2.2 Addition & Subtraction of Fractions and Mixed Numbers

2.2 Addition & Subtraction of Fractions and Mixed Numbers\n

Adding and subtracting fractions requires a clear understanding of fraction structure. Unlike multiplication, addition and subtraction can only be performed when the fractions represent parts of identical size—that is, when they share a common denominator.\n ---\n

Addition and Subtraction with Like Denominators\n

When fractions have like denominators (identical bottom numbers), they share the same unit size. Addition or subtraction is accomplished by operating directly on the numerators while keeping the denominator unchanged.\n $\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} \qquad \text{and} \qquad \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c} \qquad (c \neq 0)$\n

Worked Example 1: Like Denominators\n

Evaluate the following expressions and simplify:\n1. (\frac{7}{15} + \frac{4}{15})\n2. (\frac{11}{12} - \frac{5}{12})\n

• Problem 1 Solution:\n $\frac{7}{15} + \frac{4}{15} = \frac{7 + 4}{15} = \frac{11}{15}$\n Since 11 and 15 share no common factors other than 1, (\frac{11}{15}) is already in simplest form.\n
• Problem 2 Solution:\n $\frac{11}{12} - \frac{5}{12} = \frac{11 - 5}{12} = \frac{6}{12}$\n Simplify by dividing numerator and denominator by their GCF, 6:\n $\frac{6 \div 6}{12 \div 6} = \frac{1}{2}$\n ---\n

Addition and Subtraction with Unlike Denominators\n

When fractions have unlike denominators, they represent parts of different sizes. To add or subtract them, you must first convert them into equivalent fractions with a Least Common Denominator (LCD).\n

Finding the Least Common Denominator (LCD)\n

The LCD of two or more denominators is the Least Common Multiple (LCM) of those numbers.\n

• Method A (Listing Multiples): List the multiples of each denominator until you find the smallest shared number.\n- Method B (Prime Factorization): Write out prime factorizations, take the highest power of each prime factor, and multiply them together.\n \\n FINDING THE LEAST COMMON DENOMINATOR (LCD)\\n For 6 and 8\\n \\n Listing Multiples: Prime Factorization:\\n Multiples of 6: 6, 12, 18, [24], 30 --> 6 = 2^1 x 3^1\\n Multiples of 8: 8, 16, [24], 32, 40 --> 8 = 2^3\\n --> LCD = 2^3 x 3^1 = 8 x 3 = 24\\n\n

4-Step Framework for Operations on Unlike Fractions\n

| Step | Description | Example: Calculate (\frac{5}{6} - \frac{3}{8}) |\n| :--- | :--- | :--- |\n| 1. Find LCD | Determine LCM of denominators | (\text{LCM}(6, 8) = 24) |\n| 2. Convert | Scale each fraction to equivalent with LCD | (\frac{5 \times 4}{6 \times 4} = \frac{20}{24}) and (\frac{3 \times 3}{8 \times 3} = \frac{9}{24}) |\n| 3. Combine | Add or subtract numerators over LCD | (\frac{20 - 9}{24} = \frac{11}{24}) |\n| 4. Simplify | Reduce fraction / convert to mixed number | (\frac{11}{24}) is in simplest form |\n

Worked Example 2: Adding Fractions with Unlike Denominators\n

Calculate (\frac{3}{10} + \frac{4}{15}) and write the answer in lowest terms.\n

• Step 1: Find LCD of 10 and 15. Prime factorizations: (10 = 2 \times 5), (15 = 3 \times 5). \n $\text{LCD} = 2 \times 3 \times 5 = 30$\n- Step 2: Convert both fractions to equivalent fractions with denominator 30:\n $\frac{3 \times 3}{10 \times 3} = \frac{9}{30}$\n $\frac{4 \times 2}{15 \times 2} = \frac{8}{30}$\n- Step 3: Add the numerators:\n $\frac{9}{30} + \frac{8}{30} = \frac{9 + 8}{30} = \frac{17}{30}$\n- Step 4: 17 is a prime number, so (\frac{17}{30}) is fully simplified.\n ---\n

Adding Mixed Numbers\n

To add mixed numbers:\n1. Add the whole number parts together.\n2. Add the fractional parts together using a common denominator.\n3. If the sum of the fractional parts results in an improper fraction, convert it into a mixed number and carry the whole number over.\n

Worked Example 3: Adding Mixed Numbers with Carrying\n

Calculate (4 \frac{2}{3} + 3 \frac{3}{4}).\n

• Step 1: Find LCD for fractions (\frac{2}{3}) and (\frac{3}{4}). (\text{LCD}(3, 4) = 12).\n- Step 2: Convert fractional parts:\n $\frac{2}{3} = \frac{8}{12} \quad \text{and} \quad \frac{3}{4} = \frac{9}{12}$\n- Step 3: Add whole numbers and fractions separately:\n $\text{Whole Numbers:} \quad 4 + 3 = 7$\n $\text{Fractions:} \quad \frac{8}{12} + \frac{9}{12} = \frac{17}{12}$\n- Step 4: Convert improper fraction (\frac{17}{12}) to mixed number: (1 \frac{5}{12}).\n- Step 5: Combine whole number sum with mixed fraction:\n $7 + 1 \frac{5}{12} = 8 \frac{5}{12}$\n ---\n

Subtracting Mixed Numbers with Regrouping / Borrowing\n

When subtracting mixed numbers, if the fraction in the minuend (first number) is smaller than the fraction in the subtrahend (second number), you must borrow 1 from the whole number of the minuend.\n

Regrouping Rule: Borrowing 1 from whole number (W) converts (W \frac{a}{b}) to:\n>\n> $W \frac{a}{b} = (W - 1) + 1 + \frac{a}{b} = (W - 1) + \frac{b}{b} + \frac{a}{b} = (W - 1) \frac{a + b}{b}$\n

Worked Example 4: Mixed Number Subtraction with Borrowing\n

Calculate (8 \frac{1}{5} - 3 \frac{3}{4}).\n

• Step 1: Find LCD of 5 and 4, which is 20.\n- Step 2: Convert fractional parts:\n $8 \frac{1}{5} = 8 \frac{4}{20} \quad \text{and} \quad 3 \frac{3}{4} = 3 \frac{15}{20}$\n- Step 3: Observe that (\frac{4}{20} < \frac{15}{20}). Regroup (8 \frac{4}{20}) by borrowing 1 from 8:\n $8 \frac{4}{20} = 7 + 1 \frac{4}{20} = 7 + \frac{20}{20} + \frac{4}{20} = 7 \frac{24}{20}$\n- Step 4: Subtract whole numbers and converted fractions:\n $\text{Whole Numbers:} \quad 7 - 3 = 4$\n $\text{Fractions:} \quad \frac{24}{20} - \frac{15}{20} = \frac{9}{20}$\n- Step 5: Combine to get the final answer: (4 \frac{9}{20}).\n ---\n

ACCUPLACER Exam Traps & Common Errors\n

ACCUPLACER Exam Trap 1: Adding Denominators Together \n> Never add denominators! A widespread error is writing (\frac{1}{3} + \frac{1}{4} = \frac{2}{7}) (INCORRECT). Denominators define unit size and must remain constant once aligned. The correct calculation is (\frac{4}{12} + \frac{3}{12} = \frac{7}{12}).\n ACCUPLACER Exam Trap 2: Reverse Subtraction When Borrowing \n> When faced with (5 \frac{1}{6} - 2 \frac{5}{6}), students sometimes subtract the smaller numerator from the larger in reverse order, obtaining (3 \frac{4}{6}) (INCORRECT). You must borrow 1 from 5 to write (4 \frac{7}{6} - 2 \frac{5}{6} = 2 \frac{2}{6} = 2 \frac{1}{3}).\n ACCUPLACER Exam Trap 3: Forgetting to Convert Whole Number Borrowing \n> When subtracting from a whole number (e.g., (6 - 2 \frac{3}{5})), rewrite 6 as (5 \frac{5}{5}) before subtracting: (5 \frac{5}{5} - 2 \frac{3}{5} = 3 \frac{2}{5}).