9. How is the quotient of 4 divided by 3 expressed? What is the expression called?

10. If I divide an apple into halves, and give away of, what part of the apple do I give way? What is the expression, of called?

In the foregoing question, the unit is divided into two equal parts, and each part is of the unit. A division is again made of one of these parts into two other equal parts, and each part is of, or of the unit first divided. The expression,, as it respects the unit of which it is a part, is a fraction, (see definition, Art. 52;) but as it respects itself, or a subsequent division, it is to be regarded as itself a unit, and may be divided into halves, or any number of parts. A quarter, or of a thing, is a whole quarter; and is made up of as many parts as the thing of which it is a part. It is, therefore, in relation to a division already made, that an expression is to be regarded as a fraction. As it respects itself, or a subsequent division, it is to be considered a unit. Example.-A yard may be divided into 3 equal parts, or feet. A foot, when spoken of in relation to the yard, is ; but of a yard is one foot, and may be divided into 12 equal parts, or inches, and each inch is of a foot, or of of a yard. The inch may be divided into 3 equal parts, or barley-corns, and each barley-corn is itself a unit of less value, and it is also a fraction of a unit of a higher value; that is, 1 barley-corn is of of of a yard. That the terms, unit, and fraction, are merely relative, may be seen by the following formula :

yd.

1

12

12

in. ́bar:

1÷3=1=1, and 1÷÷121, and 1÷3=3=1.

Art. 56. To reduce a compound fraction to a simple one. 1. Reduce of to a simple fraction.

If we multiply the denominator of by 3, we obtain onethird of 4. If we multiply this numerator by 2, we obtain twothirds. Hence the

RULE.

Multiply the numerators together, and the denominators, having cancelled all the equal factors in the numerators and denominators.

Oper...

1

4 2 6

2. Reduce of of to a simple fraction.

Ans. 4.

3. Reduce of 2 of 3 of to a simple fraction. Ans. 32. Art. 57.-To change any given fraction to an equivalent fraction, which shall have any required denominator.

Change to an equivalent fraction whose denominator shall be 6. In this example, the unit is already divided into thirds, and we wish to divide it into 6ths: We have, therefore, simply to reduce thirds to sixths. 2 sixths make a third, for the unit is divided into twice as many parts, and therefore the parts are one-half as large. Hence the

RULE.

Divide the required denominator by the denominator of the given fraction, and multiply the quotient by the numerator. The product will be the required numerator.

Art. 58. To reduce a whole number to an equivalent fraction, having a given denominator.

1. Reduce 8 to a fraction whose denominator shall be 4. As in 1 unit there are 4 fourths, so in 8 units there must be 8×4=32 fourths, expressed thus: 32; therefore the

RULE.

Multiply the whole number by the given denominator, and set the product over the given denominator.

2. Reduce 16 to a fraction whose denominator shall be 7. Ans. 112.

3. Reduce 40 to a fraction whose denominator shall be 9. Ans. 360.

4. Reduce 129 to a fraction whose denominator shall be 21.

Ans. 2709

21

5. Reduce 339 to a fraction whose denominator shall be 39. Ans. 13221.

A whole number may be expressed fractionally, by writing 1 under it for a denominator.

As the expression,, is equal to 2, and to 3, the value

QUESTION.-How is a whole number reduced to an equivalent fraction, having › given denominator?

of a number is not affected by writing 1 under it, as a denominator.

To reduce improper fractions to mixed numbers, and mixed numbers to improper fractions.

Divide the numerator by the de- Multiply the whole number by nominator, and the quotient will be the denominator of the fraction, the whole number; the remainder, and to the product add the numerif any, written over the denomina-ator; under the result, place the lor, must be placed at the right | denominator of the fraction.

hand of the quotient.

Art. 61. To reduce a fraction to its lowest terms.

1. Reduce to its lowest terms.

If 4 bushels were divided equally between two persons, it is evident that one person would receive of 4 bushels, or 2 bushels; so if of a bushel be divided equally between two persons, one person will receive one half of, or of a bushel. Dividing the numerator by 2, we take one half of those parts which are contained in the fraction, while the value of each part remains the same. Therefore, To divide the numerator diminishes the value of the fraction.

If we divide the denominator of by 2, the fraction becomes. In this expression the unit is divided into half as many parts as at the first, and consequently, these parts are twice as large. It is evident, therefore, that To divide the denominator of a fraction, the numerator remaining the same, increases its value.

If we divide the terms of the fraction by 2, it becomes, which is equal to 2, or, for in either case the numerator is one half of the denominator. Hence it appears, that the value of a fraction is not affected by dividing or multiplying both the numerator and denominator by the same number. (See Art. 43.) To reducea fraction to its lowest terms, we have this

RULE.

Divide both the numerator and denominator by any number that will divide both without a remainder; and so continue to do until no number greater than 1 will divide them.

2. Reduce 84 to its lowest terms.

210

77 100 83

Ans. 19.

4. Reduce, 17, 188, 23, to their lowest terms.

Art. 62.—Were the greatest number known which would divide the terms of the fraction, a simple division would at once reduce the fraction; but, as this is not the case, the greatest divisor may be found by the following

QUESTIONS.-1. What is the rule for reducing an improper fraction to a whole or mixed number? 2. For reducing a mixed number to an improper fraction?

RULE.

Divide the denominator by the numerator, or the larger numver by the less, and if there be no remainder, the numerator, or the less number, will be that divisor; but if there be a remainder, divide the last divisor by the last remainder, and thus proceed until there be no remainder; and the last divisor will be the greatest common measure sought.

The above Rule may be illustrated in the following manner: Suppose I have two lines, and wish to obtain a third which shall be an exact measure of the two. I first apply the shorter line to the longer, and find it contains it twice, but not three times. The remainder I now apply to the shorter line, and find it contains it twice and no remainder. Therefore, this last divisor is the third line sought, and is the exact measure of the other two.

rem.

Suppose the longer line to be 40 feet and the shorter 16 feet, and we are required to find the greatest common measure, or divisor, of 16 and 40, or to reduce 16-40 to its lowest terms, we should proceed in the following manner:

Operation. •16)40(2 32

It is evident that 16 is the greatest number that will divide 16 without a remainder; and would 16 divide 40 without a remainder, it would be the greatest common measure of the 8)16(2 terms of the fraction. But we find, by trial, 16 that 16 is contained in 40 twice and 8 remainder; hence 16 is not the common measure. Dividing 16, the last divisor, by 8, the remainder, we find it contains it twice and no remainder; therefore 8 is the greatest common divisor of the terms of the fraction. For, if 8 will divide 8, it will also twice 8, which is 16, and five times 8, which is 40.

If it be required to find the greatest common measure of more than two numbers, find the greatest common measure of two of them, as before; then, of that common measure, and of one of the other numbers, and so on through the whole. The common measure last found will be the one sought.

QUESTIONS, 3. What is the rule for reducing a fraction to its lowest terms? 4. Were the greatest number known which would divide the terms of the fraction, how might you proceed? 5. When this is not the case, how may the greatest divisor be found? 6. How is the common measure, of more than two numbers found?