Understanding Fractions

• A fraction represents a part of a whole or any number of equal parts.
• A fraction consists of a numerator and a denominator.
• The numerator represents the number of equal parts of a whole.
• The denominator is the total number of parts that make up a whole.
• The denominator cannot be zero, as it would make the fraction undefined.
• Fractions are written as numerator/denominator, such as 1/2, 3/4, or 5/8.

Types of Fractions

• Proper fractions have a numerator less than the denominator; for example, 2/3.
• Improper fractions have a numerator greater than or equal to the denominator; for example, 5/3.
• Mixed numbers combine a whole number with a proper fraction; for example, 1 2/3.

Converting Between Improper Fractions and Mixed Numbers

• When converting an improper fraction to a mixed number, divide the numerator by the denominator; the quotient becomes the whole number, and the remainder is the new numerator, keeping the original denominator.
• For example: 7/3 = 2 1/3 (7 ÷ 3 = 2 with a remainder of 1).
• When converting 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.
• For example: 2 1/3 = (2 * 3 + 1)/3 = 7/3.

Equivalent Fractions

• Equivalent fractions represent the same value but have different numerators and denominators.
• They are obtained by multiplying or dividing both the numerator and denominator by the same non-zero number.
• For example: 1/2 = 2/4 = 3/6.

Simplifying Fractions

• Simplifying a fraction means reducing it to its lowest terms.
• This is done by dividing both the numerator and denominator by their greatest common divisor (GCD).
• For example: 4/6 simplified is 2/3 (GCD of 4 and 6 is 2).

Finding the Greatest Common Divisor (GCD)

• The GCD is the largest number that divides two or more numbers without leaving a remainder, also known as HCF (Highest Common Factor).
• One method to find the GCD is by listing factors.
• Another method is using the Euclidean algorithm.

Operations with Fractions

• Adding and Subtracting Fractions: Fractions must have a common denominator before they can be added or subtracted.
• If fractions do not have the same denominator, find the least common multiple (LCM) to make the denominators the same.
• Once there is a common denominator add or subtract the numerators and then place the result over the common denominator.
• Multiplying Fractions: Multiply the numerators together and denominators together.
• For example: (1/2) * (2/3) = (1 * 2) / (2 * 3) = 2/6 = 1/3.
• Dividing Fractions: Invert the second fraction (the divisor) and multiply.
• The divisor is the fraction being divided by.
• For example: (1/2) ÷ (2/3) = (1/2) * (3/2) = (1 * 3) / (2 * 2) = 3/4.

Least Common Multiple

• The least common multiple (LCM) of two integers a and b, denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b.
• Identify the prime factors of each number.
• List all prime factors with the highest exponent.
• Multiply all prime factors with the highest exponent.

Comparing Fractions

• If fractions have the same denominator, compare the numerators.
• The fraction with the larger numerator is greater.
• If fractions have different denominators, find a common denominator and then compare the numerators or convert each fraction to a decimal.
• For example: 3/4 vs. 5/6. The common denominator is 12, so 9/12 vs. 10/12. Therefore, 5/6 is greater.

Fractions on the Number Line

• Fractions can be represented on a number line.
• Divide the space between each whole number into equal parts based on the denominator.
• Place the fraction at the appropriate point.

Decimal Representation of Fractions

• Any fraction can be represented as a decimal by dividing the numerator by the denominator.
• The decimal can be terminating (e.g., 1/4 = 0.25) or repeating (e.g., 1/3 = 0.333...).

Converting Decimals to Fractions

• Terminating decimals: Write the decimal as a fraction with a denominator of 10, 100, 1000, etc., depending on the number of decimal places, and simplify.
• Repeating decimals: Use algebraic methods to convert repeating decimals to fractions.

Applications of Fractions

• Fractions are used in everyday life for cooking, measurements, and dividing quantities.
• They are also essential in algebra, geometry, and calculus.