Which is more effective, Push Ups or Dumbbell in gaining strength and muscle?
How often should I work every week?
What should I eat to gain fast?
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I'd say, you shouldn't compare the two, and implement both of them in your workout routine.
Solely for chest gains, dumbbell press would be better since you can add resistance more easily and it focusses on your chest. Doing a push-up works a lot more than just your chest.
A push-up will work your chest, shoulders, triceps and core. You're even using your legs to stabilize your body.
In my early years in the Army I noticed that those who relied solely on body weight exercises were almost always outperformed by those who used weights in the gym. I would do about 15-20 pushups before doing chest workouts to warm up and stretch, and then hit the dumbbells and barbell. I always maxed out my pushups on the PT test. If you can bench 200 lbs, then 75 pushups seems easy when the time comes for it. I'd concentrate your efforts on weights, but don't completely neglect body movement exercises. They have their place.