Understanding Fractions: Types and Conversions

• Denotes a part of a whole or any number of equal parts.
• Represented as a/b, where 'a' is the numerator and 'b' is the denominator.
• The denominator cannot equal zero.

Types of Fractions

• Proper fractions have a numerator less than the denominator; example: 2/5.
• Improper fractions have a numerator greater than or equal to the denominator; example: 7/3.
• Mixed numbers combine a whole number with a proper fraction; example: 1 2/3.
• Equivalent fractions represent the same value with different numerators and denominators; example: 1/2 and 2/4.
• Complex fractions contain a fraction in the numerator, denominator, or both; example: (1/2) / (3/4).

Converting Between Improper Fractions and Mixed Numbers

• To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient becomes the whole number, the remainder becomes the new numerator, and the denominator remains the same.
• To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator.

Simplifying Fractions

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.
• The result is the fraction in its simplest form, also known as its lowest terms.

Comparing Fractions

• Fractions sharing the same denominator can be compared by examining the numerators; the fraction with the larger numerator is the greater fraction.
• Fractions with different denominators require finding a common denominator, often the least common multiple (LCM); once converted to equivalent fractions with the common denominator, compare the numerators.

Operations with Fractions

• For addition and subtraction, fractions must have a common denominator; add or subtract the numerators while retaining the common denominator.
• Multiplication involves multiplying the numerators together and the denominators together: (a/b) * (c/d) = (ac)/(bd).
• Division involves inverting the second fraction (the divisor) and then multiplying: (a/b) / (c/d) = (a/b) * (d/c) = (ad)/(bc).

Reciprocal of a Fraction

• The reciprocal of a fraction a/b is b/a.
• A fraction multiplied by its reciprocal always equals 1.

Least Common Multiple (LCM)

• Defines the smallest multiple shared by two or more numbers.
• Used to determine the Least Common Denominator (LCD) when adding or subtracting fractions.

Greatest Common Divisor (GCD)

• Defines the largest number that divides evenly into two or more numbers.
• Used to simplify fractions by dividing both the numerator and the denominator.

Decimal Representation of Fractions

• Fractions can be represented as decimals by dividing the numerator by the denominator.
• Decimals can be terminating (ending) or repeating.
• Terminating decimals result when the denominator's prime factors consist only of 2s and/or 5s.

Applications of Fractions

• Used in cooking, measuring, and dividing quantities.
• Used in mathematical and scientific calculations.

Key Concepts for Working with Fractions

• Grasping equivalent fractions is essential for performing many operations.
• Simplifying fractions streamlines calculations.
• Proficiency in multiplication and division is important when working with fractions.
• Common denominators are required to effectively add and subtract fractions.
• Being able to convert between improper fractions and mixed numbers is a useful skill.

Learn about fractions, including proper, improper, and mixed types. Understand how to convert between improper fractions and mixed numbers. Explore equivalent and complex fractions with examples.