Ans.

and

a2m+a2n 3m3-3mn2 3a2m 3am+3amn' 3am+3amn' 3am2+3amn ART. 134.-It frequently happens, that the denominators of the fractions to be reduced, contain one or more common factors. In such cases, the preceding rule does not give the least common denominator. From the preceding Article we see, that the common denominator is a multiple of all the denominators; and, that each numerator is multiplied by a quantity which is equal to the quotient obtained, by dividing this multiple by its denominator. Thus, in the second example, nr, mr, and mn, the quantities by which each numerator is respectively multiplied, may be regarded as the quotients obtained, by dividing mnr successively, by m, n, and r. Now, if we obtain the least common multiple of the denominators, by the rule, Case III., and then divide it by each denominator respectively, and multiply the quotients by the numerators respectively, we shall obtain a new class of fractions, equivalent to the former, and having for a common denominator, the least common multiple of the given denominators. It is easily seen, that both terms of each fraction are multiplied by the same quantity, and hence, that the resulting fractions are equivalent to the given ones.

The least common multiple of the denominators is easily found to be bcd; dividing this by b, the denominator of the first fraction, the quotient is cd; then multiplying both terms of by cd, the

m

b

The process of multiplying the denominators by the quotients may be omitted, as the product in each case will be equal to the least common multiple. Hence, the

RULE,

FOR REDUCING FRACTIONS OF DIFFERENT DENOMINATORS, TO EQUIVA LENT FRACTIONS, HAVING THE LEAST COMMON DENOMINATOR.

1st. Find the least common multiple of all the denominators; this will be the common denominator.

2d. Divide the least common multiple, by the first of the given denominators, and multiply the quotient by the first of the given numerators; the product will be the first of the required numerators. 3d. Proceed, in a similar munner, to find each of the other numerators.

NOTE. Each fraction should be in its lowest terms, before commencing the operation.

Peduce the following fractions, in each example, to equivalent fractions, having the least common denominator.

Other exercises will be found in the addition of fractions.

NOTE. The two following Articles depend on the same principle as the two preceding, and are, therefore, introduced here. They will both be found of frequent use, particularly in completing the square, in the solution of equations of the second degree.

ART. 135.—To reduce an entire quantity to the form of a fraction having a given denominator.

1. Let it be required to reduce a to a fraction having b for its denominator.

α

1

Since any quantity may be reduced to the form of a fraction, by writing 1 beneath it, a is the same as ; if we multiply both terms by b, which will not change its α ab have b'

for the required fraction.

RULE,

value (See Art. 126), we Hence, the

FOR REDUCING AN ENTIRE QUANTITY TO THE FORM OF A FRACTION HAVING A GIVEN DENOMINATOR.

Multiply the entire quantity by the given denominator, and write

the product over it.

EXAMPLES.

4x

2. Reduce x to a fraction, whose denominator is 4. Ans. 3. Reduce m to a fraction, whose denominator is 9a2.

4

REVIEW.-134. How do you reduce fractions of different denominators to equivalent fractions, having the least common denominator?

4. Reduce 3c+5 to a fraction whose denominator is 16c2.

5. Reduce a-b to a fraction, whose denominator is a2-2ab+b2.

ART. 136. To convert a fraction to an equivalent one, having a denominator equal to some multiple of the denominator of the given fraction.

α

1. Reduce to a fraction, whose denominator is bc.

b

It is evident, that the terms must be multiplied by the same quantity, so as not to change the value of the fraction. It is then required to find, what the denominator, b, must be multiplied by, that the product shall become bc; but, it is evident, this multiple will be found, by dividing be by b, which gives the quotient, c. Then, multiplying both terms of the fraction by c, the result is

ac

bc'

a

α which is equal to the given fraction and has, for its denom

inator bc. Hence, the

RULE,

b'

FOR CONVERTING A FRACTION TO AN EQUIVALENT ONE, HAVING A GIVEN DENOMINATOR.

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

REMARK.-This rule is perfectly general, but it is never applied, except where the required denominator is a multiple of the given one. In other cases, it would produce a complex fraction. Thus, if it is required to convert into an equivalent fraction, whose denominator is 5, the numerator of the new fraction would be 24.

3

2. Convert to an equivalent fraction, having the denominator 16.

12

Ans.

16

REVIEW.-134. If each fraction is not in its lowest terms, before commencing the operation, what is to be done? 135. How do you reduce an entire quantity to the form of a fraction having a given denominator?

6. Converi to an equivalent fraction, having the denomi

α

b+c

nator a2(b+

CASE V.

Ans.

a3 (b+c) a2(b+c)2°

A DITION AND SUBTRACTION OF FRACTIONS.

ART. 137. -1. Let it be required to find the sum of and . Here, both parts being of the same kind, that is, fifths, we may add them together, and the sum is 6 fifths, (§).

α m

2. Let it be required to find the sum of and

b

m

Here, the parts being of the same kind, that is, mths, we may, as in the first case, add the numerators, and write the result over the common denominator.

3. Again, let it be required to find the sum of

Here, the parts not being of the same kind, that is, the denominators being different, we can not add the numerators together, and call them by the same name. We may, however, reduce them to a common denominator, and then add them together.

Reduce the fractions, if necessary, to a common denominator; add the numerators together, and place their sum over the common denominator.

ART. 138.—It is obvious, that the same principles would apply, if it were required to find the difference between two fractions; that is, if their denominators were the same, the numerators might be subtracted; but, if their denominators were different, it would be necessary to reduce them to the same denominator, before performing the subtraction. Hence, the

RULE,

FOR THE SUBTRACTION OF FRACTIONS.

Reduce the fractions, if necessary, to a common denominator; then subtract the numerator of the fraction to be subtracted from the numerator of the other, and place the remainder over the common denominator.

When entire quantities and fractions are to be added together, they may be connected by the sign of addition, or the entire quantities and the fractions may be reduced to a common denominator, and the addition then performed.

15. Add 2x, 3x+ and x+ together... Ans. 6x+

37%

45

3a together.

15a+by+9

Ans.

.

3ay

Ans. 0.

1

Ans.

1

-x

32
5

22

9

REVIEW.-136. How do you convert a fraction to an equivalent one, having a given denominator? Explain the operation by an example. 137. When fractions have the same denominator, how do you add them together? When fractions have different denominators, how do you add them together?