No particular name would encompass 1/2, -1/2, -5/2 all of which are fractions with a less than 1.
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If the numerator is less than the denominator, the fraction is less than one. For example, 5/7 is less than one. If the numerator is greater than the denominator, the fraction is greater than one. For example, 10/7 is more than one.
A proper fraction, ie: the numerator is smaller than the denominator so it is less than one.
The fraction 3/4 is not expressable as a mixed number - as it represents a value less than one.
The fraction greater than 1/2 but less than 1 is 2/3. This fraction falls between 1/2 and 1 on the number line, with a value closer to 1 than to 1/2. It represents two parts out of three equal parts, which is greater than half but less than the whole.
You can't convert a proper fraction to an improper or mixed fraction - it just doesn't make sense. By definition, a proper fraction is one that is less than one; a mixed fraction - as well as an improper fraction - is less than or equal to one.