Both of the boldfaced clauses are dependent. An independent clause has to be able to stand alone as a sentence, but a clause starting with "which," "of which," or "both of which" is modifying something else, so it can't stand alone.

A) Our understanding was that she would continue to cover the costs of school enrollment and room and board for the children, both of which she verbally agreed with the rest of the family to pay.

In this sentence, we have a little ambiguity. The phrase "both of which" introduces a noun modifier, but are we modifying costs or children? Normally, a noun modifier modifies the immediately preceding noun, so we'd say it is modifying "children" (clearly not the intent). On the other hand, since "for the children" is itself a modifier, we might say that we are modifying the whole thing--"costs of school enrollment and room and board for the children." One way we might clarify this would be to reintroduce the noun "costs":

Our understanding was that she would continue to cover the costs of school enrollment and room and board for the children--costs that she had verbally agreed with the rest of the family to pay.
(See GMAT Verbal Review 2nd ed., SC #95, for a similar use of the long dash and reintroduction of the subject to provide clarity.)

If we want to use an independent clause, it would be something like this:
Our understanding was that she would continue to cover the costs of school enrollment and room and board for the children; these were costs that she had verbally agreed with the rest of the family to pay.

B) Our understanding was that she would continue to cover the costs of school enrollment and room and board for the children, of which she verbally agreed with the rest of the family to pay.

Here, the use of "of which" is incorrect. "Of which" generally indicates that we are talking about a portion--we certainly can't be talking about a portion of the children, and the sentence doesn't go on to identify a portion of the costs. Here is a correct sentence using "of which":

We sent her a bill for $150,000, of which a third went to pay for the children's school enrollment and room and board.

Again, the second clause is dependent, as it modifies $150,000.

I hope this helps!
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Dmitry Farber | Manhattan GMAT Instructor | New York


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