Transformers Construction
Why don't we have 11 no of turns on primary and 440 turns on the
  secondary. Theoretically this would give us the required voltage
  ratios and much less weight. Why doesn't it work? What am I missing?


Why not have one turn on the primary and 40 turns on the secondary you might as well ask. With pretty much any normal size of core you might get somewhere between 1 uH and 100 uH inductance for one turn. Let's say you get 10 uH and let's say your AC voltage was at 50 Hz. The single turn primary winding has an inductance of 10 uH and at 50 Hz, this is an impedance of 3.14 milli ohm i.e., with no load attached to the secondary, the primary alone is drawing a current from your 11 volt AC supply of 3500 amps. Now nobody wants that.

With 11 turns, the inductance isn't just 11 times higher it's 11 squared times higher so now the inductance becomes 1.21 mH and the reactance is 0.38 ohms and will take an unloaded primary current of 29 amps - i.e. a lot better but still not that great but, do you see the point here - irrespective of the lump of iron being a transformer, it still has a residual primary inductance that can take too much reactive current from your 11 volt AC source and it makes sense to seek to minimize that. If you went for 44 turns the inductance would rise by 16 to 19.4 mH and the residual current would fall by 16 to about 1.8 amps - this might be more reasonable.

You might want to go higher of course but then you have a lot more copper and there becomes a trade-off between residual magnetization current (which leads to core saturation losses) and copper losses. More turns means less core loss but maybe unacceptable copper losses (\$I^2R\$).

This is why SMPS devices use switching frequencies of typically 100 kHz - 10 uH will have an impedance of 6.28 ohms (compared to 3.14 milli ohms at 50 Hz).


Using the example of 1 turn having 10 uH inductance drawing 3500 amps means a magneto motive force (MMF) of 3500 ampere turns. 
With 11 turns, the inductance is 1.21 mH and the current is 29 amps or an MMF of 319 ampere turns
With 44 turns the MMF is about 80 ampere turns.


To accomodate a lower number of turns and prevent saturation either the core cross sectional area has to increase (generates more inductance per single turn) or the core length needs to increase (decreases the H field) so when you say: -


  this would give us the required voltage ratios and much less weight


You are misled.