It is hard to decide which is a better place to live in. It depends on your plans, your interests.
Here are few points which can help you decide.
Bangalore has lot of career options for people interested to work for IT companies. There are finance firms, consultancies. Mumbai is much more diverse, you can find IT companies there but the kind of opportunities you get in Bangalore is totally different.
If you ask me about transport, I would say Mumbai. Local transport is cheap. Monthly pass, auto fares are reasonable. In Bangalore, auto drivers are in ready to fight mood all the time, but in Mumbai most of the drivers are well behaved. Now that also varies from one to another individual. But my experience has always been good when it comes to transport. Bangalore has worst traffic sense. In Mumbai , traffic is bad but at-least people try and follow traffic rules. If you are scared to travel by a local train in Mumbai and prefer taxis and autos then commuting will be a little difficult in Mumbai. Having own vehicle helps, in both cities. Bangalore has an advantage that tenants have parking lots, unlike many societies in Mumbai. Even the owner will not find a place to park his car in Mumbai forget about the tenant.
When its comes to accommodation, housing, Bangalore is the best choice again. Here you can afford to stay in a PG, hostel or rent a flat which is close to your workplace. But in Mumbai you need to choose a place which is affordable and there is no choice even if it is far from your workplace. You cannot find as many hostels and PG's as you can find in Bangalore.
Entertainment, fun, lifestyle: Mumbai and Bangalore, both have malls, multiplexes. Movie ticket fare is little cheaper in Mumbai. But if you are a person who likes to window shop or spend sometime at malls then both places have it. Bangalore does not have Bollywood and beaches. If you like to sit on Marine drive or visit Bandra-worli sea link, chowpatty beach, colaba or the juhu then you will definitely miss those in Bangalore.
Water/power supply : That depends on the area you stay. Both cities have water problems. It is the same with power supply as well, depends on the locality.
My personal experience : Mumbai is expensive I agree. You can save money if you are in Bangalore. But once in a lifetime you got to visit Mumbai, to learn some of the best values of life. The local trains, nightlife, the slums teach you a lot. There are people who aspire to become an actor and come to Mumbai and the kind of struggle they go through is terrible. Plus-point, it will give you qualities of self-confidence and also respect for the poor.
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