Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-12,\) multiply and express each product in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{2}{3} \times \frac{3}{4} $$

Step-by-Step Solution

Answer

The product is \( \frac{1}{2} \). No values make the fractions undefined.

Step 1: Multiply the Numerators

To find the product of the two fractions, first multiply the numerators. \[ 2 \times 3 = 6 \]

Step 2: Multiply the Denominators

Multiply the denominators of the two fractions.\[ 3 \times 4 = 12 \]

Step 3: Write the Product

Combine the results from Steps 1 and 2 to write the product of the fractions:\[ \frac{6}{12} \]

Step 4: Simplify the Fraction

Simplify \( \frac{6}{12} \) by finding the greatest common divisor (GCD) of 6 and 12, which is 6. Divide both the numerator and the denominator by their GCD.\[ \frac{6 \div 6}{12 \div 6} = \frac{1}{2} \]

Step 5: Identify Undefined Values

Since there are no variables in the provided fractions, there are no additional values for which the fractions are undefined beyond avoiding zero in the denominator. However, in this case, all computations were valid.

Key Concepts

Multiplying Fractions

To multiply fractions, we first multiply the numerators across the fractions and then do the same with the denominators. This method ensures that the fractions are combined properly. For example, if we have \( \frac{2}{3} \) and \( \frac{3}{4} \), the steps are straightforward:

• Multiply the Numerators: First, take the top numbers of each fraction and multiply them. In our example, \( 2 \times 3 = 6 \).
• Multiply the Denominators: Next, multiply the bottom numbers, which gives \( 3 \times 4 = 12 \).
• Combine the Products: The product of the fractions is then \( \frac{6}{12} \), as you stack the results of your previous two steps together.

This process of multiplying fractions is very straightforward. Just remember to multiply across the numerators and the denominators respectively. It is important to follow these steps before attempting to simplify the fraction or find any other specifics like undefined values.

Simplification of Fractions

After multiplying fractions, the next vital step is to simplify the resulting fraction. Simplification makes fractions easier to understand and work with. The goal is to reduce the fraction to its simplest form, where the numerator and denominator are as small as possible. In out example of \( \frac{6}{12} \), simplification is needed.

• Identify the Greatest Common Divisor (GCD): To simplify \( \frac{6}{12} \), we must find the GCD of 6 and 12. We will delve deeper into finding the GCD in the next section.
• Divide the Numerator and Denominator by the GCD: Once we know the GCD, which is 6 in this example, we divide both the numerator and the denominator by it. This gives us \( \frac{6 \div 6}{12 \div 6} = \frac{1}{2} \).

A fraction is deemed simplified when both numbers have no common factors other than 1. This makes working with fractions in operations and equations much easier.

Finding Greatest Common Divisor

The Greatest Common Divisor (GCD) is central to simplifying fractions. It is the largest integer that divides both the numerator and the denominator without leaving a remainder. To find the GCD of two numbers, you can use one of several methods, like listing factors or using the Euclidean algorithm.

• List the Factors: One way is to list all factors of the numbers and identify the largest one they share. For example, for numbers 6 and 12, the factors are:\[ 6: 1, 2, 3, 6 \] \[ 12: 1, 2, 3, 4, 6, 12 \]Here, the largest common factor is 6.
• Use the Euclidean Algorithm: Other methods, like the Euclidean algorithm, involve dividing, yet listing factors is often simplest for smaller numbers.

Finding the GCD allows you to simplify fractions quickly and is an essential tool in working with fractions. It ensures that fractions are reduced to their simplest form, which is easier for mathematical operations and comprehension.