions of different integers to those of the same, and all of them to a common denominator ; then the suin of the numerators, written over the common denominator, will be the sum of the fractions required.

Examples. 1. Add 8, 73, and į of toge. 2. What is the sum of ik of a ther.

week, # of a day, and 3 of an First 71=ų, and of = i. hour ?

Ans. 2d. 144h. Then, ys, and in are the tractions.

3. What is the sum of 3 of 67, 5 X 2 x 12 120

of 2, and 72 ?

Ans. 107 15 x 8 x12=1440 3X8 X 2= 48

4. Adds of a yard, of a foot, 1608

and f of a mile. =817,3=83

Ans. 660 yds. 2 ft. 9 in. 8x2x12 192

Ans.

* By reducing fractions to a common denominator, they are made to express similar parts of the same unit, and as each numerator shows how many of ihose parts are signified by the fraction, the sum or difference of the womerators written over the common denominator, is evidently the sum or difference of the fractions,

3. SUBTRACTION OF VULGAR FRACTION'S. Rule.- Prepare the fractions as for addition, and the difference of the numerators written over the common denominator will be the difference of the fractions required.

Examples. 1. From 2 take of 3.

4. From 7 weeks take 9% days 011=6= and f-=

Ans. 5 w. 4 d. 7 h. 12 m. 13-0= Hi=11 Ans.

5. From € take is. 2. From 961 take 144.

Ans. 9s. 3d. Ans. 81.11.

6. From 141 take 3 of 19. 3. From 48 take z.

Ans. 112 Ans. 196

4. MULTIPLICATION OF VULGAR FRACTIONS.

ICATION

Rule.--Reduce compound fractions to single ones, and mixed numbers to improper fractions; then multiply the numerators together for the numerator, and the denominators together for the de. nominator of the fraction required.

Ans. 4.

Examples. 1. Multiply 41, 4 of 4, and 184 | 4. Multiply š of by 8 of 54. continually together. 44=1. of ans, and 184-94. The X X4=2389266

5. Multiply ty by 4. Ans. I'y. 918 Aus.

6. Multiply of 5 by 1 of 4.

Ans. 5. 2. Multiply 54 by 4. Ans. 7. S. Multiply 44 by . Ans.

7. Multiply d by }.

Ans.

5. DIVISION OF VULGAR FRACTIONS.

RULE.-Prepare the fractions as for Multiplication, then invert the divisor, and proceed exactly as in Multiplication.

* Fractions are sometimes most conveniently brought to a common denominator by Multiplication or Division. In the first example s is brought to a common denominator with 1 by multiplying both its terms by 7.

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