- Define the total cost equation as \(180 + 45a = 495\).
- Subtract the fixed charge from the total cost to isolate the variable term: \(45a = 315\).
- Divide the result by the cost per acre to find the number of acres: \(a = 315 / 45\).
- The number of acres prepared is \(\boxed{7}\).
Explanation
- Problem Analysis
The problem involves determining the number of acres, denoted by \(a\), prepared by a farmer given a specific cost structure. The total cost is composed of a fixed charge and a variable cost dependent on the number of acres prepared.
- Fixed charge: GH\(\phi\) 180
- Variable cost per acre: GH\(\phi\) 45
- Total bill: GH\(\phi\) 495
- Number of acres: \(a\)
The relationship between these values can be expressed as: \(\text{Fixed charge} + (\text{Variable cost per acre} \times a) = \text{Total bill}\).
- Solving the Linear Equation
Based on the identified costs, we can write the following linear equation:
\[180 + 45a = 495\]
To isolate the variable \(a\), we first subtract the fixed charge of GH\(\phi\) 180 from both sides of the equation:
\[45a = 495 - 180\]
\[45a = 315\]
Next, to solve for \(a\), we divide both sides by the variable cost per acre, which is 45:
\[a = \frac{315}{45}\]
Performing the division, we find:
\[a = 7\]
Thus, the number of acres prepared is 7.
- Conclusion
The linear equation representing the total cost is \(180 + 45a = 495\). Solving this equation demonstrates that the farmer prepared 7 acres. The final answer is \(\boxed{7}\).
Examples
Understanding how to set up and solve linear equations is essential for budgeting and managing resources in everyday life. For instance, if you are planning to hire a service that charges a flat fee for the visit plus an hourly rate, you can use the same algebraic method to determine how many hours of work you can afford within a specific budget. By defining your variables and forming an equation, you gain control over financial planning and decision-making.