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So if I want to get fitter and be able to run faster, does trying to lose weight at the same time hinder this?
I've read that on a calorie restricted diet, one has to take care to make sure they are still eating enough protein. Other than that, can trying to lose weight (or I guess, more specifically, trying to lose fat since that's just 'dead weight') by calorie restriction hinder attempts to become fitter and run faster?
The common lore is that you gain between 2-4 seconds per mile in pace per pound lost for the same amount of effort. So, if you run an 8 minute mile at about a 80% of race pace tempo, and you lose 10 pounds, you can generally expect to be at 7:40 to 7:20 per mile for the same amount of effort.
As far as the calorie restrictions, if you are attempting to be a faster runner (I'm assuming that you are talking about endurance, such as the 5k and up, not sprinting), then your carbohydrates are more important. Yes, sufficient protein is necessary, but it's not quite as critical as it would be for a lifting/bulking type of program. I'd shoot for 50-60% carbs, 20-30% protein and the rest in fats. As shown by one of the linked articles in this answer, .75g / lb is about the highest limit that you would need.
As far as improving at running, the best way to do that is to run more. As Scout7 (great poster on another forum) puts it, Run. Run lots. Mostly slow, sometimes fast. If you want a great program to progress, check out these articles, most especially the top 3, "The Program" 1 2 and 3. I've detailed the plan in brief here in a few different answers, but it's a great progressive running method. Save the weights for when you're at the last nth degree and needed that little extra.
Losing weight can be achieved through running and losing weight would probably give you advantage of running faster. Furthermore the more you practice running (although try to not overdo it seems your knees might hurt) would help you run faster.
Also another great option is to perform exercises like squats and deadlifts that would strengthen and give you a lot more power to put with your legs when running.
