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: Find a Common Denominator Fractions can only be added directly if they have the same denominator. The denominators in our case are 4 and 5. To add these fractions, we need to find the least common denominator (LCD). - The denominators are 4 and 5. - The least common multiple (LCM) of 4 and 5 is 20. This is because 20 is the smallest number that both 4 and 5 can divide into without leaving a remainder. ### Step 2: Convert Each Fraction Now that we have the common denominator, we need to convert each fraction to have this denominator. 1. **Convert \( \frac{3}{4} \)**: - To convert \( \frac{3}{4} \) to a fraction with 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 fraction with 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 3: Add the Converted Fractions Now that both fractions have the same denominator, we can add them together: \[ \frac{15}{20} + \frac{8}{20} = \frac{15 + 8}{20} = \frac{23}{20} \] ### Step 4: Final Answer The sum of \( \frac{3}{4} \) and \( \frac{2}{5} \) is \( \frac{23}{20} \). ### Explanation of Options - **Option A: \( \frac{1}{2} \)** - This is incorrect because \( \frac{1}{2} \) is much smaller than the sum we calculated. - **Option B: \( \frac{23}{20} \)** - This is the correct answer, as we derived it through proper addition of the fractions. - **Option C: \( \frac{17}{20} \)** - This is incorrect. It seems to be a miscalculation, possibly from adding the numerators incorrectly or using the wrong common denominator. - **Option D: \( 1 \)** - This is incorrect because \( 1 \) is equivalent to \( \frac{20}{20} \), which is less than \( \frac{23}{20} \). ### Summary - To add fractions, find a common denominator. - Convert each fraction to have the same denominator. - Add the numerators and keep the common denominator. - The sum of \( \frac{3}{4} \) and \( \frac{2}{5} \) is \( \frac{23}{20} \). ### Revision Summary - Always find the least common denominator when adding fractions. - Convert each fraction to the common denominator before adding. - Add the numerators and keep the common denominator for the result. - Check your work by ensuring the final answer makes sense in relation to the original fractions.