11. Find two mixed numbers with unlike denominators that have a sum of \( 4 \frac{5}{8} \).

To find two mixed numbers with **unlike denominators** that add up to \( 4 \frac{5}{8} \), let's follow these steps: ### Understanding the Problem A **mixed number** consists of a whole number and a fractional part, such as \( 2 \frac{1}{3} \). **Unlike denominators** means that the fractional parts of the two mixed numbers have different denominators (e.g., one has a denominator of 3 and the other has a denominator of 4). ### Step-by-Step Solution 1. **Express the Total Sum as an Improper Fraction:** \( 4 \frac{5}{8} = \frac{4 \times 8 + 5}{8} = \frac{37}{8} \) 2. **Choose a Whole Number for One Mixed Number:** Let's choose a whole number for one of the mixed numbers to simplify calculations. For example, let’s select \( 1 \frac{1}{2} \). 3. **Convert \( 1 \frac{1}{2} \) to an Improper Fraction:** \( 1 \frac{1}{2} = \frac{3}{2} \) 4. **Subtract This from the Total Sum to Find the Other Mixed Number:** \[ \frac{37}{8} - \frac{3}{2} = \frac{37}{8} - \frac{12}{8} = \frac{25}{8} = 3 \frac{1}{8} \] 5. **Convert \( \frac{25}{8} \) Back to a Mixed Number:** \( \frac{25}{8} = 3 \frac{1}{8} \) 6. **Verify the Denominators Are Unlike:** - \( 1 \frac{1}{2} \) has a denominator of **2**. - \( 3 \frac{1}{8} \) has a denominator of **8**. Since **2 ≠ 8**, the denominators are unlike. ### Conclusion One such pair of mixed numbers that satisfies the condition is: \[ 1 \frac{1}{2} \quad \text{and} \quad 3 \frac{1}{8} \] **Verification:** \[ 1 \frac{1}{2} + 3 \frac{1}{8} = \frac{3}{2} + \frac{25}{8} = \frac{12}{8} + \frac{25}{8} = \frac{37}{8} = 4 \frac{5}{8} \] ### Additional Example Another pair could be: \[ 2 \frac{1}{4} \quad \text{and} \quad 2 \frac{3}{8} \] **Verification:** \[ 2 \frac{1}{4} + 2 \frac{3}{8} = \frac{9}{4} + \frac{19}{8} = \frac{18}{8} + \frac{19}{8} = \frac{37}{8} = 4 \frac{5}{8} \] Both denominators (**4** and **8**) are unlike. ### Note There are infinitely many pairs of mixed numbers with unlike denominators that add up to \( 4 \frac{5}{8} \). The examples above are just two possible solutions.