4 Maria packs her backpack for a hike. In one pocket, she puts in a \( \frac{1}{2} \)-pound bag of trail mix, a water bottle weighing \( 2 \frac{1}{8} \) pounds, and a flashlight weighing \( \frac{1}{4} \) pound. How much weight do these three items add to her backpack? Show your work.

To determine the total weight Maria adds to her backpack, we'll sum the weights of the three items: 1. **Trail mix:** \( \frac{1}{2} \) pound 2. **Water bottle:** \( 2 \frac{1}{8} \) pounds 3. **Flashlight:** \( \frac{1}{4} \) pound **Step 1: Convert Mixed Numbers to Improper Fractions** First, convert any mixed numbers to improper fractions to make addition easier. - **Water bottle:** \( 2 \frac{1}{8} \) \[ 2 \frac{1}{8} = \frac{2 \times 8 + 1}{8} = \frac{16 + 1}{8} = \frac{17}{8} \] **Step 2: Find a Common Denominator** To add the fractions, they must have the same denominator. The denominators we have are 2, 8, and 4. The least common denominator (LCD) for these is **8**. - **Trail mix:** \( \frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8} \) - **Water bottle:** \( \frac{17}{8} \) (already has denominator 8) - **Flashlight:** \( \frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8} \) **Step 3: Add the Fractions** Now, add all the fractions with the common denominator: \[ \frac{4}{8} + \frac{17}{8} + \frac{2}{8} = \frac{4 + 17 + 2}{8} = \frac{23}{8} \] **Step 4: Simplify the Result** Convert the improper fraction back to a mixed number: \[ \frac{23}{8} = 2 \frac{7}{8} \] **Conclusion** The three items add a total of **\( 2 \frac{7}{8} \) pounds** to Maria's backpack.