1. I think it right that you are helping the poor.

2. I think it right that you help the poor.

3. I think it right that you helped the poor.

4. I think it right that we should help
the poor.

5. I think it right that you should help the poor.
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Which ones are grammatical and natural? Can we omit 'that' in each sentence?

All are fine, yes.

No, in these sentences "that" is needed because it is introducing a noun clause in each sentence. You can often omit "that" when it's introducing a relative clause (an adjective clause).

1. "I think it right that you are helping the poor." - This sentence is grammatically correct and natural. It expresses the present continuous tense, indicating an ongoing action of helping the poor.

2. "I think it right that you help the poor." - This sentence is also grammatically correct and natural. It uses the base form of the verb "help" to express a general statement or habitual action.

3. "I think it right that you helped the poor." - This sentence is grammatically correct, but it may not be as natural as the previous two. It uses the past simple tense to refer to a completed action of helping the poor.

4. "I think it right that we should help the poor." - This sentence is grammatically correct and natural. The inclusion of "should" indicates an obligation or recommendation to help the poor.

5. "I think it right that you should help the poor." - This sentence is also grammatically correct and natural. It adds the modal verb "should" to indicate that helping the poor is seen as the proper or expected course of action.

Regarding the use of "that," it is necessary in all of these sentences to introduce the subordinate clause that expresses the action or intention being discussed. Omitting "that" would result in incomplete or unclear sentences.