b. The number of fractional units which equal the unit. c. The kind and denomination of the fractional units. 3. When the numerator of a fraction is equal to the denominator, the fraction is equal to 1.

4. When the numerator of a fraction is less than the denominator, the fraction is less than 1.

5. When the numerator of a fraction is greater than the denominator, the fraction is greater than 1.

NUMERATION AND NOTATION.

213. Numeration of Fractions is the art of reading a fraction when expressed by figures.

Rule. Read the number of fractional units expressed by the numerator, and give them the name indicated by the denominator.

214. Notation of Fractions is the art of expressing fractions by means of figures.

Rule. Write the number of fractional units, draw a line beneath, under which write the number which indicates the kind of fractional units.

Write the following fractions:

1. Two-thirds.

2. Three-fourths.

3. Ten-twelfths.

4. Five twenty-firsts.

5. Nineteen twenty-seconds.
6. Fifty-three hundredths.
7. Fifty three-thousandths.
8. Forty-five ten-millionths.

ANALYSIS OF FRACTIONS.

215. To Analyze a fraction is to explain what is expressed by the fractional notation.

1. Analyze the fraction

SOLUTION. In the fraction, the denominator 5 indicates that the unit is divided into 5 equal parts, and the numerator 4 denotes that 4 of these parts are taken.

216. The Number of Cases of common fractions is eight

217. There are Two Methods of treating common fractions, which may be distinguished as the Inductive and Deductive Methods.

218. By the Inductive Method we solve all the different cases by analysis, and derive the rules or methods of operation from these analyses by inference or induction.

219. By the Deductive Method we first establish a few general principles, and then derive the rules or methods of operation from these general principles.

NOTE. These two methods are entirely distinct in principle and form, and demand attention. We have given both methods, and teachers may use either or both, as they choose. The first solution given under each case is by the inductive method, the second solution is by the deductive method.

PRINCIPLES OF FRACTIONS.

1. Multiplying the numerator of a fraction by any number multiplies the value of the fraction by that number.

If we multiply the numerator of any fraction by any number, as 5, the resulting fraction will express 5 times as many fractional units, each of the same size as before, hence the value of the fraction is 5 times as great. Therefore, etc.

2. Dividing the numerator of a fraction by any number divides the value of the fraction by that number.

If we divide the numerator of a fraction by any number, as 4, the resulting fraction will express as many fractional units, each of the same size as before, hence the value of the fraction is divided by 4.

3. Multiplying the denominator of a fraction by any number divides 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 multiply the denominator of a fraction by any number, as 5, the unit will be divided into 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 6, the unit will be divided into as many equal parts, hence each fractional unit will be 6 times as large as before, and the same number of fractional units being taken, the value of the fraction will be 6 times as great.

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

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 frac tion 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.

7. A fraction is equal to the quotient of its numerator divided by its denominator.

For the fraction is the same as 4 times; but 4 times is equal to of 4; and of 4 is equal to 4 divided into 5 equal parts; and to divide a number into 5 equal parts we must divide it by 5; hence is equal to 4 divided by 5; and since this is general, therefore the principle is correct.

NOTE.-Authors usually assume this principle as true, but it is clear that it is not an immediate inference from the explanation that the denominator denotes the number of equal parts into which the unit is divided, and the numerator expresses the number of equal parts taken.

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

General Principle.-A change in the NUMERATOR by multiplication 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.

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

222. 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 and a Common Numerator are included in these six cases.

CASE I.

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

1. How many fifths in 73?

SOLUTION. In one there are 5 fifths, and in 7 there are 7 times 5 fifths, or 35 fifths, which added to 3 fifths equals 38 fifths. Therefore 73 = 32.

SOLUTION 2D.-7 equals 7, and multiplying both terms by 5, we have 7=35 (Prin. 5); and 35+3= 33. Therefore 73=33.

8

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

Reduce the following to improper fractions:

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

1. How many units in 25?

SOLUTION.-In one there are 7, hence in 25 there are as many ones as 7 is contained times in 25, which are 3. Therefore 2 equals 34.

OPERATION

2,5=34.

SOLUTION 2D.-Since by Prin. 6, dividing both terms by the same number does not change the value of the fraction, by dividing both

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

Reduce to whole or mixed numbers,

225. To reduce fractions to higher terms.

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

1. How many twentieths in ?

OPERATION.

SOLUTION.-In one there are 20, and in there are of 20, which are, and in there are 4 times, which are 16; therefore = 18. SOLUTION 2D.-Since multiplying both numerator and denominator of a fraction by the same number does not alter its value (Prin. 5), we multiply both terms by 4, which gives us 18 6

- 20⚫

4x4

=

5x4

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

227. To reduce fractions to lower terms.

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