Theory:

Proper fraction:
1. A fraction in which the absolute value of numerator (top number) is smaller than the absolute value of denominator (bottom number).
 
2. It is represented as, \(p / q\); \(q ≠ 0\) and \(|p|< |q|\), where \(p\) is the numerator, \(q\) is the denominator.
 
3. Proper fractions are greater than \(0\) but lesser than \(1\).
1. Negative proper fraction:
A negative proper fraction will have a negative sign either in numerator or denominator.
 
It can be represented as pqorpqwhereq0and|p|<|q|
Example:
916; where \(p = -9\), \(q = 16 ≠ 0\) and \(|-9| < |16|\)
2. Positive proper fraction:
A positive proper fraction will have a positive sign in both numerator and denominator.
 
It can be represented as pqorpqwhereq0and|p|<|q|
Example:
45; \(p = 4\), \(q = 5 ≠ 0\) and \(|4| < |5|\)
Improper fractions:
1. Fractions where the numerator (the top number) is greater than or equal to the denominator (the bottom number).
 
2. It is represented as pq; \(q ≠ 0\) and \(|p| ≥ |q|\); where \(p\) is the numerator, \(q\) is the denominator.
 
3. Improper fractions will be always \(1\) or greater than \(1\).
Example:
31,42,132
1. Positive improper fraction:
A positive improper fraction will have a positive sign in both numerator and denominator and the value of a positive improper fraction will always be equal or greater than \(1\).
 
It can be represented as pq;whereq0and|p||q|
Example:
1/1=15/3=1.6>19/8=1.125>1
2. Negative improper fraction:
A negative improper fraction will have a negative sign in either numerator or denominator and the value of a negative improper fraction will always be equal or greater than \(-1\).
 
It can be represented as, pqorpq;whereq0and|p||q|
Example:
6/3=2<19/5=1.8<12/1=2<1
Important!
All the mixed fractions (234,638) and integers (\(-1\), \(3\), \(7\)) are improper fractions.