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Half would be 50% of something. So half of half would be 25% of something.
A fraction is a part of a whole of something. If a whole is 100%, and 50% is taken away, there remains 50% or a half.
If you have half of something and you cut it in half again, you get the fraction of 1/4.
In the UK, having half and half (in a pub) is having half a pint of bitter with half a pint of mild, or stout, in a pint glass. Or, it is having half of one thing with half of something else. For instance, in a restaurant, ordering half chips with half rice rather than a full portion of chips, as part of a main meal.
Fractions are a part of the whole. So 1/2 is a half of something. A circle divided into 10 parts, each part is 1/10th of the whole circle.