Write + or - in eaeh \( O \) to make eaeh equation true. (a) \( \frac{13}{18} \bigcirc \frac{1}{9} \bigcirc \frac{1}{2} \bigcirc \frac{2}{3}=1 \) (b) \( \frac{3}{4} \bigcirc \frac{13}{24} \bigcirc \frac{5}{3} \bigcirc \frac{1}{6}=1 \)

Let's solve each part by determining where to place the \( + \) or \( - \) signs to satisfy the given equations. --- ### **Problem (a)** \[ \frac{13}{18} \bigcirc \frac{1}{9} \bigcirc \frac{1}{2} \bigcirc \frac{2}{3} = 1 \] **Objective:** Assign \( + \) or \( - \) to each \( \bigcirc \) so that the equation holds true. **Solution:** We need to find: \[ \frac{13}{18} \; \text{op}_1 \; \frac{1}{9} \; \text{op}_2 \; \frac{1}{2} \; \text{op}_3 \; \frac{2}{3} = 1 \] where each \( \text{op}_i \) is either \( + \) or \( - \). By testing possible combinations, the following assignment works: \[ \frac{13}{18} + \frac{1}{9} - \frac{1}{2} + \frac{2}{3} = 1 \] **Verification:** \[ \frac{13}{18} + \frac{1}{9} = \frac{13}{18} + \frac{2}{18} = \frac{15}{18} = \frac{5}{6} \] \[ \frac{5}{6} - \frac{1}{2} = \frac{5}{6} - \frac{3}{6} = \frac{2}{6} = \frac{1}{3} \] \[ \frac{1}{3} + \frac{2}{3} = 1 \] **Answer for (a):** \[ \frac{13}{18} \mathbf{+} \frac{1}{9} \mathbf{-} \frac{1}{2} \mathbf{+} \frac{2}{3} = 1 \] --- ### **Problem (b)** \[ \frac{3}{4} \bigcirc \frac{13}{24} \bigcirc \frac{5}{3} \bigcirc \frac{1}{6} = 1 \] **Objective:** Assign \( + \) or \( - \) to each \( \bigcirc \) so that the equation holds true. **Solution:** We need to find: \[ \frac{3}{4} \; \text{op}_1 \; \frac{13}{24} \; \text{op}_2 \; \frac{5}{3} \; \text{op}_3 \; \frac{1}{6} = 1 \] where each \( \text{op}_i \) is either \( + \) or \( - \). By testing all possible combinations of \( + \) and \( - \), none of the assignments satisfy the equation. Here's a summary of the results: - \( +, +, + \): Greater than 1 - \( +, +, - \): Greater than 1 - \( +, -, + \): Less than 0 - \( +, -, - \): Less than 0 - \( -, +, + \): Greater than 1 - \( -, +, - \): Greater than 0 - \( -, -, + \): Less than 0 - \( -, -, - \): Less than 0 **Conclusion for (b):** No combination of \( + \) and \( - \) signs will satisfy the equation. --- ### **Final Answer** **Problem (a) Answer:** \[ \frac{13}{18} + \frac{1}{9} - \frac{1}{2} + \frac{2}{3} = 1 \] **Problem (b) Answer:** No combination of \( + \) and \( - \) signs makes the equation true.