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These two upper-body calisthenics are classics for good reason - they are simple, and efficient by stressing many muscle groups. These are mainly guy exercises, because males have greater, or greater potential, upper body strength and power than most women (so women commonly do assisted pull-ups and knee push-ups. But women have boobs, so there's that...).
I've been focusing on Push-ups lately because our Saturday morning class expects 200 of them in the course of an exercise session (for the guys and the gals). That's a lot, for me anyway, with or without my shoulder's damn traumatic arthritis. "Push-ups are basic strength-building total body exercises that strengthen the upper body and improve the core strength. Several muscle groups in the chest, arms, shoulder, triceps, back, and neck work simultaneously during a push-up." Yes, you tense your core (belly and back) like a board, but you have to breathe too. Little puffs.
Here's a simple challenge program to get you to 60 pushups in one go - or at least in one day. A healthy male under age 75 or 80 can quickly get to 50 pushups in a row, and work up from there if they want. Lots of guys aim to do 100 in a row. Why not 99?
Next, inclined pull-ups, aka inverted rows. These are mainly back, but secondarily shoulder and arm stressors (as are dumbell rows). Like regular pull-ups, you can do these overhand or underhand. I do them with TRX straps and alternate overhand and underhand.
RegularPull-ups are primarily back muscle exercises (you do not pull with your arms - you thrust your elbows down with your back muscles). Chin-ups (palms facing you) engage more arm strength. Unassisted pullups are difficult for most middle-aged men, and for most women. Some tips for working up to your first pull-up.
If a guy can do 10 pull-ups, he's got darn good upper body fitness. I can't - but I could when I was 16. I will try but I doubt I will get to ten again. I saw a middle-aged gal do three sets of ten (unassisted) last week. She was a fit lady, no bulging muscles at all and only slight kipping.
The thing about exercises with multiple muscle groups is that the weak link fails first. Thus, to make your back work hard with pull-ups, you need to be able to keep a hold on to the darn bar with your forearms. Many humans are like T-Rex: puny arms, strong thighs.