Problem 87 - Prealgebra

Question

Subdivision A 6 -acre piece of land is subdivided into \(\frac{3}{5}\) acre lots. How many lots are there?

Step-by-Step Solution

Answer

There are 10 lots.

Step 1: Understanding the Problem

We have a total of 6 acres of land and want to divide it into smaller lots, each with an area of \(\frac{3}{5}\) of an acre. Our goal is to find how many of these smaller lots we can create from the original 6-acre plot.

Step 2: Setting up the Division Problem

To find the number of \(\frac{3}{5}\)-acre lots that can be subdivided from 6 acres, we can set up the division problem as follows: \( 6 \div \frac{3}{5} \). This problem involves dividing a whole number by a fraction.

Step 3: Dividing by a Fraction

When dividing by a fraction, it is equivalent to multiplying by its reciprocal. Therefore, the division problem \( 6 \div \frac{3}{5} \) can be rewritten as a multiplication problem: \( 6 \times \frac{5}{3} \).

Step 4: Performing the Multiplication

Now, multiply the whole number by the reciprocal of the fraction: \( 6 \times \frac{5}{3} = \frac{6 \times 5}{3} = \frac{30}{3} \).

Step 5: Simplifying the Fraction

Simplify the fraction \( \frac{30}{3} \) by dividing the numerator by the denominator, which gives \( 10 \). This means there are 10 lots.

Key Concepts

Fraction Multiplication

Multiplying fractions might seem complex at first, but it's quite straightforward once you get the hang of it. When you multiply fractions, you simply multiply the numerators (top numbers) together and the denominators (bottom numbers) together. Here's how it works:

• If you have the problem \( \frac{a}{b} \times \frac{c}{d} \), then you multiply \( a \times c \) to get the new numerator.
• Next, multiply \( b \times d \) to get the new denominator.
• The result is a new fraction: \( \frac{a \times c}{b \times d} \).

This simple rule makes fraction multiplication quite intuitive. In our exercise, we turned the division problem into a multiplication problem by using the reciprocal, then we followed these steps to find the answer.

Reciprocal

A reciprocal is like a mirror image of a fraction. Simply put, to find the reciprocal of a fraction, you switch its numerator and denominator. For example, the reciprocal of \( \frac{3}{5} \) is \( \frac{5}{3} \).

• The reciprocal of any nonzero number or fraction \( x \) is \( \frac{1}{x} \).
• Finding a reciprocal can help transform division into multiplication, making problems easier to solve.

In our exercise, dividing by \( \frac{3}{5} \) is equal to multiplying by its reciprocal, \( \frac{5}{3} \). This clever trick turns division into a task of fraction multiplication.

Simplifying Fractions

After performing operations with fractions, simplifying them can make your answers much clearer and easier to understand. Simplifying means reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1.

• Start by dividing both the numerator and the denominator by their greatest common factor (GCF).
• In our example, we simplified \( \frac{30}{3} \) by dividing both 30 and 3 by their GCF, which is 3.
• Once simplified, \( \frac{30}{3} \) becomes 10, the simplest form in this case.

Simplifying fractions helps ensure your answers are as neat and concise as possible, making them easier to work with in further calculations.

Word Problems in Mathematics

Word problems are scenarios that present math concepts in real-world contexts. They often seem challenging at first, but with practice, you can break them down into manageable parts. Here’s how to approach them:

• Identify the question: What is being asked? In our problem, we needed to find out how many \( \frac{3}{5} \)-acre lots we could make from 6 acres.
• Translate words into math: Convert the problem into a mathematical equation. Here, the division \( 6 \div \frac{3}{5} \) was the key operation.
• Solve step-by-step: Use operations such as reciprocals, multiplication, and simplification to find the answer.

Practicing word problems enhances problem-solving skills and deepens your understanding of mathematical concepts.