Short answers, pulled from the story.
The word fraction comes from the Latin fractus, meaning broken. A fraction represents a part of a whole or any number of equal parts, such as one-half, eight-fifths, or three-quarters. A simple fraction has an integer numerator above the line and a non-zero integer denominator below it.
When the numerator and denominator are both positive, a fraction is proper if the numerator is less than the denominator and improper otherwise. More generally, a fraction is proper if its absolute value is strictly less than one and improper, or top-heavy, if that value is at least 1. The improper fraction was a late development, since fraction means piece and a proper fraction was held to be less than 1.
The horizontal fraction bar is first attested in the work of Al-Hassar, a Muslim mathematician from Fez, Morocco, who was active around 1200 and specialized in Islamic inheritance jurisprudence. The same notation, with the fraction given before the integer, appears soon after in the work of Leonardo Fibonacci in the 13th century.
Decimal fractions became a common computational practice through the Flemish pamphlet De Thiende, published at Leyden in 1585 with a French translation, La Disme, by Simon Stevin, who lived from 1548 to 1620. Decimals predated him, however, used by the Chinese many centuries earlier and by the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century.
An Egyptian fraction is the sum of distinct positive unit fractions, and every positive rational number can be expanded this way in infinitely many forms. The Egyptians used Egyptian fractions around 1000 BC, dividing with least common multiples and unit fractions about 4000 years ago.
A mixed number is the sum of a non-zero integer and a proper fraction written side by side. To convert it to an improper fraction, multiply the whole number by the denominator, add the numerator to get the new numerator, and keep the same denominator. Conversely, an improper fraction converts to a mixed number using division with remainder, as when 4 goes into 11 twice with 3 left over.