Package \( A \) weighs \( 4 \frac{1}{2} \) pounds. Package \( B \) weighs \( 2 \frac{1}{8} \) pounds more than Package \( A \). What is the total weight of the two packages?

To find the total weight of the two packages, we first need to calculate the weight of Package \( B \). Package \( A \) weighs \( 4 \frac{1}{2} \) pounds, which can be converted into an improper fraction: \[ 4 \frac{1}{2} = \frac{9}{2} \text{ pounds.} \] Now, \( B \) weighs \( 2 \frac{1}{8} \) pounds more than Package \( A \). We convert \( 2 \frac{1}{8} \) into an improper fraction as well: \[ 2 \frac{1}{8} = \frac{17}{8} \text{ pounds.} \] Now, we find the weight of Package \( B \): \[ \text{Weight of Package } B = \text{Weight of Package } A + 2 \frac{1}{8} = \frac{9}{2} + \frac{17}{8}. \] To add these fractions, we need a common denominator. The least common multiple of \( 2 \) and \( 8 \) is \( 8 \). Convert \( \frac{9}{2} \) to eighths: \[ \frac{9}{2} = \frac{36}{8}. \] Now we can add the weights: \[ \text{Weight of Package } B = \frac{36}{8} + \frac{17}{8} = \frac{53}{8} \text{ pounds.} \] Now, let's calculate the total weight of both packages: \[ \text{Total weight} = \text{Weight of Package } A + \text{Weight of Package } B = \frac{9}{2} + \frac{53}{8}. \] Convert \( \frac{9}{2} \) to eighths again: \[ \frac{9}{2} = \frac{36}{8}. \] Now we can add these: \[ \text{Total weight} = \frac{36}{8} + \frac{53}{8} = \frac{89}{8} \text{ pounds.} \] Converting \( \frac{89}{8} \) into a mixed number: \[ \frac{89}{8} = 11 \frac{1}{8} \text{ pounds.} \] So, the total weight of both packages is \( 11 \frac{1}{8} \) pounds.