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For example:

$$ \frac{\ \dfrac{3}{5}\ }{\dfrac{7}{8}} $$

I was wondering if such a situation had a name, wherein both the numerator and the denominator of a fraction consist of fractions themselves. I was also wondering if this was something common, or at least in some way useful.

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I learned the name "compound fraction" for things like this. One way they appear is when you have a fraction $\frac xy$ and substitute $\frac 35$ for $x$ and $\frac 78$ for $y$ (for reasons elsewhere in the problem). then Joe Johnson 126 has shown the next step. – Ross Millikan Feb 21 '12 at 1:33
Tutoring middle and high schoolers, I've seen these called "complex fractions," a term that I find confusing. – Shaun Ault Feb 21 '12 at 3:16

Any fraction of that form can be simplified:

$$ \dfrac{\ \dfrac{a}{b}\ }{\dfrac{c}{d}}=\dfrac{a}{b}\cdot \dfrac{d}{c}=\dfrac{ad}{bc}. $$

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An aside: \dfrac displays well! – user21436 Feb 21 '12 at 1:32
@KannappanSampath: Thanks. – Joe Johnson 126 Feb 21 '12 at 1:39

Your example is a complex fraction like Robert Israel said, if it is clear where the longest fraction line is. In your case one can see it is the middle one (between $5$ and $7$). If that is not the case, it is not well-defined and therefore meaningless: $$ \frac{\frac{1}{3}}{\frac{1}{3}} \text{ is not defined!} $$

In essence this way of displaying fractions is useful since it fits perfectly into our view of division and fractions: $$ \dfrac{1}{3}\times 1 = \dfrac{1}{3} \Leftrightarrow 1 = \dfrac{\dfrac{1}{3}}{\dfrac{1}{3}} $$ which is obviously true.

This kind of fraction (complex) are used mainly when solving equations, and should be considered very basic. If you go one step further and ask yourself the question, what happens when I begin to nest fractions. You might come across something like this: $$a+\dfrac{b}{c+ \dfrac{d}{e+\dfrac{f}{\vdots}}}$$ Now things are getting interesting (and useful!). This is called a continued fraction, it can be infinite or finite. These continued fractions can represent every real number, obviously $\pi$ needs an infinite continued fraction to display.

Amazingly one can (there are several ways to write a infinite continued fraction) write: $$ \pi = \dfrac{4}{1+\dfrac{1^2}{2+\dfrac{3^2}{2+\dfrac{5^2}{2+\dfrac{7^2}{2+\dots}}}}} $$ This is a regular structure for a non-periodic number! (Cheer here!)

But continued fractions have also a close relation to the Euclidean algorithm, which is an very efficient ($O(h^2)$) method for computing the greatest common divisor of two integers, which is again used in a lot of practical applications (think RSA).

This might not answer, whether complex fractions have any practical use (imho just a way to display/write), but if you take one step back there are deeper connections between the mere way to display something and factorization of prime numbers.

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Sometimes called a complex fraction.
See e.g.

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