(104.) The Quotient of two Fractions is equal to the dividend multiplied by the reciprocal of the divisor, (80).

For since the value of the dividend is not altered by multiplying each of its terms by both terms of the divisor, (81), the quotient is equal to

And by dividing the numerator of this dividend by the numerator of the divisor, and the denominator by the denominator, we have the

Observe that the Quotient multiplied into the divisor, produces

the dividend, (46.)

Complex or Mixed Fractions.

(105.) When the dividend or the divisor is a Fraction or a mixed quantity, the dividend over the divisor, with a line between them, forms a complex or mixed fraction.

From the nature of Division, and the proposition before demonstraed, (104), we have

RULE XVI.

(106.) For the Division of Fractions.

1. Divide the numerator of the divisor into the numerator of the dividend, and the denominator into the denominator; or multiply the dividend by the reciprocal of the divisor.

2. A fraction is divided by an integral quantity, by dividing the numerator, or multiplying the denominator, by the integer.

3. An integral quantity is divided by a fraction, by dividing the integer by the numerator, and multiplying by the denominator; or by multiplying the integer by the reciprocal of the fraction.

4. A mixed quantity may be taken in division under the form of an improper fraction: or the integral and the fractional part may be divided separately

If we cancel the factor a-x from a2-2ax+x2, (60), and from a2x2, (58); and also cancel the factor 5c2, the operation will be, (103),

By thus canceling common factors, we find the Quotient in its lowest terms.

67

CHAPTER V.

SIMPLE EQUATIONS.

(107.) An EQUATION is an expression denoting the equality of two quantities by means of the sign =, equal to, placed between them.

=

The quantity on the left of the sign is called the first member, or side, and that on the right the second member, or side, of the Equation.

Thus 3x+ab=5x+8d-9 is an equation,

in which 3x+ab is the first member, and 5x+8d-9 is the second.

(108.) Equations are employed in the solution of particular mathematical questions, or in the investigation of general mathematical principles.

In the solution of questions, the unknown or required quantity is represented by a letter, usually x, or y, &c., and an Equation is then formed which expresses the relation between this and the known or given quantities.

To give a simple example;-Suppose we wish to find a number the third and fourth of which shall together make 35.

Let x represent the number to be found, and the Equation

(109.) The solution of an Equation consists in finding the value of the unknown quantity in the equation.

The value found for the unknown quantity is verified, or the Equation satisfied, when this value, substituted for its symbol in the equation, makes the first member the same as the second.

The value of x in the preceding equation is 60, since this number, substituted for x, satisfies the equation; thus

Different Degrees of Equations.

(110.) A simple Equation, or an equation of the first degree, is one which contains no power of the unknown quantity but its first power.