what's the difference between algebraically and arithmetically?

My answer would be:

---   to solve a problem algebraically, you use an equation, involving one or more unknowns;

---   to solve a problem arithmetically, you don't write an equation.

Example:

If 5 dogs eat 10 pounds of dog food , how many dogs (same size) would it take to eat 40 pounds of dog food?

Algebraically:  5/10  =  x/40   --->   cross multiply   --->   10x  =  200   --->   x  =  20 dogs.

Arithmetically: Think this way: if 5 dogs ate 10 pounds, each dog ate 2 pounds, and 40 / 2  =  20 dogs.