Correct Option
Let the initial length of the rectangular garden be L and the initial width be W. The initial area (A_initial) is given by L × W.
The length is increased by 40%, resulting in a new length (L_new) = L + 0.40L = 1.40L.
The width is decreased by 20%, resulting in a new width (W_new) = W - 0.20W = 0.80W.
The new area (A_new) is calculated as L_new × W_new = (1.40L) × (0.80W) = 1.12LW.
To determine the percentage change in area, the formula is:
Percentage Change = ((A_new - A_initial) / A_initial) × 100
Percentage Change = ((1.12LW - LW) / LW) × 100
Percentage Change = (0.12LW / LW) × 100
Percentage Change = 0.12 × 100 = 12%.
Therefore, the area of the new garden has increased by 12%.
Incorrect Options
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Option 1 (has increased by 20%): This outcome would imply a new area of 1.20LW. The product of the length and width multipliers (1.40 × 0.80 = 1.12) does not support a 20% increase. This error often arises from incorrectly adding or subtracting percentage changes directly (40% - 20% = 20%).
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Option 3 (has increased by 8%): This would correspond to a new area of 1.08LW. The calculated new area of 1.12LW indicates a 12% increase, not 8%.
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Option 4 (is exactly the same as the old area): This implies no change in area, meaning the new area would be LW. However, the calculation demonstrates a net increase of 12%.