In your first group 'could' is the past tense of 'can' when it means "to know how to do something; or be physically or mentally able to do something. 1-He could solve the problem…………. This means that he had the ability or knew how to solve the problem. This does not mean that he solved the problem. 2-He was able to solve the problem…… This means that he actually solved the problem. When talking about one time or particular actions that occurred in the past, like solving a problem, "able to" is preferred. In the negative form both (couldn’t) and (wasn’t able to) have about the same meaning. In the second group ‘can’ is a modal verb. One of the uses of can as a modal verb is to express a polite request. Could is the past tense of modal can. A) Can you give me a pen?…………. Present tense “can” is used here as a helping verb to express an informal polite request between friends or people who know each other well. Better would be…. Can I borrow a pen? B) Could you give me a pen?……… Could here is a helping verb used to express a polite request. Better would be…. Could I borrow a pen? For more explanation see: http://www.tolearnenglish.com/exercises/exercise-english-2/exercise-english-7233.php http://unamsanantoniolab.com/eng/modals_past_ability/ex2.htm

'Could' and 'was able' used for past ability. 1- For ability only, either can be used (we can use 'could' or 'was able'): 'When I was young I COULD / WAS ABLE to climb any tree in the forest.' ' He could solve the problem.' 'He was able to solve the problem.' 2- For ability + a particular action (or situation), use 'was able': 'He was badly hurt but he WAS ABLE to explain what had happened.' This rule, however, is relaxed in the negative when the action did not take place, and with verbs of the senses: 'He read the message but he COULDN'T / WAS NOT ABLE to understand it.' 'He COULD / WAS ABLE to see him through the window.'

'could' and 'able to' are of the same meaning. when asking for a favor, the use of "could" is more formal and polite than "can" :)

The other comments here are good and they answer your question specifically, however I believe they are neglecting to point out the fact that using "could" or "can" in a polite request is not proper English. It is the same as using "would I be able" or "am I able" - which is not the correct question. For example: "Could I borrow your pen?" is the same as asking "Am I able to borrow your pen?" which does not ask the question you intend to ask. The question you want to ask is "May I borrow your pen", this is polite, formal and proper English. It should not be a question of your ability, it should be a question of the other persons willingness to allow you to borrow the pen. "Could you give me a pen?" asks if the person is able to give you a pen, of course he/she is able! Are they willing to give you pen? that is a different question... Many native English speakers use "Could I" or "Can I" in place of "May I" - so it is good to recognize and understand their meaning, however the proper and formal usage is "May I". When you want to question ones ability to perform a task it is proper to use "Can" - "Can you carry that heavy object by yourself?" - as you are questioning their ability, not asking for permission. "Can" is also appropriate when questioning your own ability: "Can I get there by turning left?" - that *is* a question of ability, so "can" is appropriate. "Could" should almost always be used in past tense, and almost always represents a potential - "that could have been a disaster", "I could have called you - if I had a phone". "I could have arrived earlier by turning left.", or "You could have used 'may' instead of 'can' in that sentence."

"Could" refers to possibility while "can" refers to ability. However, in questions, "could" is often used in the place of "can" to ask someone a favor in a polite way.