duced to. Now, as 1 and 3 are prime to each other, the fraction is in its lowest terms.

SECOND OPERATION.

16) 19

=

Ans.

The same result is often more readily obtained by dividing the terms of the fraction by their greatest common divisor, as by the second operation.

Since dividing the numerator and denominator of a fraction by the same number, or cancelling equal factors in both, changes only the form of the fraction, while the value expressed remains unchanged (Art. 217).

RULE. - Divide the numerator and denominator by any number greater than 1 that will divide them both without a remainder, and thus proceed until they are prime to each other. Or,

Divide both the numerator and denominator by their greatest common divisor.

whole or mixed number.

Ex. 1. How many yards in 17 of a yard?

OPERATION.

19) 117 (6, Ans.

14

3

Ans. 6.

Since 19 nineteenths make one yard, it is evident there will be as many yards in 117 nineteenths as 19 is con tained times in 117, which is 63 times. Therefore, 6 yards is the answer required.

RULE. · Divide the numerator by the denominator.

NOTE. -Should there a remainder occur,

make this fraction a part of the answer.

write it over the denominator, and

222. To reduce a whole or mixed number to an improper fraction.

Ex. 1. Reduce 19 to a fraction whose denominator shall be 7.

RULE.

Ans. 88.

Since there are 5 fifths in 1 whole one, in 17 whole ones there are 85 fifths, and adding fifths for the fraction, we have 88 as the equivalent of 173. Hence the

Multiply the whole number by the given denominator, and to the product add the numerator of the fractional part, if any; and write the result over the denominator.

NOTE. A whole number may be expressed in its simplest fractional form, by taking it for a numerator with 1 for a denominator. Thus, 4 may be written , and read 4 ones.

3. Change 5 to a fraction whose denominator shall be 17.

Ans. 우 Ans. 9554.

Ans. 1101.

9. Reduce 984 to an improper fraction. 10. Reduce 11631 to an improper fraction. 11. 718 equal how many ninety-sevenths? 12. Reduce 100188 to an improper fraction. 13. Reduce 7 to an improper fraction. 14. Reduce 19 to a fraction whose denominator shall be 13.

Ans. 69691. Ans. 20g 20000.

Ans. 247.

15. 116 yards equal how many fourths of a yard?

Ans. 465 fourths.

223. To reduce a compound fraction to a simple fraction. Reduceof to a simple fraction.

Ex. 1.

OPERATION.

, Ans.

Ans.

By multiplying the denominator of by 4, the denominator of, it is evident, we obtain of =2, since the parts into which the number is divided are 4 times

as many, and consequently only as large as before; and since of will be 3 times

} =

=

7

32 of

RULE.

=

Multiply all the numerators together for a new numerator, and all the denominators for a new denominator.

NOTE 1. All whole and mixed numbers in the compound fraction must be reduced to improper fractions, before multiplying.

NOTE 2. When there are factors common to both numerator and denomi nator, they may be cancelled in the operation.

EXAMPLES.

2. Reduce of 175 of 15 of 23 to a simple fraction. Ans. T

6. What is the value of of § of 1 of 21? ΤΙ

7. What is the value of of 15% of 5 of 100?

8. What is of 4 of 11?

Ans. 57581.

9. What is the value of of of 1⁄2 of $72?

Ans. $1.75.

10. What is the value of of of 1 of 33 gallons?

224.

A COMMON DENOMINATOR.

Fractions have a common denominator when all their denominators are alike.

225. A common denominator of two or more fractions is a common multiple of their denominators; and their least common denominator is the least common multiple of their denomi

nators.

226.

To reduce fractions to a common denominator. Ex. 1. Reduce, 12, and 1 to other fractions of equal value, having a common denominator.

=

FIRST OPERATION.

7 X 12 X 16 13 4 4 new numerator. 5 X 8 X 16 = 640 66 11 X 8 X 12

66

1056

66

66

1344 = 1536 12=640 Ans. 18 = 1888.

8 X 12 X 16 = 15 3 6 common denominator.

1536

1056

We first multiply the numerator of by the denominators 12 and 16, and obtain 1344 for a new numerator. We next multiply the numerator of by the denominators 8 and 16, and obtain 640 for a new numerator; and then we multiply the numerator of by the denominators 8 and 12, and obtain 1056 for a new numerator. Finally, we multiply all the denominators together for a common denominator, and write it under the several numerators, as in the operation.

By this process, since the numerator and denominator of each fraction are multiplied by the same numbers, their relation to each other is not changed, and the value of the fraction remains the same. (Art. 217.)

16 X 3 = tor.

48, least common multiple, and least common denomina.

Having first obtained the least common multiple of all the denominators of the given fractions, we assume this to be their least common denominator. We then take such a part of this number, 48, as is expressed by each of the fractions separately for their respective new numerators. Thus, to get a new numerator for, we take of 48, the least common denominator, by dividing it by 8, and multiplying the quotient 6 by 7. We proceed in like manner with each of the fractions, and write the numerators thus obtained over the least common denominator. In this process the value of each fraction remains unchanged, as both terms are multiplied by the same number. (Art. 217.)

The method used in the second operation, it will be perceived, expresses the fractions of the result in lower terms than that used in the first. On this account it is often to be preferred to the other.

RULE. Find the least common multiple of the denominators for the LEAST COMMON denominator.

Divide the least common denominator by the denominator of each of the given fractions, and multiply the quotients by their respective numerators, for the new numerators. Or,

Multiply each numerator by all the denominators except its own, for the new numerators; and all the denominators together for A COMMON denominator.

NOTE 1. - Compound fractions must be reduced to simple ones, whole and mixed numbers to improper fractions, before finding a common denominator, and all to their lowest terms, before finding the least common denominator.

NOTE 2. - Fractions may sometimes be reduced to a common denominator most readily by multiplying both terms of one or more of them by such a number as will make all the denominators alike. Thus and may be brought to a common denominator simply by multiplying both terms of the by 2, and changing in that way its form to 2.

NOTE 3.- Fractions may often be reduced to lower terms, without destroying their common denominator, by dividing all their numerators and denominators by a common divisor.

EXAMPLES.

Reduce the following fractions to their least common denominator:

2. Reduce, 5, 1, and 11⁄2.

19

3. Reduce, 5, 18, and . Ans. 185, 24.

40

Ans. 무솔, 녹물, 우표, 우울.

7%, 7, 138, 1720.

720 704 1254 1320 1320•