Thus,,,, and 4, are each improper fractions.

A fraction of a fraction, connected by the word of, is called a compound fraction.

Thus, of of of 7, of of, and of 11 of 3, are 8 compound fractions.

A fraction is said to be inverted, when the numerator and the denominator change places.

Thus, the fractions 1,, and, when inverted, become 17, 1, and g.

Any integer may take the form of an improper fraction, by writing a unit for its denominator.

Thus, 6, 5, 3, and 11, are the same as the improper frac-, tions,,, and .

A number consisting of an integer and fraction, is called a mixed number.

Thus, 4, 57, 6, and 131, are mixed numbers. They may also be written 4+, 5+, 6+, and 13+}}.

REDUCTION OF FRACTIONS.

16. Since the value of a fraction is the quotient arising from dividing the numerator by its denominator, we may infer the following propositions.

I. That, multiplying the numerator of a fraction by any number, is the same as multiplying the value of the fraction by the same number.

II. That, multiplying the denominator of a fraction by any number, is the same as dividing the value of the fraction by the same number.

III. That, multiplying both numerator and denominator by the same number, does not alter the value of the fraction.

IV. That, dividing the numerator of a fraction by any number, is the same as dividing the value of the fraction by the same number.

V. That, dividing the denominator of a fraction by any number, is the same as multiplying the value of the fraction by the same number.

VI. That, dividing both numerator and denominator by any number, does not alter the value of the fraction.

17. When the numerator and denominator of a fraction have no common measure, it is said to be in its lowest terms. To reduce simple fractions to their lowest terms, we have the following

RULE.

Divide both numerator and denominator by their greatest common measure, (found by one of the rules under Art. 8, or 9.) This division will not alter the value of the fraction. (Prop. VI. Art. 16.)

Examples.

1. Reduce 21 to its lowest terms.

In this example we find the greatest common measure of 375 and 425, to be 25.

Dividing both numerator and denominator by 25, we find 275=14

2. Reduce 1942 to its lowest terms.

8392

Ans. 1.

Ans.

Ans. 3.

Ans. 178.

18. To reduce an improper fraction to a mixed number, we have this

RULE.

Divide the numerator by the denominator, the quotient will be the integral part of the mixed number. The remainder placed over the denominator of the improper fraction, will fórm the fractional part.

The correctness of the above rule is obvious from considering that the value of a fraction is the quotient arising from dividing the numerator by the denominator.

Dividing 17 by 4, we obtain the quotient 4, with the remainder 1; ... the mixed number equivalent to is 41 or 4+1.

2. What mixed number is equivalent to 11?

7. What mixed number is equivalent to 2348?

8. What mixed number is equivalent to 48597354 ?

37538

9. What mixed number is equivalent to 227229 ?

433

4583

10. What mixed number is equivalent to 220072 ? 11. What mixed number is equivalent to 3573 ?

4927

19. To reduce a mixed number to its equivalent improper fraction, we have this

RULE.

Multiply the integral part of the mixed number by the denominator of the fractional part, to the product add the numerator of the fractional part, the sum will be the numerator of the improper fraction; under which place the denominator of the fractional part.

This rule is obviously correct, since it is the reverse of the rule under Art. 18, where a reverse operation was required to be performed.

Examples.

1. Reduce 13 to an improper fraction.

Multiplying the integer 13 by the denominator 7, we obtain 91; to which, adding the numerator 6, we get 97 for the numerator of the improper fraction; ... the improper fraction equivalent to 134 is 27.

2. What improper fraction is equivalent to 12781?

Ans. 3835.

3. What improper fraction is equivalent to 189104 ? Ans. 132374.

4. What improper fraction is equivalent to 4925361} ?

Ans. 64 02979

6402979.

38783

5. What improper fraction is equivalent to 5,2729? 6. What improper fraction is equivalent to 114? 7. What improper fraction is equivalent to 23. ? 8. What improper fraction is equivalent to 375,43? 9. What improper fraction is equivalent to 6833,7443? 10. What improper fraction is equivalent to 11223344?

៖

6

20. Reduce the compound fraction of to its equivalent simple fraction.

of

can be obtained by dividing the value of the fraction by 4, which (by Prop. II., Art. 16) can be effected by

1 7

multiplying the denominator by 4; ...of.

=

4 11 4 x 11'

Again of is obviously three times as great as of;

... to obtain of we must multiply

7

4 × 11

by 3, which

(by Prop. I. Art. 16) can be done by multiplying the numer

Hence, to reduce compound fractions to their equivalent simple ones we have this

RULE.

Consider the word OF, which connects the fractional parts, as equivalent to the sign of multiplication. Then multiply all the numerators together for a new numerator; and all the denominators together for a new denominator. Always observing to reject, or cancel, such factors as are common to the numerators and denominators; which is the same as dividing both numerator and denominator by the same quantity, and (by Rule under Art. 17) does not change the value of the fraction.