Let's analyze the problem step-by-step. --- ### Step 1: Understand the problem Gan and Julia each think differently about how taping two rectangular pieces of art paper affects the total perimeter. - **Gan's idea:** Tape along the *shorter sides*. - **Julia's idea:** Tape along the *longer sides*. We need to determine who is correct regarding which method results in a *greater* perimeter of the combined shape. --- ### Step 2: Define variables Let: - \( L \) = length of the longer side of each paper. - \( S \) = length of the shorter side of each paper. Assume both pieces are identical rectangles. --- ### Step 3: Understand what happens when taping When the two papers are taped together: - The *perimeter of the combined shape* depends on how they are joined. - Taping along a side effectively *merges* that side, removing it from the perimeter calculation, since it's no longer an exposed edge. --- ### Step 4: Calculate the perimeter in each case --- #### Case 1: Taping along the shorter sides (Gan's idea) - Before taping, each paper's perimeter is: \[ P_{\text{single}} = 2L + 2S \] - When taping along the shorter sides (length \( S \)): - The two shorter sides are joined, removing 2 \( S \) edges from the perimeter. - The combined shape is a rectangle of dimensions \( L + L = 2L \) (lengthwise) and \( S \) (width). **Resulting perimeter:** \[ P_{\text{Gan}} = 2 \times (2L) + 2 \times S = 4L + 2S \] Alternatively, think as: - The total perimeter of the combined figure is: \[ \text{Perimeter} = 2 \times (\text{total length}) + 2 \times (\text{total width}) - 2 \times (\text{taped sides}) \] - But it's easier to think directly: Taping along the shorter sides merges the two \( S \) sides into one, so the perimeter becomes: \[ \text{Perimeter} = 2L + 2S + 2L + 2S - 2S = 4L + 2S \] --- #### Case 2: Taping along the longer sides (Julia's idea) - Taping along the longer sides (length \( L \)): - The two longer sides (\( L \)) are joined, removing 2 \( L \) edges from the perimeter. - The combined shape becomes a rectangle of dimensions \( S + S = 2S \) (width) and \( L \) (length). **Resulting perimeter:** \[ P_{\text{Julia}} = 2 \times (2S) + 2 \times L = 4S + 2L \] Or similarly, after merging: \[ \text{Perimeter} = 2S + 2L + 2S + 2L - 2L = 4S + 2L \] --- ### Step 5: Compare the two perimeters - **Gan's perimeter:** \[ P_{\text{Gan}} = 4L + 2S \] - **Julia's perimeter:** \[ P_{\text{Julia}} = 4S + 2L \] --- ### Step 6: Determine which is greater Compare: \[ P_{\text{Gan}} - P_{\text{Julia}} = (4L + 2S) - (4S + 2L) = 4L - 2L + 2S - 4S = (2L) - (2S) = 2(L - S) \] - If \( L > S \), then \( P_{\text{Gan}} - P_{\text{Julia}} > 0 \), meaning **Gan's perimeter is greater**. - If \( S > L \), which is impossible since \( L \) and \( S \) are defined as longer and shorter sides respectively, so typically \( L > S \). ### **Conclusion:** - Since the longer side \( L \) is greater than \( S \), the difference \( 2(L - S) \) is positive. - **Thus, Gan's method of taping along the shorter sides results in a larger perimeter.** --- ### **Final Answer:** **Gan is correct. Taping along the shorter sides produces a greater total perimeter than taping along the longer sides, assuming the longer side \( L \) is greater than the shorter side \( S \).** --- ### **Summary:** - Taping along shorter sides yields perimeter \( 4L + 2S \). - Taping along longer sides yields perimeter \( 4S + 2L \). - Since \( L > S \), \( 4L + 2S > 4S + 2L \), so Gan's method results in a larger perimeter.