REDUCTION OF FRACTIONS.

Reduction changes the terms of a fraction without changing its value. The change is to Higher Terms, to Lower Terms, or to Lowest Terms.

Reduction of Fractions to Higher Terms.

Is not $

=

$2?

May we not multiply the terms of by 2 and thus obtain 4?

PRINCIPLE.

Multiplying both terms of a fraction by the same number does not change the value of the fraction. (See p. 68.)

NOTE. The pupil should perceive that fractions are, by their very nature, subject to the principles of division.

Divide the required denominator by the given denominator, and multiply both terms of the fraction by the quotient.

May we not divide both terms of by 2 and obtain ?

PRINCIPLE.

Dividing both terms of a fraction by the same number does not change the value of a fraction.

(See page 68.)

EXERCISES.

1. Reduce to eighths.

Process.

Explanation.

The division of 16 by 8 shows that both terms of the fraction must be divided by 2 to change sixteenths to eighths. Dividing both 12 and 16 by 2, we have §.

RULE.

Divide the given denominator by the required denominator, and divide both terms of the fraction by the quotient.

Reduction of Fractions to Lowest Terms.

Reduction to lowest terms requires the terms of the fraction to be divided by their greatest common factor (G. C. D.).

The G. C. D. of 1760 and 5280 is 1760. Dividing both terms of the fraction by 1760 we obtain, the lowest terms. (See Principle, p. 95.)

RULE.

Divide both terms of the given fraction by their G. C. D. Continued division by a common factor will secure lower or lowest terms.

The terms are the lowest when they are prime to each other.

Reduction of Integers and Mixed Numbers.
We have learned that, 3, 4, or §, etc., equal one.
How many halves in one whole thing?

How many thirds? Sixths? Tenths?
How many thirds in two? In 23?

Multiply the integer by the denominator, to the product add the numerator, and write the sum over the denominator.

2. Reduce:

1. 61 to fourths.

2. 2 to thirds.
3. 121 to halves.
4. 9 to sevenths.
5. 161 to fourths.

6. 135 to eighths.
7. 3147 to twenty-firsts.
8. 673 to twelfths.
9. 7021 to elevenths.

10. 122 to fifteenths.

11. 15 to fifths.

12. 13 to sixths.

13. 18 to elevenths.
14. 5 to ninths.
15. 5 to eighteenths.
16. 2725 to elevenths.
17. 2784 to ninths.
18. 946 to thirteenths.
19. 615 to fifths.

20. 2414 to twenty-firsts.

Have your results been proper or improper fractions?

Reduction of Improper Fractions.

How many dollars in $? In $18?

3

How many units in 16? In 13? In 37? In 48? In 51? What kind of numbers are your results?

EXERCISES.

1. Reduce 594 to an integer and 595 a mixed number.

Explanation.

Since 594 indicates the division of 504 by 7, we divide and obtain the integer 72.