Complex Fractions
~all numbers can be considered fractions. Whole numbers have denominators of 1.
~if there are “smaller” fractions in the numerator and also in the denominator,
then the expression is called a complex fraction.
~Examples: (8/9 + 1)/(7/6), (5/6 + 7/8)/(3/7 - 4/5), in arithmetic, (2/x - x/y)/(7+5/x) in algebra.
~to simplify any complex fraction, we need to find the LCD (also called LCM) of all
denominators of the “smaller” fractions involved in the top & bottom.
~after finding this, we multiply the top & bottom of our complex fraction by this LCD.
~if done correctly, all “smaller” fractions will disappear, leaving just one fraction.
~Example: simplify the first complex fraction above. Note that the “smaller” fractions
involved are 8/9, 1/1, & 7/6. The LCD of 9,1, & 6 is 18.
Multiplying the top & bottom of our complex fraction by 18, we get:
[(18)(8/9) + (18)(1/1)]/[(18)(7/6)] =[(2)(8) + (18)]/(3)(7)= [16+18]/21 = 34/21 (reduced)
~Example: simplify the 2nd complex fraction above: Note that the “smaller” fractions
involved are 5/6, 7/8, 3/7, & 4/5.
The LCD of 6,8,7, & 5 is (2)(3)(4)(5)(7).
Multiplying the top & bottom of our complex fraction by this number, we get:
[(4)(5)(7)(5)+(3)(5)(7)(7)]/[(2)(3)(4)(5)(3)-(2)(3)(4)(7)(4)]=[700+735]/[360-672] = 1435/(-312) = -1435/312 (reduced
~note: when the LCD is large (like above), it’s best to leave it in factored form when
multiplying the top & bottom. The “smaller denominators” will be part of that
LCD & will cross out, leaving the rest of the factors.
~note: to determine that -1435/312 is reduced, just x-ray each & you will see that they are relatively prime (i.e., no factors in common).
~Example: simplify the third algebraic complex fraction
.
Notice that the “smaller fractions” involved are 2/x, x/y, 7/1, & 5/x.
The LCD of all denominators is (x)(y)(1) = xy. So, multiplying the top & bottom of our complex fraction by xy, we get: [(xy)(2/x) - (xy)(x/y)]/ [(xy)(7/1) + (xy)(5/x)]= [2y - x2]/[7xy + 5y] (reduced)
~note: reducing an algebraic fraction is much easier, if you understand the techniques of algebra involved. (nothing is easy without understanding)