Would someone explain to me why my method didn't work?

Well u method is perfect till this step
Unmowed lawn 16-10/12 = 6/12 = 1/2

But u have missed the final step...
The question asks u that what fraction is unmowed

So the ans = Unmowed / Total lawn
=> 1/2 / 16/12
=> 1/2 * 12/16
=> 6/16 => 3/8

So u just missed the last step..

I followed a different approach

Let Back lawn be x
Hence front lawn = 1/3x
Now its given that John mows 1/2 of his front lawn and 2/3 of his back
Hence Back lawn (unmowed) = 1/2*1/3x
Front lawn (unmowed) = 1/3x

Hence [1/2*1/3x+1/3x] / 4/3x
Solving we get 3/8

Hope this helps...

F+B=L
F=1/3B
Unmowed=B/2
Fraction= Unmowed/Total
(B/2) / (4/3B) =3/8

If you assume data & solve this question you will be able to solve it much quicker than the conventional method that I have described below...

Total lawn(T)=Front Lawn(F) + Back Lawn(B)
Given: F=B/3
then T= 4B/3

Given that 1/2 of the front lawn and 2/3 of the back lawn are mowed...

i.e 1/2(B/3)+2/3(B) = 5B/6
therefore lawn unmowed is given by (4B/3)-(5B/6) = 3B/6 = B/2

Fraction of lawn unmoved is given by (B/2)/(4B/3) = 3/8

Easy explanation
Total = 12

Front = 3
Back = 9

Mowed
Front = 3 / 2 = 1.5
Back = 9 * 2 /3 = 6

Left
Front = 1.5
Back = 3

Fraction
4.5 / 12 = 3/8