I. A fractional is said to be in its simplest form when the numer ator and denominator do not contain a common factor. Now, since both terms of a fraction may be divided by the same quantity without altering its value, we have for the reduction of a fraction to its simplest form the following RULE. Resolve both numerator and denominator into their simple factors (Art. 59); then, suppress all the factors common to both terms, and the fraction will be in its simplest form. REMARK.-When the terms of the fraction cannot be resolved into their simple factors by the aid of the rules already given, resort must be had to the method of the greatest common divisor, yet to be explained. EXAMPLES. 3ab + 6ac 1. Reduce the fraction 3ad + 12a tots simples. form We see, by inspection, that 3 and a factoTM of the merator, hence, 3ab+6ac3a (b + 2c) We also see, that 3 and a are factors the domina nu II. From what was shown in Art. 63, it follows that we may reduce the entire part of a mixed quantity to a fractional form with the same fractional unit as the fractional part, by multiply. ing and dividing it by the denominator of the fractional part. The two parts having then the same fractional unit, may be reduced by adding their numerators and writing the sum obtained over the common denominator. Hence, to reduce a mixed quantity to a fractional form, we have the RULE. Multiply the entire part by the denominator of the fraction: then add the product to the numerator and write the sum over the denominator of the fractional purt. REMARK. We shall hereafter treat mixed quantities as though they were fractional, supposing them to have been reduced to a fractional form by the preceding rule. III. From Art. 64, we deduce the following rule for reducing a fractional to an entire or mixed quantity. RULE. Divide the numerator by the denominator, and continue the oper ation so long as the first term of the remainder is divisible by the first term of the divisor: then the entire part of the quotient found, added to the quotient of the remainder by the divisor, will be the mixed quantity required. If the remainder is 0, the division is exact, and the quotient is an entire quantity, equivalent to the given fractional expression. EXAMPLES. ax + a2 1. Reduce to a mixed quantity. Ans. = a + IV. To reduce fractions having different denominators to equiv alent fractions having a common denominator. It is evident that both terms of the first fraction may be mul adf tiplied by df giving and that this operation does not bdf" change the value of the fraction (Art. 67). In like manner both terms of the second fraction may be bef multiplied by bf, giving bdf; also, both terms of the fraction If now we examine the three fractions and bde adf bef bdf' bdf bdf we see that they have a common denominator, bdf, and that each numerator has been obtained by multiplying the numerator of the corresponding fraction by the product of all the denom inators except its own. Since we may reason in a similar manner upon any fractions whatever, we have the following |