Solution:

Given the relation between temperature scales \( A \) and \( B \):

\[
\frac{A - 42}{100} = \frac{B - 7}{220}
\]

To find the temperature at which both scales show the same reading, set \( A = B = x \):

\[
\frac{x - 42}{100} = \frac{x - 7}{220}
\]

Cross-multiplying:

\[
220(x - 42) = 100(x - 7)
\]

Expanding and solving for \( x \):

\[
220x - 9240 = 100x - 700
\]
\[
120x = 8540
\]
\[
x = \frac{8540}{120} = 12
\]

Thus, the temperature at which both scales read the same is 12.