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To find out in about how many weeks their savings in the bank will be equal, we can calculate the difference in their initial amounts, which is $595.25 - $147.25 = $448.
Then, we find out the difference in their weekly transactions, which is $27.50 - $23.70 = $3.80.
To estimate how many weeks it will take for their savings to be equal, we divide the difference in their initial amounts by the difference in their weekly transactions, which is $448 / $3.80 = 118.
Therefore, the most reasonable estimate is that their savings in the bank will be equal in about 118 weeks. Answer: \boxed{118}.