O Fracions Addition or subtraction of mixed numbers with different denominators... Add. Write your answer as a mixed number in simplest form. \[ 5 \frac{2}{3}+2 \frac{1}{7} \]

To add the mixed numbers \( 5 \frac{2}{3} \) and \( 2 \frac{1}{7} \), follow these steps: ### Step 1: Add the Whole Numbers Start by adding the whole number parts of the mixed numbers. \[ 5 + 2 = 7 \] ### Step 2: Add the Fractional Parts Next, add the fractional parts \( \frac{2}{3} \) and \( \frac{1}{7} \). #### a. Find a Common Denominator To add the fractions, they must have the same denominator. The denominators here are 3 and 7. The least common denominator (LCD) of 3 and 7 is 21. \[ \text{LCD of } 3 \text{ and } 7 = 21 \] #### b. Convert Fractions to Equivalent Fractions with the LCD Convert each fraction to an equivalent fraction with the denominator of 21. \[ \frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21} \] \[ \frac{1}{7} = \frac{1 \times 3}{7 \times 3} = \frac{3}{21} \] #### c. Add the Equivalent Fractions Now, add the two fractions: \[ \frac{14}{21} + \frac{3}{21} = \frac{17}{21} \] ### Step 3: Combine the Whole Number and Fractional Parts Combine the sum of the whole numbers with the sum of the fractions to get the final result. \[ 7 + \frac{17}{21} = 7 \frac{17}{21} \] ### Step 4: Simplify if Possible Check if the fractional part can be simplified. - **Numerator:** 17 (a prime number) - **Denominator:** 21 (factors are 3 and 7) Since 17 is a prime number and does not share any common factors with 21 other than 1, the fraction \( \frac{17}{21} \) is already in its simplest form. ### Final Answer \[ 5 \frac{2}{3} + 2 \frac{1}{7} = 7 \frac{17}{21} \]