Simplifying Complex Fractions

What do we mean by complex fractions?

Complex fractions are fractions whose numerator and/or denominator themselves are expressions containing other fractions.

Can a complex fraction become a simple fraction?

Yes, it can! The goal of simplifying complex fractions is to rearrange them into equivalent simple fractions which are in simplest form.

And can a simple fraction be simplified?

You should already know it can. The goal of simplifying a simple fraction is to obtain an equivalent fraction that looks simpler (i.e. it has less terms in its numerator and denominator).

Sometimes the numerator and denominator of a complex fraction are just single simple fractions themselves. Then, for the first step in simplifying the complex fraction, we just use the wellknown “invert and multiply” rule: multiply the fraction in the numerator by the reciprocal of the fraction in the denominator:

You see that the initial complex fraction on the left has been turned into a single simple fraction on the right. This step is justified only if the numerator and denominator of the original complex fraction are both single simple fractions. When the pattern in the box above is valid, all that is left to do in simplifying the original complex fraction is to use methods already illustrated many times in the last few documents in this series to check whether the simple fraction on the right can be simplified any further.

Example:

Simplify:

solution:

We include this simple example just to emphasize that you must treat all complex fractions as complex fractions and apply the general procedure illustrated in the previous examples whenever you deal with a complex fraction.

(By the way, the subscript labels, 1 and 2, on the R’s are just labels. In this case, consider R ₁ and R ₂ each to be distinct simple single literal symbols as a whole, just as you might use the even simpler symbols such as x and y to represent numerical values. In this instance, the subscripts are used to distinguish the two symbols, and have no deeper mathematical significance.)

This example seems very simple! Often people think: dividing by a fraction means “invert and multiply,” and so here we just do

But as “natural” as this method appears to be here, it is completely incorrect! To see this, just try a simple numerical experiment with the initial and proposed final simplified form. For instance, if R ₁ = 2 and R ₂ = 4, the original expression evaluates to

However, if we use the result of the erroneous “invert and multiply” method shown above, we get

which is very different from the correct answer of 1.33333 (to five decimal places).

The mistake above is that “invert and multiply” applies only when the expression in the denominator is a single simple fraction. Here, our denominator is a sum of two simple fractions. So, following the general strategy for simplifying complex fractions, we must first reorganize the expression in the denominator into a simple fraction.

So now,

as the correct simplification of the original complex fraction.

By the way, the little numerical experiment that demonstrated the presence of error in the incorrect approach earlier works fine here. Substituting R ₁ = 2 and R ₂ = 4 in the final expression just above gives

which we know is the value we should expect to get. (This numerical experiment is not a proof that our final result is correct, since it might be that the final result gives the same value as the original complex fraction for these values of R ₁ and R ₂ just by coincidence. The equivalence of two mathematical expressions cannot be “proven” with a single numerical experiment – certainty that a result is correct requires careful checking of the algebraic connection between the two expressions. However, if two expressions do not give the same numerical result when the same values are substituted for the literal symbols they contain, that is certain proof that they are not equivalent mathematical expressions.)

You might be wondering: why spend all this time learning how to simplify complex fractions? Surely such strange expressions are found only in math books and would never be encountered in actual technical applications!

But this is not so. Recall that a fraction is one way of writing the division of one mathematical expression by another mathematical expression. Whenever either or both of the expressions involved in such a division operation also contain fractions themselves, then writing the division as a fraction generates a complex fraction. Since division is a very common mathematical operation in all technical applications, complex fractions are not uncommon in the analysis of actual technical problems. (For instance, the complex fraction considered in Example above arises in describing one of the very simplest types of electrical circuits.)

And, since complex fractions are “complex,” there is value in being able to simplify them to make their use in subsequent steps of the problem solving process that much easier.