A rectangular field is to be fenced on three sides, leaving one side of length 20 feet unfenced. If the area of the field is 680 sq. ft, how many feet of fencing are required for the remaining three sides?

Introduction / Context:
This problem combines the concepts of area and perimeter of a rectangle and introduces a practical scenario of fencing only part of the boundary. Such questions test the ability to interpret given information correctly, especially when not all sides are treated in the same way. Careful reading is essential to ensure that the correct sides are used when calculating the length of fencing required.

Given Data / Assumptions:

• The field is rectangular in shape.
• One side that is 20 feet long is left unfenced.
• The total area of the field is 680 square feet.
• Fencing is required only on the remaining three sides of the rectangle.
• The field has right angles at all corners, as is the case with rectangles.

Concept / Approach:
Area of a rectangle is length * breadth. If one side is given as 20 feet and the area is known, the other side can be found by dividing the area by 20. Once both dimensions are known, the total length of the three sides that need fencing is simply the sum of the two vertical sides plus the horizontal side opposite the unfenced side. Using these basic formulas gives the total required fencing length.

Step-by-Step Solution:
Let the side left unfenced be the length L = 20 ft.Area = L * B = 680 sq. ft, so breadth B = 680 / 20 = 34 ft.The three sides to be fenced are the two breadths and the opposite length: B + L + B.Total fencing length = 34 + 20 + 34 = 88 ft.Hence, 88 feet of fencing are required.

Verification / Alternative check:
If we imagine the rectangle as 20 ft by 34 ft, the full perimeter would be 2 * (20 + 34) = 108 ft. Since one 20 ft side is left open, the fencing must cover 108 − 20 = 88 ft. This alternative perimeter-based reasoning leads to the same result as the earlier direct calculation, confirming that 88 is the correct total length of fencing.

Why Other Options Are Wrong:
Option 44 ft is half of 88 ft and comes from mistakenly taking only one breadth and one length. Option 72 ft results from incorrect combination of sides or miscomputed breadth. Option 66 ft may arise from using 680 / 10 = 68 or mixing up dimensions. Option 22 ft is far too small and does not correspond to any logical combination of the sides of a 680 sq. ft rectangle with one side of 20 ft.

Common Pitfalls:
Students often forget that two breadths must be added when three sides are fenced. Another typical mistake is to misidentify which side is 20 ft, or to assume that the 20 ft side is the breadth rather than the length, though it does not matter if the reasoning is consistent. Confusing area with perimeter by adding or multiplying the wrong quantities also leads to incorrect answers.

Final Answer:
The total length of fencing required is 88 feet.