To find the sum of the fractions \( \frac{3}{4} \) and \( \frac{2}{5} \), we need to follow a systematic approach. Let's break it down step-by-step. ### Step 1: Identify the Fractions We have two fractions: - \( \frac{3}{4} \) - \( \frac{2}{5} \) ### Step 2: Find a Common Denominator To add fractions, they must have the same denominator. The denominators here are 4 and 5. The least common multiple (LCM) of 4 and 5 will be our common denominator. - The multiples of 4 are: 4, 8, 12, 16, 20, ... - The multiples of 5 are: 5, 10, 15, 20, ... The smallest common multiple is 20. Therefore, our common denominator is 20. ### Step 3: Convert Each Fraction Next, we need to convert each fraction to have this common denominator of 20. 1. **Convert \( \frac{3}{4} \)**: - To convert \( \frac{3}{4} \) to a denominator of 20, we multiply both the numerator and the denominator by 5 (since \( 4 \times 5 = 20 \)): \[ \frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20} \] 2. **Convert \( \frac{2}{5} \)**: - To convert \( \frac{2}{5} \) to a denominator of 20, we multiply both the numerator and the denominator by 4 (since \( 5 \times 4 = 20 \)): \[ \frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20} \] ### Step 4: Add the Converted Fractions Now that both fractions have the same denominator, we can add them: \[ \frac{15}{20} + \frac{8}{20} = \frac{15 + 8}{20} = \frac{23}{20} \] ### Step 5: Final Answer The sum of the fractions \( \frac{3}{4} \) and \( \frac{2}{5} \) is \( \frac{23}{20} \). ### Explanation of Options - **Option A: \( \frac{23}{20} \)** - This is the correct answer as we calculated. - **Option B: \( \frac{31}{20} \)** - This option is incorrect. It suggests that the sum is larger than \( \frac{23}{20} \), which is not supported by our calculations. - **Option C: \( \frac{13}{20} \)** - This option is also incorrect. It is less than both fractions, which cannot be the case when adding positive fractions. - **Option D: \( \frac{17}{20} \)** - This option is incorrect as well. It is less than \( \frac{23}{20} \) and does not reflect the correct sum. ### Common Pitfalls - **Not finding a common denominator**: This is a crucial step in adding fractions. Always ensure the denominators are the same before proceeding. - **Incorrectly converting fractions**: Be careful with multiplication when converting fractions to a common denominator. - **Adding numerators without a common denominator**: This will lead to incorrect results. ### Revision Summary - To add fractions, find a common denominator. - Convert each fraction to have the common denominator. - Add the numerators and keep the common denominator. - Simplify if necessary, but in this case, \( \frac{23}{20} \) is already in simplest form. The correct answer is **A. \( \frac{23}{20} \)**.