Мы обрадовались новости за неделю до отпуска.

Questions & Answers about Мы обрадовались новости за неделю до отпуска.

How does the grammar in за неделю до отпуска translate to "a week before the vacation"?
This sentence uses the exact structure your current lesson covers for expressing time before an event: за + Accusative case (for the amount of time) + до + Genitive case (for the event itself). Here, неделю is the Accusative of неделя (a week), and отпуска is the Genitive of отпуск (vacation).
Why is the word for "news" in the form новости? What case is that?
It is in the Dative case. In Russian, the verb обрадоваться (to be delighted / to rejoice) requires the Dative case for the thing that causes the joy. Because новость (news) is a feminine noun ending in a soft sign (-ь), its Dative singular form is новости.
Could I use the imperfective verb радовались instead of the perfective обрадовались here?
You could, but it changes the focus. The perfective обрадовались shows a sudden reaction—the exact moment we received the news and became delighted. The imperfective радовались would mean we were in a continuous state of rejoicing for a period of time.
Is новости plural or singular in this sentence?
In this sentence, it is singular. It means "the piece of news" or "the specific news we just heard." It can be confusing because the Nominative plural of this word is also новости, but here the grammar is strictly driven by the verb needing the Dative singular form.

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