Problem 84 - Elementary Algebra

Question

Problem: Complete the "Understand the problem," "Make a plan," and "Complete the plan" steps. A 20 -acre field is going to be divided into building lots. The area of each lot is \(\frac{2}{5}\) acre. Find the number of lots in the development. Incorrect Answer: The unknown is the number of lots: \(L\) = number of lots. To find the number of lots, multiply the area by the number of lots. $$ \begin{aligned} &L=\left(\frac{2}{5}\right)(20) \\ &L=8 \text { lots } \end{aligned} $$

Step-by-Step Solution

Answer

There are 50 lots in the development.

Step 1: Understand the Problem

Identify the given information and what needs to be found. The total area of the field is 20 acres, and each lot has an area of \(\frac{2}{5}\) acre. The objective is to find the number of lots that can be made from the 20-acre field.

Step 2: Make a Plan

Determine the mathematical operation needed to find the number of lots. Since dividing the field into lots gives their total area, divide the total area of the field by the area of each lot.

Step 3: Complete the Plan

Divide the total area of the field (20 acres) by the area of each lot (\(\frac{2}{5}\) acre) to determine the number of lots:\[ L = \frac{20}{\frac{2}{5}} \]Simplify by multiplying by the reciprocal of \(\frac{2}{5}\):\[ L = 20 \times \frac{5}{2} \]Calculate the result:\[ L = 50 \]%Thus, the number of lots is 50.

Key Concepts

Area Calculation

Area calculation is an essential skill in various real-life scenarios, including construction, agriculture, and landscaping. In this exercise, we need to calculate the area of smaller sections (lots) from a larger area (field). We know the total area of the field is 20 acres, and each smaller section, or lot, is \(\frac{2}{5}\) acre. To solve such problems, it's vital to understand how to manipulate fractions and use basic division. Here, area calculation involves determining how many fractional parts fit into a whole number. This approach can be applied to different shapes and sizes, making it broadly useful in practical situations.

Field Division

Field division refers to breaking down a large area into smaller, more manageable sections. This concept is frequently applied in agricultural planning, urban development, and land management. In our problem, we divided a 20-acre field into smaller lots of \(\frac{2}{5}\) acre each.
To perform the division, we need to divide the total area of the field by the area of each individual lot. This will tell us how many lots can be made from the field. Field division can be performed using various units, and understanding the relationship between different measures (like acres and square feet) is crucial. The goal is to allocate the land efficiently, ensuring equal use of the available area.

Mathematical Operations

Mathematical operations are fundamental in solving real-world problems. In this example, we used division and multiplication to find the number of lots in a 20-acre field. First, we set up the division:
\(\frac{20}{\frac{2}{5}}\)
This means we are dividing 20 by \(\frac{2}{5}\). To simplify, we multiply 20 by the reciprocal of \(\frac{2}{5}\), which is \(\frac{5}{2}\). This changes our equation to:
\(\frac{20}{1} \times \frac{5}{2} = \frac{100}{2} = 50\)
Understanding these basic operations ensures that we can approach and resolve more complex problems efficiently. Division, especially with fractions, becomes simpler with practice, and it's a critical skill for students and professionals alike.