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CHAPTER XVI.

COMPLEX FRACTIONS. MIXED EXPRESSIONS.

136. WE now propose to consider some miscellaneous questions involving fractions of a more complicated kind than those already discussed

In the previous chapters on Fractions, the numerator and denominator have been regarded as integers; but cases frequently occur in which the numerator or denominator of a fraction is itself fractional.

137. DEFINITION. A fraction whose numerator and denominator are whole numbers is called a Simple Fraction.

A fraction of which the numerator or denominator is itself a fraction is called a Complex Fraction.

a

0 Thus

:

are Complex Fractions.

a In the last of these types, the outside quantities, a and d, are sometimes referred to as the extremes, while the two middle quantities, b and c, are called the means.

138. DEFINITION. If we divide the unit into any number of equal parts, each part is called a Sub-unit.

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If this is to obey the definition, it must represent

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parts,

å of which make up the unit.

Divide the unit into bc sub-units.

Thus parts make up the unit, and are therefore equivalent to bc sub-units.

Hence c parts are equivalent to bcd sub-units; 1 part is

bd sub-units; 7

parts are ...... ad sub-units;

a

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a 140. From the preceding article we deduce an easy method of writing down the simplified form of a complex fraction.

Multiply the extremes for a new numerator, and the means for a new denominator.

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b ab (a +x) Example.

a?
a? - b (a- x)

ab
by cancelling common factors in numerator and denominator.
141. We have proved that

d ad Õ 7

;

[Art. 127.] d

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We have previously shown that, when the numerator and denominator are integers, a fraction may be regarded as representing the quotient of the numerator by the denominator. We now see that a complex fraction may be regarded in the same

sense.

142. The student should especially notice the following cases, and should be able to write down the results readily.

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143. We now proceed to show how complex fractions can be reduced by the rules already given.

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NOTE. To ensure accuracy and neatness, when the numerator and denominator are somewhat complicated, the beginner is advised to simplify each separately as in the above example.

In the case of Continued Fractions, we begin from the lowest fraction, and simplify step by step.

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4 (9x2 – 64)

=4 (3x+8).

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14.

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15.

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16.

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18.

ax + x2

a2 + ax + x2

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a? - 2
y2 – x2
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y2 + x2

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x+y

a

21.

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+
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+
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a + b 4-6
a-ba+b

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(a + b)2 1

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