"Half and half" is half milk and half cream. All the cream is taken from the milk and then a 1:1 ratio of fatless milk and cream are mixed. half and half has more fat than whole milk, so they are not interchangeable.
"half the half" is one quarter. One quarter = 1/4 One and a half = 11/2 = 3/2 One quarter of 1 and a half = 1/4 x 3/2 = 3/8 So, half the half of one and a half = three eighths
Half of any amount is still a 'half'
In mathematics the word 'of' is the equivalent of the symbol X or 'times' so it follows that a half of a half is 1/2 X 1/2 so the answer is 1/4 or one quarter.