Adding and Subtracting Fractions

• Adding and subtracting fractions is a fundamental arithmetic skill involving rational numbers
• Mastery requires understanding of numerators, denominators, equivalent fractions, and common denominators

Adding Fractions with Like Denominators

• When fractions have the same denominator, addition involves adding the numerators and keeping the denominator the same
• Example: 1/5 + 2/5 = (1+2)/5 = 3/5
• The denominator represents the total number of equal parts of a whole, while the numerator represents the number of those parts you have
• The result should be simplified if possible, by reducing the fraction to its lowest terms

Subtracting Fractions with Like Denominators

• Subtraction with like denominators follows a similar process
• Subtract the numerators and keep the denominator the same
• Example: 4/7 - 1/7 = (4-1)/7 = 3/7
• Ensure the result is simplified to its lowest terms

Adding Fractions with Unlike Denominators

• When fractions have different denominators, a common denominator must be found before adding
• The least common denominator (LCD) is the smallest multiple that both denominators share
• One method to find the LCD is listing the multiples of each denominator until a common multiple is found
• Example: For 1/4 and 2/6, multiples of 4 are 4, 8, 12, 16..., and multiples of 6 are 6, 12, 18...
• Thus, the LCD is 12
• Once the LCD is found, each fraction must be converted into an equivalent fraction with the LCD as the new denominator
• Multiply both the numerator and denominator of each fraction by the factor that makes the original denominator equal to the LCD
• Once both fractions have the same denominator, add the numerators and keep the common denominator
• Simplify the resulting fraction if necessary

Subtracting Fractions with Unlike Denominators

• Subtraction with unlike denominators is similar to addition, but involves subtracting the numerators after finding the LCD and converting the fractions
• Example: To subtract 1/3 from 1/2, find the LCD of 3 and 2, which is 6
• Convert 1/2 to 3/6 (multiply numerator and denominator by 3)
• Convert 1/3 to 2/6 (multiply numerator and denominator by 2)
• Subtract the numerators: 3/6 - 2/6 = (3-2)/6 = 1/6

Adding Mixed Numbers

• A mixed number consists of a whole number and a proper fraction (where the numerator is less than the denominator)
• To add mixed numbers, there are generally two methods: adding whole numbers and fractions separately, or converting mixed numbers to improper fractions
• Method 1: Add the whole numbers together and then add the fractions together
• If the sum of the fractions results in an improper fraction, convert it to a mixed number and add the whole number part to the sum of the whole numbers
• Method 2: Convert each mixed number to an improper fraction (numerator is greater than or equal to the denominator)
• To convert, multiply the whole number by the denominator and add the numerator, then place the result over the original denominator
• Then, add the improper fractions (after finding a common denominator if necessary)
• Convert the resulting improper fraction back to a mixed number if needed

Subtracting Mixed Numbers

• Subtracting mixed numbers also involves two main methods: subtracting whole numbers and fractions separately, or converting to improper fractions
• Method 1: Subtract the whole numbers and fractions separately
• Borrow from the whole number if the fraction being subtracted is larger than the fraction being subtracted from
• Method 2: Convert each mixed number to an improper fraction
• Subtract the improper fractions (after finding a common denominator if necessary)
• Convert the resulting improper fraction back to a mixed number if needed

Simplifying Fractions and Mixed Numbers

• Simplification involves reducing a fraction to its lowest terms
• Divide both the numerator and the denominator by their greatest common factor (GCF)
• The GCF is the largest number that divides both the numerator and denominator without leaving a remainder
• Keep simplifying until numerator and denominator have no common factors other than 1
• For Mixed Numbers, ensure the fractional part is simplified

Real-World Applications

• These operations with fractions are essential in various real-world scenarios
• Cooking: Recipes often involve fractional quantities of ingredients
• Measurement: Construction, woodworking, and sewing often require precise measurements using fractions
• Finance: Calculating portions of investments, interest rates, or discounts
• Time Management: Scheduling and planning activities often involve breaking down time into fractions of hours or minutes