To find the result of adding the fractions \( \frac{3}{4} \) and \( \frac{1}{6} \), we need to follow a systematic approach. Let's break it down step-by-step. ### Step 1: Find a Common Denominator When adding fractions, we need a common denominator. The denominators in our fractions are 4 and 6. To find the least common denominator (LCD), we can list the multiples of each denominator: - Multiples of 4: 4, 8, 12, 16, 20, ... - Multiples of 6: 6, 12, 18, 24, ... The smallest common multiple is 12. Therefore, our common denominator will be 12. ### Step 2: Convert Each Fraction Next, we need to convert each fraction to have this common denominator of 12. 1. **Convert \( \frac{3}{4} \)**: - To convert \( \frac{3}{4} \) to a fraction with a denominator of 12, we can multiply both the numerator and the denominator by 3 (since \( 4 \times 3 = 12 \)): \[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \] 2. **Convert \( \frac{1}{6} \)**: - To convert \( \frac{1}{6} \) to a fraction with a denominator of 12, we can multiply both the numerator and the denominator by 2 (since \( 6 \times 2 = 12 \)): \[ \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \] ### Step 3: Add the Converted Fractions Now that both fractions have the same denominator, we can add them: \[ \frac{9}{12} + \frac{2}{12} = \frac{9 + 2}{12} = \frac{11}{12} \] ### Final Answer The result of adding \( \frac{3}{4} \) and \( \frac{1}{6} \) is \( \frac{11}{12} \). ### Explanation of Options Now, let's analyze the options provided: - **A. \( \frac{5}{12} \)**: This is incorrect because it does not represent the sum of the two fractions. It is too small. - **B. \( \frac{9}{12} \)**: This is the value of \( \frac{3}{4} \) alone, not the sum of both fractions. - **C. \( \frac{11}{12} \)**: This is the correct answer, as we calculated. - **D. \( \frac{7}{12} \)**: This is incorrect because it is less than both fractions and does not represent their sum. ### Common Pitfalls - **Not finding a common denominator**: This is a common mistake when adding fractions. Always ensure the denominators are the same before adding. - **Incorrectly converting fractions**: Be careful when multiplying the numerator and denominator to ensure they remain equivalent fractions. ### Revision Summary - To add fractions, find a common denominator. - Convert each fraction to have the same denominator. - Add the numerators and keep the common denominator. - Simplify if necessary, but in this case, \( \frac{11}{12} \) is already in simplest form. The correct answer is **C. \( \frac{11}{12} \)**.