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. a If this is to obey the definition, it must represent ū 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 that is the fraction represents ad sub-units, bc of which make ad up the unit; but this, by definition, is the fraction bc ad bc 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. atx b ab (a +x) Example. a? ab d ad Õ 7 ; [Art. 127.] d bc a a-C a a с 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. 1 6 1: =1x a a a 143. We now proceed to show how complex fractions can be reduced by the rules already given. |