choice a. based on the account ....., scholars... it sounds as if scholars were based on the account. instead when we read further, the sketchy pic is supposed to be based on the accounts, not the scholars.
thus , the rule that right after a clause, we should have it's right subject. (since a PHRASE lacks a subject)
choice e. using the accounts......, scholars..... it makes sense as it implies that the scholars were using the account and scholars is the subject as well.

Glad to see that this discussion has stood the test of time!

With C, there are a few problems:

Based on accounts of various ancient writers, scholars have painted a sketchy picture of the activities of an all-female cult that, perhaps as early as the sixth century B.C., worshipped a goddess known in Latin as Bona Dea, "the good goddess."

(A) Based on accounts of various ancient writers
(B) Basing it on various ancient writers' accounts
(C) With accounts of various ancient writers used for a basis
(D) By the accounts of various ancient writers they used
(E) Using accounts of various ancient writers

1) Compared to E, C is really wordy and awkward, so as a general rule if you're down to two and don't see a huge grammatical difference, the shorter, more-direct one is your best bet.

2) E starts with "using accounts" to modify "scholars", so it's abundantly clear that scholars used these accounts. C isn't just awkward and long, it's passive "with accounts used for a basis". That isn't as clear as to who is using them for a basis. Even "a" in basis is vague...are these scholarly renderings based primarily on these accounts, or just somewhat?

Because of those problems, C is a clumsy, not-altogether-accurate sentence...it leaves room for misconception. When compared to E - E leaves no room for doubt as to who used the accounts, how they used them, etc., so E is a much, much stronger sentence.

I hope that helps...

Brian has explained nicely on this question. He say

There are really three ways that they can do this:

Can anyone put examples to explain three types to use modifier? I am little confused about type 3.

Quote:
(A) Based on accounts of various ancient writers, scholars have painted a sketchy picture of the activities of an all-female cult that, perhaps as early as the sixth century BC, worshipped a goddess known as Bona Dea, "the good goddess".


Maybe some of you will disagree, but I think (A) sounds good. This absolutely sounds like a sentence that a smart history professor would include in a paper or a lecture.

Trouble is, this is the GMAT, and your ear is not your friend. Welcome to GMAT SC: correct answers will “sound bad”, and wrong answers often “sound good.”

In this case, you’ll want to be strict and literal when you analyze the “-ed” modifier and its relationship to the noun it modifies. (A) is saying that scholars themselves are “based on accounts of various ancient writers”, and that doesn’t make sense. (A) is gone.

(And for more on “-ed” modifiers, please check out our Topic of the Week.)

Quote:
(B) Basing it on various ancient writers' accounts

This one is easier to eliminate, since “it” has no antecedent. “Picture” would be the closest singular noun, but that’s too far ahead in the sentence to be reasonable. (B) is gone.

Quote:
(C) With accounts of various ancient writers used for a basis

If you wanted to be conservative, I suppose that you could hang onto this one until you see a better option. I don’t think that it’s grammatically wrong, exactly, but it’s definitely wordy, and I don’t love the use of “with” in this sentence –- it just seems like a very clunky way to express the fact that scholars used accounts of various ancient writers to accomplish their work. And as we’ll see in a moment, (E) is much better.

Quote:
(D) By the accounts of various ancient writers they used

This one seems to be saying that the scholars used the various ancient writers, and that doesn’t make a whole lot of sense. I’m also not sure that the structure makes sense in general: the scholars painted a sketchy picture “by the accounts of various ancient writers”? Sure, the scholars used those accounts, or based their picture on those accounts – but it wouldn’t make sense to say that they painted the picture “by the accounts” of the ancient writers. We can eliminate (D).


Quote:
(E) Using accounts of various ancient writers

The “-ing” modifier is perfect here: the scholars used the accounts of various ancient writers to paint their sketchy picture. And there’s no extra nonsense in (E), so it’s much clearer than (C). (E) wins.