Natural Blogarithms

Skin a cat with complex fractions

September 5th, 2006 by SplineGuy

Isn’t there an old saying that says there’s more than one way to skin a cat? Was cat skin ever such a valuable commodity that these methods were enumerated?

Today in College Algebra, I explained the technique that I use to simplify complex fractions. By complex fractions, I am referring to the rational expressions that have fractions inside fractions. The goal of the technique is reduce a complex fraction into an equivalent form, $\displaystyle \frac{p}{q}$ where $p$ and $q$ are polynomials. Primarily due to the restrictions of time in class, I limited myself to explaining a single technique for reduction, namely, multiply through the numerator and denominator by the LCD (lowest common denominator) of all the fractions in the complex fraction.

EX: $\displaystyle \frac{\displaystyle a + \frac{3}{b}}{\displaystyle b+\frac{3}{a}}$

LCD: $ab$

$= \displaystyle \frac{\displaystyle a + \frac{3}{b}}{\displaystyle b+\frac{3}{a}} \cdot \frac{ab}{ab}$

$= \displaystyle \frac{a^2 b + 3a}{a b^2 + 3b} = \frac{a(ab+3)}{b(ab+3)} = \frac{a}{b}$

There is an alternate technique to accomplish the same result. One can also simplify the numerator and denominator separately by combining them into a single fraction. In other words, in the numerator and denominator separately, find the LCD, rewrite each fraction with the LCD and then add the results. At this point we simply invert and multiply to reduce the complex fraction.

EX: $\displaystyle \frac{\displaystyle a + \frac{3}{b}}{\displaystyle b+\frac{3}{a}}$

$= \displaystyle \frac{\displaystyle \frac{ab+3}{b}}{\displaystyle \frac{ab+3}{a}}$

$= \displaystyle \frac{ab+3}{b} \cdot \frac{a}{ab+3} = \frac{a}{b}$

I claimed to have left this second technique out primarily because of time, but to be completely honest, I fear that providing students with too many techniques leaves them confused. Many of them, especially those lacking confidence in their mathematical abilities, feel overloaded and give up too easily in mastering the multiple techniques. One strategy I have used is to give them the multiple techniques and allow them to choose which one they’d like to use. I have also approached this by actively motivating the students to develop their own technique. In almost every case, they develop the second version of this, since the layout of this unit has complex fractions following up the section on addition and subtraction of rational expressions.

Open question: When do you teach multiple techniques? Is it justifiable to leave out multiple approaches to the problem even when both of them are reasonable and commonly used?

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