Find the total length of the set of real numbers satisfying \[\frac{x^2 - 80x + 1500}{x^2 - 55x + 700} < 0.\]
1. **Factor the given expression:**
   \[
   \frac{x^2 - 80x + 1500}{x^2 - 55x + 700}
   \]
   We need to factor both the numerator and the denominator.

   - For the numerator \(x^2 - 80x + 1500\):
     \[
     x^2 - 80x + 1500 = (x - 30)(x - 50)
     \]

   - For the denominator \(x^2 - 55x + 700\):
     \[
     x^2 - 55x + 700 = (x - 20)(x - 35)
     \]

   Thus, the expression becomes:
   \[
   \frac{(x - 30)(x - 50)}{(x - 20)(x - 35)}
   \]

2. **Determine the critical points:**
   The critical points are the roots of the numerator and the denominator:
   \[
   x = 20, 30, 35, 50
   \]

3. **Perform a sign analysis:**
   We need to analyze the sign of the expression in the intervals determined by the critical points. The intervals are:
   \[
   (-\infty, 20), (20, 30), (30, 35), (35, 50), (50, \infty)
   \]

   - For \(x < 20\):
     All factors \((x - 30)\), \((x - 50)\), \((x - 20)\), and \((x - 35)\) are negative.
     \[
     \frac{(-)(-)}{(-)(-)} = \frac{+}{+} = +
     \]

   - For \(20 < x < 30\):
     \((x - 30)\) and \((x - 50)\) are negative, \((x - 20)\) is positive, and \((x - 35)\) is negative.
     \[
     \frac{(-)(-)}{(+)(-)} = \frac{+}{-} = -
     \]

   - For \(30 < x < 35\):
     \((x - 30)\) is positive, \((x - 50)\) is negative, \((x - 20)\) and \((x - 35)\) are positive.
     \[
     \frac{(+)(-)}{(+)(-)} = \frac{-}{-} = +
     \]

   - For \(35 < x < 50\):
     \((x - 30)\) and \((x - 50)\) are positive, \((x - 20)\) is positive, and \((x - 35)\) is negative.
     \[
     \frac{(+)(-)}{(+)(+)} = \frac{-}{+} = -
     \]

   - For \(x > 50\):
     All factors \((x - 30)\), \((x - 50)\), \((x - 20)\), and \((x - 35)\) are positive.
     \[
     \frac{(+)(+)}{(+)(+)} = \frac{+}{+} = +
     \]

4. **Identify the intervals where the expression is negative:**
   From the sign analysis, the expression is negative in the intervals:
   \[
   (20, 30) \text{ and } (35, 50)
   \]

5. **Calculate the total length of these intervals:**
   \[
   (30 - 20) + (50 - 35) = 10 + 15 = 25
   \]

Conclusion:
\[
\boxed{25}
\]