A large field has a total area of 700 hectares and is divided into two parts. The difference between the areas of these two parts is equal to one-fifth of the average of their areas. What is the area of the smaller part, in hectares?

Introduction:
This is an algebra-based word problem that involves total, average, and difference of two quantities. The total area of the field is given, and we know a special relation between the difference of the two parts and their average. Such questions check a student's ability to translate words into equations, use averages correctly, and solve simultaneous equations to find the required quantity, in this case the smaller area.

Given Data / Assumptions:

• Total area of the field = 700 hectares. • The field is divided into two parts, with areas A and B hectares. • Without loss of generality, assume A is the larger part and B is the smaller part. • A + B = 700. • The difference of the two areas, A - B, is equal to one-fifth of the average of A and B.

Concept / Approach:
The average of the two areas is (A + B) / 2. We are given that A - B = (1/5) * ((A + B) / 2). Since A + B is known to be 700 hectares, the average becomes 350 hectares. The difference A - B is thus (1/5) * 350. After computing this difference, we can solve the system of equations: A + B = 700 and A - B = difference, using the standard method of adding and subtracting equations.

Step-by-Step Solution:
Step 1: Use the total area. A + B = 700. Step 2: Compute the average of the two areas. Average = (A + B) / 2 = 700 / 2 = 350 hectares. Step 3: Use the given relation for the difference. A - B = (1/5) * 350 = 70 hectares. Step 4: Solve the system of equations: A + B = 700. A - B = 70. Step 5: Add the two equations. (A + B) + (A - B) = 700 + 70 => 2A = 770. Step 6: Compute A. A = 770 / 2 = 385 hectares. Step 7: Compute B. B = 700 - 385 = 315 hectares. Thus, the smaller part has an area of 315 hectares.

Verification / Alternative check:
Compute the average again: (385 + 315) / 2 = 700 / 2 = 350 hectares. The difference between the two areas is 385 - 315 = 70 hectares. One-fifth of the average is (1/5) * 350 = 70 hectares. This exactly matches the given condition that the difference equals one-fifth of the average, so our values for A and B are consistent and correct.

Why Other Options Are Wrong:
Option 385 hectares: This is the larger part, not the smaller one. Option 415 hectares: Then the other part would be 700 - 415 = 285 hectares, giving an average of 350 and a difference of 130, which is not one-fifth of 350. Option 485 hectares: This would make the other part 215 hectares, giving a difference far larger than the required one-fifth of the average. Option 350 hectares: That would imply both parts are equal, giving difference 0, which contradicts the given condition.

Common Pitfalls:
Some students confuse the average with half of one of the areas instead of half of the sum. Others mistakenly set the difference equal to one-fifth of the total rather than one-fifth of the average. There can also be careless errors when solving the two equations, such as subtracting incorrectly or mixing up which area is larger. Writing down the equations carefully and solving stepwise avoids these mistakes.

Final Answer:
The area of the smaller part of the field is 315 hectares.