He was going at a rate of 50 miles per hour.
He was going at the rate of 50 miles per hour.
Which one is the most appropriate grammatically speaking and why?
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They are both correct. Generally speaking, which is appropriate depends on the context.
In many contexts, including those you quote, you can use either. The distinction is a subtle one.
For example, it sounds more natural to say:
This device works best at a rate of between 100 and 200 rpm.
Because it can be any rate within that range. But you could use either article in:
This device works best at a/the rate of 100 rpm.
The following example requires the because it refers to a specific rate and context:
The rate at which he was travelling posed a danger to other traffic.
In short, you simply have to decide which article fits your sentence better.