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Fractions are numerical expressions that represent a part of a whole. They consist of a numerator and a denominator. Think of the numerator as the number of parts that you have, and the denominator as the total number of parts in the whole. For example, the fraction \( \frac{3}{4} \) indicates 3 parts out of a total of 4 parts.
Fractions can be proper, improper, or mixed. A proper fraction is when the numerator is less than the denominator, like \( \frac{3}{4} \). An improper fraction has a numerator greater than or equal to the denominator, such as \( \frac{7}{4} \).
Fractions can express various mathematical situations, including parts of a collection, division of quantities, and ratios. Being able to work with fractions is crucial since they often appear in different areas of mathematics and real-world problems.

Repeating decimals are decimals in which a digit or a group of digits repeats infinitely. For example, the decimal \( 0.272727... \) is a repeating decimal because the "27" sequence repeats endlessly. These can be represented using a bar notation, like \( 0.\overline{27} \).
Any rational number can be expressed as either a terminating decimal or a repeating decimal. Repeating decimals occur when the denominator of the fraction, in its simplest form, has prime factors other than 2 and 5. This is because fractions with prime factors of only 2 and 5 in the denominator will terminate.
To convert a repeating decimal into a fraction, identify the repeating section and use algebraic techniques to achieve an exact fractional representation. This involves setting up an equation where the repeating decimal equals a variable and using multiplication to isolate the repeating portion.

Simplifying fractions means reducing them to their simplest form. A fraction is simplified when the numerator and denominator have no common factors other than 1. For example, \( \frac{27}{99} \) can be simplified to \( \frac{3}{11} \) because both 27 and 99 are divisible by 9.
To simplify a fraction, follow these steps:

• Find the greatest common divisor (GCD) of the numerator and the denominator. This is the largest number that evenly divides into both.
• Divide both the numerator and the denominator by their GCD.

For instance, the GCD of 27 and 99 is 9, so dividing the numerator and the denominator by 9 simplifies \( \frac{27}{99} \) to \( \frac{3}{11} \).
Simplifying fractions is essential for comparing fractions, performing arithmetic operations, and understanding the relationships between different numbers.