The length of a rectangular plot is increased by 50 percent. To keep its area unchanged, by what percentage should the width of the plot be decreased?

Introduction / Context:
This question tests understanding of how changing one dimension of a rectangle affects the other dimension when the area is kept constant. When one side is increased by a certain percentage, the other side must change proportionally in the opposite direction so that the product, which represents area, remains unchanged. This type of problem appears often in percentage and mensuration topics.

Given Data / Assumptions:

• Original length of the rectangle is L and original width is W.
• Original area A = L * W.
• Length is increased by 50 percent, so new length is 1.5L.
• Area must remain the same after the change.
• We must find the required percentage decrease in width.

Concept / Approach:
Area of a rectangle is the product of its length and width. If the length is multiplied by a factor, then the width must be multiplied by the reciprocal factor for the product to stay equal to the original area. Here the length factor is 1.5. The width factor must be 1 / 1.5. After finding this factor, we convert it to a percentage of the original width and then determine the percentage reduction from 100 percent.

Step-by-Step Solution:
Step 1: Original area A = L * W.Step 2: After the change, new length L_new = 1.5L.Step 3: Let new width be W_new. Since area is unchanged, L_new * W_new = L * W.Step 4: Substitute L_new = 1.5L: 1.5L * W_new = L * W.Step 5: Divide both sides by 1.5L to get W_new = (L * W) / (1.5L) = W / 1.5.Step 6: Simplify W / 1.5. Since 1.5 = 3/2, W / 1.5 = W * (2/3) = (2/3)W.Step 7: The new width is two-thirds of the old width, which is 66.67 percent of the original. Therefore, the decrease in width is 100 percent minus 66.67 percent = 33.33 percent.

Verification / Alternative check:
Use simple numbers. Suppose the original rectangle has length 2 units and width 3 units, so area is 2 * 3 = 6 square units. Increase length by 50 percent: new length is 3 units. To keep area at 6, width must be 6 / 3 = 2 units. Since original width was 3, the width has changed from 3 to 2, which is a decrease of 1 out of 3. In percentage terms, that is (1 / 3) * 100 = 33.33 percent decrease. This matches our algebraic result.

Why Other Options Are Wrong:
A decrease of 29.87 percent or 22.22 percent or 19.5 percent would not produce the reciprocal factor needed for area preservation. For example, a 30 percent decrease would leave width at 70 percent of its original, leading to area 1.5 * 0.70 = 1.05 times the original, which is a 5 percent increase, not unchanged area. Only a reduction to two-thirds of the original width, that is a 33.33 percent decrease, keeps the area constant.

Common Pitfalls:
One common mistake is to subtract percentages directly, such as assuming that if one side goes up by 50 percent, the other must go down by 50 percent. This ignores the multiplicative nature of area. Another mistake is to invert the fraction incorrectly when calculating W / 1.5. Remember that dividing by 3/2 is the same as multiplying by 2/3, not by 3/2.

Final Answer:
The width must be decreased by 33.33% to keep the area unchanged, so the correct option is 33.33%.