Problem 4 - Prealgebra

Question

Find each quotient. Use an area model if necessary. $$\frac{1}{2} \div \frac{6}{7}$$

Step-by-Step Solution

Answer

The quotient is \( \frac{7}{12} \).

Step 1: Identify the Problem

We need to find the quotient of the two fractions \( \frac{1}{2} \div \frac{6}{7} \). In division, the quotient is the result we get after dividing one number by another.

Step 2: Reciprocate the Divisor

To divide by a fraction, multiply by its reciprocal. The reciprocal of \( \frac{6}{7} \) is \( \frac{7}{6} \). Thus, the operation becomes \( \frac{1}{2} \times \frac{7}{6} \).

Step 3: Multiply the Fractions

To multiply fractions, multiply the numerators together and the denominators together. This gives \( \frac{1 \times 7}{2 \times 6} = \frac{7}{12} \).

Step 4: Simplify if Necessary

Check if \( \frac{7}{12} \) can be simplified further. Since 7 and 12 have no common factors other than 1, \( \frac{7}{12} \) is already in its simplest form.

Key Concepts

Using the Area Model for Division

The area model is a powerful visual tool that helps students understand division of fractions on a more intuitive level. To apply the area model to the division of fractions, like \( \frac{1}{2} \div \frac{6}{7} \), visualize each fraction as a part of a rectangle.

Let's say you have a rectangle representing the whole, divided into two equal parts. Half of this rectangle will depict \( \frac{1}{2} \). Now, within each of these two parts, imagine dividing it into seven more equal parts to represent \( \frac{6}{7} \).

The question essentially asks: How many \( \frac{6}{7} \)-sized parts fit into a \( \frac{1}{2} \)-sized part? By rearranging and counting the smaller sections visually, you can see how many times \( \frac{6}{7} \) fits into \( \frac{1}{2} \). This helps students understand not just the procedure of dividing fractions but also the logical reasoning behind it.

Understanding Reciprocals

The concept of reciprocals is crucial when dealing with division of fractions. The reciprocal of a number is another number which, when multiplied with the original number, results in 1.

For fractions, to find the reciprocal, you simply swap the numerator and the denominator. Therefore, the reciprocal of \( \frac{6}{7} \) is \( \frac{7}{6} \).

• Reciprocals are essential in fraction division because they allow us to transform division into multiplication.
• This is because dividing by a fraction is equivalent to multiplying by its reciprocal.

Understanding this transformation simplifies the process considerably and highlights the versatile nature of fractions.

Steps for Multiplying Fractions

Once you have converted the division problem into a multiplication problem by using the reciprocal, you need to multiply the fractions. In our example, \( \frac{1}{2} \times \frac{7}{6} \), you multiply the numerators together and the denominators together.

This results in:

• Numerator: \( 1 \times 7 = 7 \)
• Denominator: \( 2 \times 6 = 12 \)

Hence, the product is \( \frac{7}{12} \). Multiplying fractions may sound simple, but it’s the key step in dividing them effectively.

Simplifying Fractions

After obtaining the result of the division, it's important to check if the fraction can be simplified. Simplifying a fraction involves reducing it to its simplest form, where the numerator and denominator have no common divisors other than 1.

For \( \frac{7}{12} \), check the greatest common divisor (GCD) of 7 and 12. Since the only common divisor they share is 1, \( \frac{7}{12} \) is already simplified.

• Simplification ensures that the fraction is expressed in the most concise form.
• A fraction is considered simplified when both numerator and denominator are as small as possible while maintaining the same value.

This skill is essential for clearer mathematical communication and for performing further calculations efficiently.