second fraction (3), the parts are 3 times the size of those in the first (3): but, =} of } (Art. 130); consequently, the value of the second fraction is 3 times that of the first. Hence,

PROP. IV. If the denominator be divided, without changing the numerator, the value of the fraction will be multiplied as many times as there are units in the divisor.

Hence, a fraction is multiplied, by dividing its denominator.

ART. 135. If the numerator of a fraction be multiplied by any number, its value (PROP. 1,) will be multiplied by that number; if the denominator be multiplied, the value (PROP. III,) will be divided by that number.

Hence, if both terms are multiplied by the same number, the increase from multiplying the numerator, equals the decrease from multiplying the denominator: and the value is not changed. 3X2 6 3 X 3 9 5X2 10' 5X3 15'

Thus,

and

ILLUSTRATION.—Multiplying both terms of the fraction by 2, gives, in which the parts are twice as many, but only one-half the size. Multiplying both terms of by 3, gives; three times as many parts, each part one-third the size. Hence,

PROP. V. Multiplying both terms by the same number, changes its form, but does not alter its value.

ART. 136. If the numerator of a fraction be divided by any number, its value (PROP. II,) will be divided by that number; if the denominator be divided, the value (PROP. IV,) will be multiplied by that number.

Hence, if both terms are divided by the same number, the decrease from dividing the numerator, equals the increase from dividing the denominator: and the value is not changed.

REVIEW.-132. What is the effect of dividing the numerator of a fraction, without changing the denominator? 133. What of multiplying the denominator without changing the numerator?

134. What of dividing the denominator, without changing the numerator? 135. What of multiplying both terms by the same number?

ILLUSTRATION.-Dividing both terms of the fraction by 2, it gives; in which there are one-half as many parts, but each part is twice the size. Dividing both terms of by 3, gives, third as many parts, each part three times the size. Hence,

12

one

PROP. VI. Dividing both terms by the same number, changes its form, but does not alter its value.

TO TEACHERS.-By considering the numerator a dividend, the denominator a divisor, and the value of the fraction the quotient (Art. 125), the preceding propositions may be regarded as inferences from Art. 57, 58, 59. This short method is not best adapted to young pupils.

ART. 137. REDUCTION OF FRACTIONS

Is changing their form without altering their value.

CASE I.

ART. 138. To reduce a fraction to its lowest terms.

A fraction is in its lowest terms, when the numerator and denominator are prime to each other. Art. 110, Def. 5.

Thus, is in its lowest terms, while 34 is not.

0

2);

Ans.

30

15 5

factor, divide both terms by it; the fraction then becomes 13. Again, since 3 is a factor of 12 and 15, divide them both by it; the result,, can not be reduced lower.

Instead of dividing by 2, and then by 3, divide at once by 6, the greatest com. divisor of the two terms, and the result is the same.

SECOND OPERATION.

6)

24

30

4

Ans. 5

Solve the two following Examples by both methods.

NOTE. All subsequent Examples having a star, *, are intended to illustrate the principles on which the next succeeding rule is founded. The pupil should solve them and explain the operation, referring, at the conclusion of the exercise, to the rule which follows.

Rule for Case I.-Divide the numerator and denominator by any common factor; divide the resulting fraction in the same manner, and so on till no number greater than 1 will exactly divide both terms.

Or, Divide the numerator and denominator by their greatest common divisor; the resulting fraction will be in its lowest terms.

REM. When the terms of a fraction are small, the first method is most convenient; when large, the second method.

EXPRESS IN ITS SIMPLEST FORM,

16. The quotient of 391 divided by 667. 17. The quotient of 585 divided by 1287. 18. The quotient of 796 divided by 14129.

CASE II.

Ans..

Ans. T

・Ans. •

ART. 139. To reduce an improper fraction to a whole or mixed number.

1. In 4 halves (4) of an apple, how many apples? in 6 thirds (9)? in 8 fourths (8)? ing? in 12?

in of a bushel, how many

4

2. In 8 pecks, that is, bushels? in? in 40? in 11?

4

in 13?

REVIEW.-136. What is the effect of dividing both terms of a fraction by the same number? 187. What is Reduction of Fractions?

138. When is a fraction in its lowest terms?

is a fraction reduced to its lowest terms, Rule?

Give an example. How

OPERATION.

3. In 9 fourths (2) of a dollar, how many dollars? SOLUTION. Since 4 fourths make one dollar, there are as many dollars as there are times 4 fourths in 9 fourths; that is, 21 dollars.

4. Reduce to a mixed number.

SOLUTION. Since 5 fifths make 1 (unit), there will be as many ones as there are times 5 in 17; that is, 32.

*5. In 3 of a dollar, how many dollars?

23

*6. Reduce 25 to a mixed number.

4)9 Ans. $21.

OPERATION.

5)17 Ans. 3.

Ans. 23.

Ans. 81.

Rule for Case II.-Divide the numerator by the denominator: the quotient will be the whole or mixed number.

4

7. In 53 of a dollar, how many dollars?

Ans. 131.

11.

19

18.

REDUCE TO WHOLE OR MIXED NUMBERS,

Ans. 1. 15. 6437.

298.

Ans. 21178

79

ART. 140. To reduce a whole or mixed number to an

REVIEW.-188. Why is the value of a fraction not altered by being reduced to its lowest terms? 139. How is an improper fraction reduced to a whole or mixed number, Rule?

6. In 5 dollars, how many fourths? Or, reduce 53 to an improper fraction.

Ans. 35.

Ans. 63.

*7. In 8 apples, how many fourths?

*8. Reduce 123 to an improper fraction.

Rule for Case III.-Multiply the whole number by the denominator of the fraction; to the product add the numerator, and write the sum over the denominator.

REM. The analysis of question 6, shows that the whole number is really the multiplier, and the denominator the multiplicand; but the result will be the same (Art. 30), if the denominator be taken as the multiplier.

9. In 5 dollars, how many tenths? 10. In 15 yards, how many sixths? 11. In 2613 days, how many 24ths?

REDUCE TO IMPROPER FRACTIONS,

53

Ans. 8. Ans. 93. Ans. 637.

6

24

ART. 141. To reduce a whole number to a fraction

having a given denominator.

1. Reduce 3 to a fraction whose denominator is 4.