Q: What is the difference between “can” and “could” in these two pairs? (1) “You did the best you could” … “You did the best you can.” (2) “Can you please wash the floor?” … “Could you please wash the floor?”
A: “Could” differs from “can” in two respects: it’s not only the past-tense form of “can” but it sometimes represents a change in attitude as well.
The two sets of examples you’ve given illustrate these aspects very nicely.
Let’s start with your first pair: “You did the best you could” and “You did the best you can.”
The first sentence is correct. It’s written in the past tense (“You did the best …”), and the tense of the auxiliary verb matches (“you could”).
But in the second sentence, the sequence of tenses is askew. If the sentence began with the present (“You do the best …”), the tense of the auxiliary verb would match (“… you can”).
An English teacher would tell you that “can” and “could” are modal auxiliary verbs. The term isn’t as intimidating as it sounds.
They’re “auxiliary” because they’re what people sometimes call helping verbs, always used with another verb whether present or implied (in your first set of examples, there’s an implied “do” at the end).
And they’re “modal” because (like the pairs “will/would,” “shall/should,” and “may/might”) they add a dimension or modality to the verb they help along. They add the notion of probability, necessity, permission, or obligation.
Here’s how the different forms of these verbs are used to refer to present and to past time:
Present: “They say you can [will/shall/may] get there on time.”
Past: “They said you could [would/should/might] get there on time.”
So much for the past-tense aspect of “could.” Now it’s time for a change of attitude.
Let’s look at your second set of sentences: “Can you please wash the floor?” and “Could you please wash the floor?”
Both sentences are correct. “Can you …?” and “Could you …?” here mean “Is it possible for you to …?”
But the two sentences reflect different attitudes on the part of the speaker. We sometimes use “could” instead of “can” when we want to be more tactful or polite.
As Sidney Greenbaum writes in the Oxford English Grammar, the past-tense forms “could” and “would” may be used to refer to present or future action “as a more tentative or more polite alternative to the present tense.”
For example, “Could you pass the salt?” is more polite than “Can you pass the salt?” And “Would you help me?” is more deferential than “Will you help me?”
We touched on this subject once before, in a posting about the phrase “would like.”
As we wrote in that blog item, the authors of The Cambridge Grammar of the English Language say that “would” often “introduces a rather vague element of tentativeness, diffidence, extra politeness, or the like.” The same might be said of “could.”
We’ve barely scratched the surface here, but we hope this answers your questions.
Check out our books about the English language