Since the denominator denotes the number of equal parts into which the unit is divided, if we multiply the denominator of a fraction by any number, as 5, the unit will be divided into 5 times as many equal parts, hence each fractional unit will be as large as before, and the same number of fractional units being taken, the value of the fraction is as great.

4. Dividing the denominator of a fraction by any number multiplies the value of the fraction by that number.

Since the denominator denotes the number of equal parts into which the unit is divided, if we divide it by any number, as 4, the unit will be divided into as many equal parts, hence each fractional unit will be 4 times as large as before, and the same number of fractional units being taken, the value of the fraction will be 4 times as great.

5. Multiplying both numerator and denominator of a frac tion by the same number does not change the value of the fraction..

Since multiplying the numerator multiplies the value of the fraction, and multiplying the denominator divides the value of the fraction, multiplying both numerator and denominator both multiplies and divides the value of the fraction by the same number, and hence does not change its value.

6. Dividing both numerator and denominator of a fraction by the same number does not change its value.

Since dividing the numerator divides the value of the fraction, and dividing the denominator multiplies the value, dividing both numerator and denominator both divides and multiplies the value of the fraction, and hence does not change its value.

157. These principles may be embodied in one general law as follows:

General Principle.—A change in the NUMERATOR by mul tiplication or division produces a SIMILAR change in the value of the fraction, but such a change in the DENOMINATOR produces an OPPOSITE change in the value of the fraction.

REDUCTION OF FRACTIONS.

158. The Reduction of Fractions is the process of changing their form without altering their value.

159. There are Six Cases of reduction:

1st Numbers to fractions. | 4th. To lower terms.

2d. Fractions to numbers. 5th. Compound to simple 3d. To higher terms.

6th. Complex to simple.

NOTE.-Reducing to a Common Denominator is included in these stx

cases.

CASE I.

160. To reduce whole or mixed numbers to improper fractions.

MENTAL EXERCISES.

1. How many fifths in 4?

SOLUTION.-In one there are 5 fifths, and in 4 there are 4 times 5 fifths, or 20 fifths, which added to 3 fifths, equal 23 fifths; therefore 48=23. 2. How many fourths in 7? in 54? in 9? in 12? 3. How many fifths in 63? in 7? in 8? in 10? 4. How many sixths in 43? in 88? in 9? in 11g?

5. How many sevenths in 74? in 94? in 84? in 124?

6. Describe the operation we perform in reducing a mixed number to a fraction.

WRITTEN EXERCISES.

1. Reduce 27 to fourths.

SOLUTION.-In one there are 4 fourths, and in 27 there are 27 times 4 fourths, or 108, which added to the, equals 111. Therefore, etc.

Rule.-Multiply the whole number by the denominator of the fraction, add the numerator to the product, and write the denominator under the sum.

5. 11.

6. 27.

Ans. 11.

Ans. 431.

Ans. 312; 970.

Ans. 104. 10. 8214; 100,9%. Ans. 1744; 10882.
Ans. 221. 11. 4948; 23547. Ans. 2482; 15817

CASE II.

161. To reduce improper fractions to whole or mixed numbers.

MENTAL EXERCISES.

1. How many units in 23?

SOLUTION.-In one there are, hence in 23 fourths there are as many ones as 4 is contained times in 23, which are 53. Therefore 2354.

2. How many units in 18? in 26 ? in 35? 3. How many units in ? in 2? in 25? 4. How many units in 30? in ? in t§?

5. How many units in ? in ? in §?

6. Describe the process of reducing an improper fraction to a mixed number.

WRITTEN EXERCISES.

1. How many units in 2? SOLUTION. Since dividing both terms of a fraction by the same number does not change its value (Prin. 6), by dividing both terms by 5, we have 15, or 15.

1

OPERATION.

79-151.

Rule.-Divide the numerator by the denominator, and the quotient will be the whole or mixed number.

Reduce to whole or mixed numbers,

162. To reduce fractions to higher terms.

163. Reducing a Fraction to higher terms is the process of reducing it to an equivalent fraction, having a greater numerator and denominator.

MENTAL EXERCISES.

1. Change to twelfths.

SOLUTION.-In one there are 14, and in there are of 12, which are and in there are 2 times, which are; therefore =

2. Change and to twentieths; and to fifteenths.

3. Change and § to thirtieths; and to sixteenths.

4. Change and to seventieths; and to eighteenths.

5. Change and to seventy-seconds;

and to 120ths.

6. Describe the process of reducing a fraction to higher terms.

WRITTEN EXERCISES.

1. How many twentieths in ?

SOLUTION. Since multiplying both terms of a fraction by the same number does not change its value (Prin. 5), we multiply both terms by the number which will give the required denominator, which we see is 4; hence, 3=18.

Rule.-Multiply both numerator and denominator by the number which will give the required denominator.

164. To reduce fractions to lower terms.

165. Reducing a Fraction to lower terms is the process of reducing it to an equivalent fraction having a smaller numerator and denominator.

Principle.-A fraction is in its lowest terms when the numerator and denominator are prime to each other.

MENTAL EXERCISES.

1. Reduce to fourths.

SOLUTION.-One equals 1, and

equals of 1, or; since equals

equals as many fourths as 3 is contained times in 9, which is 3; hence equals. Therefore, etc.

8. Describe the process of reducing a fraction to lower terms.

WRITTEN EXERCISES.

1. Reduce to fifths.

SOLUTION. Since dividing both terms of a fraction by the same number does not change its value (Prin. 6), we may reduce to lower terms by dividing both numerator and denominator by 6; dividing, we have

equal to ; and since the terms 4 and 5 are prime to each other, the fraction is in its lowest terms. Therefore, etc.

Rule I.-Divide both terms successively by their common factors.

Rule II.-Divide both terms by their greatest common divisor.

SOLUTION. of is one of the three equal parts into which may be divided; if each fourth is divided into 3 equal parts, 4 fourths, or the unit, will be divided into 4 times 3, or 12 equal parts; hence each part is of a unit. Therefore, ofis.

7. In reducing a compound fraction to a simple one, what do we do with the numerators, and what with the denominators?

Rule.-Multiply the numerators together and the denomi nators together, cancelling the factors common to both terms.

NOTE.-Reduce whole or mixed numbers to fractions before commencing the reduction to a simple fraction. To reduce complex fractions to simple ones, see Art. 183.