Problem 42 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$\frac{2}{3} \div \frac{2}{3}$$

Step-by-Step Solution

Answer

The value is 1.

Step 1: Understand the Division of Fractions

When dividing fractions, we use the rule: to divide by a fraction, multiply by its reciprocal. This means that \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \). In this problem, we're dividing \( \frac{2}{3} \) by \( \frac{2}{3} \).

Step 2: Determine the Reciprocal

The reciprocal of a fraction is obtained by swapping its numerator and denominator. For \( \frac{2}{3} \), the reciprocal is \( \frac{3}{2} \).

Step 3: Set Up the Multiplication

Using the division rule, transform the original division problem \( \frac{2}{3} \div \frac{2}{3} \) into a multiplication problem: \(\frac{2}{3} \times \frac{3}{2} \).

Step 4: Multiply the Fractions

To multiply fractions, multiply the numerators together and the denominators together: \[\frac{2 \times 3}{3 \times 2} = \frac{6}{6}.\]

Step 5: Simplify the Result

The fraction \( \frac{6}{6} \) simplifies to 1 since any number divided by itself is 1.

Key Concepts

Reciprocal of a Fraction

To divide one fraction by another, you need to understand the concept of a reciprocal. A reciprocal is simply a fraction flipped upside down. In other words, it means switching the numerator (top number) and the denominator (bottom number).
For example, the reciprocal of the fraction \( \frac{2}{3} \) is \( \frac{3}{2} \). Why? Because we've swapped the 2 (numerator) with the 3 (denominator).

• This step is crucial when dividing fractions.
• Finding the reciprocal turns the division problem into a multiplication one, which is often simpler to solve.

Remember, when you multiply any number by its reciprocal, the result is always 1. This is why the reciprocal is such a handy tool when dealing with fractions.

Multiplication of Fractions

Once you find the reciprocal of the second fraction in a division problem, you're ready to convert the problem into a multiplication one. Let's break it down.
When multiplying fractions, the rule is quite straightforward: multiply the numerators together and then the denominators together.

• For example, in our exercise, we take the first fraction, \( \frac{2}{3} \), and multiply it by the reciprocal of the second, \( \frac{3}{2} \).
• Multiply the numerators: \( 2 \times 3 = 6 \).
• Then multiply the denominators: \( 3 \times 2 = 6 \).

Putting this into a fraction gives us \( \frac{6}{6} \).
This method of multiplication is essential because it not only solves the division of fractions but also introduces the concept of cross-canceling when applicable.

Simplifying Fractions

Simplifying fractions is an important final step in solving fraction problems. It involves reducing the fraction to its most basic form. This makes it easier to understand and interpret.
In our multiplication from earlier, we ended up with \( \frac{6}{6} \).

• Whenever the numerator and denominator are the same, like \( \frac{6}{6} \), it simplifies to 1.
• This is because any number divided by itself is equal to 1.
• Simplifying makes clear what the ultimate result of an operation is without any excess complexity.

This step is not just a mathematical nicety; it's crucial for accurately conveying and understanding the results of fraction operations in their simplest forms.