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In german elementary schools you learn "In Summen kürzen nur die Dummen.", i.e. it is dumb to cancel sums in quotients. But later in algebra or category theory you may learn

$\sum_{i \in I} A_i / \sum_ {i \in I} B_i = \sum_{i \in I} A_i / B_i$,

which is correct.

$\sum_{i \in I} A_i / \sum_ {i \in I} B_i = \sum_{i \in I} A_i / B_i$,