Understanding Fractions

• A fraction represents a part of a whole, or any number of equal parts in general.
• It is written as one number over another, separated by a line.
• The number above the line represents the numerator.
• The number below the line represents the denominator.
• The denominator cannot be zero.

Types of Fractions

• Proper fractions have a numerator less than the denominator, representing a value less than 1 (e.g., 2/3).
• Improper fractions have a numerator greater than or equal to the denominator, representing a value greater than or equal to 1 (e.g., 5/3).
• Mixed numbers combine a whole number with a proper fraction (e.g., 1 2/3).
• Unit fractions have a numerator of 1 (e.g., 1/4).
• Equivalent fractions represent the same value, with different numerators and denominators (e.g., 1/2 and 2/4).

Converting Between Improper Fractions and Mixed Numbers

• Steps to convert an improper fraction to a mixed number:
  • Divide the numerator by the denominator.
  • Use the quotient as the whole number.
  • Use the remainder as the numerator of the fractional part.
  • Keep the original denominator.
  • For example: 7/3 = 2 1/3 (7 divided by 3 gives a quotient of 2 and a remainder of 1).
• Steps to convert a mixed number to an improper fraction:
  • Multiply the whole number by the denominator.
  • Add the numerator to this result.
  • Place the sum over the original denominator.
  • For example: 2 1/3 = (2 * 3 + 1) / 3 = 7/3.

Simplifying Fractions

• Simplifying means reducing a fraction to its lowest terms.
• To simplify, divide both the numerator and the denominator by their greatest common factor (GCF).
• A fraction exists in its simplest form when the numerator and denominator share no common factors other than 1.
• For example: 4/6 can be simplified to 2/3 by dividing both 4 and 6 by 2.

Finding Equivalent Fractions

• To find equivalent fractions, multiply (or divide) both the numerator and denominator by the same non-zero number.
• Multiplying creates an equivalent fraction using larger numbers.
• Dividing simplifies the fraction to an equivalent fraction using smaller numbers.
• For example: 1/2 is equivalent to 2/4 (multiply both 1 and 2 by 2).

Adding and Subtracting Fractions

• Fractions must have a common denominator to be added or subtracted.
• Find the least common multiple (LCM) of the denominators if they are different.
• Convert each fraction to an equivalent fraction with the common denominator.
• Add or subtract the numerators, keeping the denominator the same.
• Simplify the resulting fraction if necessary.
• For example: 1/4 + 2/4 = (1+2)/4 = 3/4.
• For example: 1/3 + 1/2 = 2/6 + 3/6 = 5/6 (LCM of 3 and 2 is 6).

Multiplying Fractions

• Multiply the numerators to get the new numerator.
• Multiply the denominators to get the new denominator.
• Simplify the resulting fraction if necessary.
• For example: 1/2 * 2/3 = (1 * 2) / (2 * 3) = 2/6 = 1/3.

Dividing Fractions

• To divide fractions, invert the second fraction, also known as the divisor, then multiply.
• Inverting a fraction swaps the numerator and the denominator.
• Simplify the resulting fraction if necessary.
• For example: 1/2 ÷ 2/3 = 1/2 * 3/2 = (1 * 3) / (2 * 2) = 3/4.

Comparing Fractions

• With the same denominator, the fraction with the larger numerator is larger.
• With different denominators, find a common denominator and convert the fractions, then compare the numerators.
• Alternatively, convert the fractions to decimals and compare the decimal values.
• For example: Comparing 1/3 and 1/4: 1/3 = 4/12, 1/4 = 3/12, thus 1/3 > 1/4.

Fractions on a Number Line

• Fractions are representable on a number line.
• The denominator indicates the number of equal parts the interval between two whole numbers is divided into.
• The numerator indicates the number of those parts to count from zero.
• To represent 1/4 on a number line, divide the interval between 0 and 1 into four equal parts, and mark the first part.

Real-World Applications

• Fractions are applicable in various real-world scenarios:
  • In cooking and baking for measuring ingredients.
  • In construction for measuring lengths.
  • In finance for calculating percentages and ratios.
  • In calculations of time, like representing parts of an hour.
  • In maps and scales.

Common Mistakes to Avoid

• Adding or subtracting fractions without a common denominator.
• Forgetting to simplify fractions to their lowest terms.
• Incorrectly inverting fractions when dividing.
• Misunderstanding the relationship between the numerator and denominator.

Reciprocal of a fraction

• Swapping the numerator and denominator of a fraction yields its reciprocal.
• The product of a fraction and its reciprocal always equals 1.
• For example, the reciprocal of 2/3 is 3/2, and (2/3) * (3/2) = 1.