April 1991 price of silver was $3.9707/oz troy, gold was $358.38/oz troy. That's a 90:1 silver-gold ratio, ounce to ounce.

Now, the Sickle-Galleon ratio is 17:1. But a Galleon is larger than a Sickle, by a significant amount, as can be seen here; , and gold is denser than silver, by a significant amount. Assuming the coins are similar thickness, the Galleon is about 1.7 times larger than the Sickle, and about 3.1 times heavier. So the ratio by weight is around 5.4:1 silver-to-gold.

That means each cycle of arbitrage is a multiplication by, well, we'll round down to 16 for various transaction costs in our Fermi estimate. 100 Galleons becomes 1,600 after one cycle, and 1,600 becomes 25,600 after cycle 2.

Okay, that "couple times" didn't quite get us all the way there from 100. Harry needs to manage to get his hands on (considering the uncertainties on transaction costs) more than 234 but almost certainly less than 300 Galleons and run through the arbitrage cycle twice to get the 60,000 galleons.