I remember going all over South America (we lived in Peru) with my parents and two sisters at exactly those ages. Never a problem.

However (and I am not being biased even though I lived most of my life in Peru), out of the countries/cities you mentioned:

Rio - Is know for having a high crime rate
Bolivia - Has recently had a lot of civil unrest.
Quito - Has also had some civil unrest

Peru is going through a very good period with a good economy, no civil unrest, and low crime. If you use common sense, you should not have a problem.