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 fractions 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 to the Common Denominator Now that we have the common denominator of 20, we need to convert each fraction. 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: \[ \frac{15}{20} + \frac{8}{20} = \frac{15 + 8}{20} = \frac{23}{20} \] ### Step 4: Final Answer The sum of \( \frac{3}{4} \) and \( \frac{2}{5} \) is \( \frac{23}{20} \). ### Explanation of Other Options Now, let's analyze the other options to understand why they are incorrect: - **Option A: \( \frac{7}{9} \)**: This fraction does not relate to the sum of \( \frac{3}{4} \) and \( \frac{2}{5} \) at all. It is not a common result when adding these fractions. - **Option C: \( \frac{11}{20} \)**: This fraction is less than both \( \frac{3}{4} \) and \( \frac{2}{5} \). Since both fractions are positive and greater than zero, their sum must also be greater than either of them, making this option incorrect. - **Option D: \( \frac{17}{20} \)**: This fraction is also less than \( \frac{23}{20} \). It does not represent the correct sum of the two fractions. ### Summary - To add fractions, find a common denominator. - Convert each fraction to have that common denominator. - Add the numerators while keeping the common denominator. - Simplify if necessary (though \( \frac{23}{20} \) is already in simplest form). ### Revision Summary - The sum of \( \frac{3}{4} \) and \( \frac{2}{5} \) is \( \frac{23}{20} \). - Always find the least common denominator before adding fractions. - Convert each fraction to the common denominator before performing the addition. - Check your answer against the original fractions to ensure it makes sense.