Learn English – “The sale goes on through the 16th” Why is this correct

You would have to choose #1 , because #2 contains the diparaged construction "the reason why".

However, neither #1 nor #2 is a good rephrasing of the incorrect example.

For one thing, "what is the reason" was changed to "is this the reason", which changes the meaning.

For another thing, "neglecting doing" is awkward. Of course, simply taking out "doing", as user8399 suggests, changes the meaning. "Neglecting to do..." would be better.

For another, If you treat "neglecting" as a noun, it should be your neglecting. . . " rather than you neglecting. . ."

So really, the correct answer is that neither #1 nor #2 is a good rephrasing.

Feel free to tell your teacher this; and that the question should be thrown out (not counted), because it was so badly written; and that you deserve extra credit for this analysis, not a downcheck for whichever answer you chose.

Ciao, Federico

You have used through (definition 5, by means of) correctly in your sentence. Good job.

I will say that it is not all that natural. In other words, it's grammatical and understandable, but not idiomatic, not the most natural way to express the thought.

Through is often used as a result of a middleman, or to signify a process. Neither of these apply to your tips.

Also, are you thanking Alex directly? If so, you need commas before and after Alex:

Thanks, Alex, for all the money saved through your tips.

In addition, tips is ambiguous without further context, as the word can mean either (monetary) tips gained from working as a waiter, for example, or tips as in suggestions. I assume you mean the latter, and in any case it does not much affect the choice of preposition.

Changing no word except through, the following are perhaps preferable:

Thanks, Alex, for all the money saved from your tips.

and

Thanks, Alex, for all the money saved by your tips.

But, frankly, two more natural sentences are

Thanks, Alex, for all the money you (have) saved me by (means of) your tips.

and

Thanks, Alex, for all the money I (have) saved from/by (using/following) your tips.

Just as many native speakers would not normally choose the preposition through in either of these two sentences, we wouldn't normally choose it in your original sentence. There is no explanation as to Why, rather it's simply a matter of being familiar with which preposition sounds better, or "works" more naturally in a particular sentence.

I am a native speaker of American English.