Problem 61 - Prealgebra

Question

Lot Size. A land developer wants to subdivide 5 acres of property into lots suitable for building a home. If each lot is to be \(\frac{1}{4}\) of an acre in size how many lots can be made?

Step-by-Step Solution

Answer

20 lots can be made.

Step 1: Understand the Total Area

The total area of the property is given as 5 acres. This is the total amount of land available to be subdivided into smaller lots.

Step 2: Determine the Size of Each Lot

Each new lot will be of size \( \frac{1}{4} \) of an acre. This means each lot uses one-fourth of an acre of land.

Step 3: Calculate the Number of Lots

To find how many \( \frac{1}{4} \, \text{acre} \) lots can fit into 5 acres, divide the total area by the size of each lot: \( \frac{5}{\frac{1}{4}} \).

Step 4: Simplify the Division

Dividing by a fraction \( \frac{1}{4} \) is equivalent to multiplying by its reciprocal. Thus, \( \frac{5}{\frac{1}{4}} = 5 \times 4 = 20 \).

Step 5: Determine the Final Count of Lots

The calculation shows that 20 lots of \( \frac{1}{4} \, \text{acre} \) each can be made from 5 acres of land.

Key Concepts

Fraction Division

Dividing with fractions can seem tricky at first, but it becomes straightforward with a bit of practice. When you divide a number by a fraction, you are determining how many of the fractional parts fit into the number. In this case, we are dividing 5 acres by \(\frac{1}{4}\) acre lots.

To perform fraction division, you follow this simple rule: multiply the dividend by the reciprocal of the divisor. Here, the dividend is 5 (the total area in acres), and the divisor is \(\frac{1}{4}\). By applying the rule:

• First, find the reciprocal of \(\frac{1}{4}\), which is 4.
• Then, multiply 5 (the total number of acres) by 4 (the reciprocal), which equals 20.

Thus, you can fit 20 lots into 5 acres, when each lot is \(\frac{1}{4}\) of an acre.

Reciprocal of a Fraction

Understanding the reciprocal is crucial when dealing with fraction division. A reciprocal of a fraction is simply a flipped version of that fraction. If you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).

Here's how it works in practice:

• If you have \(\frac{1}{4}\), the reciprocal is 4 because you essentially switch the numerator and denominator.
• This flipping process allows you to multiply instead of divide, transforming a division problem into a multiplication problem.

Using the reciprocal simplifies many fraction division problems. It's one of the key techniques to master in prealgebra.

Land Subdivision Math

Land subdivision math involves breaking down a large piece of land into smaller plots. It's often necessary for developers or city planners, and understanding the math behind it is crucial for accurate planning.

In our example, we are tasked with dividing 5 acres into \(\frac{1}{4}\) acre lots. Using division, we can determine how many lots will fit:

• Calculate the total lots: Divide the total area (5 acres) by the size of each lot (\(\frac{1}{4}\) acre).
• Simplify the division: Recognize that dividing by a fraction is akin to multiplying by its reciprocal.
• Distribute the plots: This means you found out you can create 20 smaller lots from the large land.

Such calculations ensure that the land is subdivided efficiently and according to requirements.