A large field of 700 hectares is divided into two parts. The difference of the areas of the two parts is one-fifth of the average of the two areas. What is the area of the smaller part in hectares?

### Concept & Linear Equations Translate the word problem into a system of two linear equations. You need to express the total area, the average area, and the difference between the two parts. ### Step-by-Step Solution * Let the areas of the two parts be $x$ and $y$, where $x$ is the larger part. * The total area is 700 hectares: $$ x + y = 700 $$ * The average of the two areas is: $$ \text{Average} = \frac{x + y}{2} = \frac{700}{2} = 350 $$ * The problem states the difference ($x - y$) is one-fifth of the average: $$ x - y = \frac{1}{5} \times 350 $$ $$ x - y = 70 $$ * Now we have a simple system of equations: 1) $x + y = 700$ 2) $x - y = 70$ * We want to find the smaller area ($y$). Subtract equation (2) from equation (1): $$ (x + y) - (x - y) = 700 - 70 $$ $$ 2y = 630 $$ $$ y = 315 $$ ### Exam Strategy & Shortcut You know the total is 700, so $x + y = 700$. The difference $x - y = \frac{1}{5} \times (\frac{700}{2}) = 70$. To find the smaller number when given a sum and a difference, subtract the difference from the sum and divide by 2: $\frac{700 - 70}{2} = \frac{630}{2} = 315$. ### Common Pitfall A common error is taking one-fifth of the *total* area (700) instead of the *average* area (350), which would incorrectly make the difference 140, leading to a wrong smaller area of 280 (which is a trap option). ### Final Answer Therefore, the correct answer is **315**.