To calculate the ratio \( \frac{A}{B} \), we need to evaluate the sums \( A \) and \( B \) given in the problem.

Given:
\[ A = \frac{1}{1 \cdot 21} + \frac{1}{2 \cdot 22} + \frac{1}{3 \cdot 23} + \ldots + \frac{1}{80 \cdot 100} \]
\[ B = \frac{1}{1 \cdot 81} + \frac{1}{2 \cdot 82} + \frac{1}{3 \cdot 83} + \ldots + \frac{1}{20 \cdot 100} \]

To find the ratio \( \frac{A}{B} \), let's first simplify the general term of each series.

For \( A \):
\[ \frac{1}{k \cdot (k + 20)} = \frac{1}{k(k + 20)} \]

For \( B \):
\[ \frac{1}{k \cdot (k + 80)} = \frac{1}{k(k + 80)} \]

Now, let's express each term in partial fractions.

For \( A \):
\[ \frac{1}{k(k + 20)} = \frac{1}{20} \left( \frac{1}{k} - \frac{1}{k + 20} \right) \]

For \( B \):
\[ \frac{1}{k(k + 80)} = \frac{1}{80} \left( \frac{1}{k} - \frac{1}{k + 80} \right) \]

Now, we sum these series.

For \( A \):
\[ A = \sum_{k=1}^{80} \frac{1}{k(k + 20)} = \frac{1}{20} \sum_{k=1}^{80} \left( \frac{1}{k} - \frac{1}{k + 20} \right) \]

For \( B \):
\[ B = \sum_{k=1}^{20} \frac{1}{k(k + 80)} = \frac{1}{80} \sum_{k=1}^{20} \left( \frac{1}{k} - \frac{1}{k + 80} \right) \]

Notice that in both sums, most terms will cancel out, leaving us with:

For \( A \):
\[ A = \frac{1}{20} \left( \sum_{k=1}^{80} \frac{1}{k} - \sum_{k=21}^{100} \frac{1}{k} \right) \]

For \( B \):
\[ B = \frac{1}{80} \left( \sum_{k=1}^{20} \frac{1}{k} - \sum_{k=81}^{100} \frac{1}{k} \right) \]

Now, let's simplify the remaining sums:

For \( A \):
\[ A = \frac{1}{20} \left( \sum_{k=1}^{20} \frac{1}{k} - \sum_{k=81}^{100} \frac{1}{k} \right) \]

For \( B \):
\[ B = \frac{1}{80} \left( \sum_{k=1}^{20} \frac{1}{k} - \sum_{k=81}^{100} \frac{1}{k} \right) \]

Since the sums \( \sum_{k=1}^{20} \frac{1}{k} \) and \( \sum_{k=81}^{100} \frac{1}{k} \) are the same in both \( A \) and \( B \), we can see that:

\[ \frac{A}{B} = \frac{\frac{1}{20} \left( \sum_{k=1}^{20} \frac{1}{k} - \sum_{k=81}^{100} \frac{1}{k} \right)}{\frac{1}{80} \left( \sum_{k=1}^{20} \frac{1}{k} - \sum_{k=81}^{100} \frac{1}{k} \right)} = \frac{1/20}{1/80} = 4 \]

Therefore, the ratio \( \frac{A}{B} \) is \( 4 \).