Problem 35 - Algebra A Combined Function

Question

Multiply or divide as indicated. See Examples 11 through 14 and \(21 .\) $$ \frac{2}{3} \cdot \frac{3}{4} $$

Step-by-Step Solution

Answer

\( \frac{1}{2} \)

Step 1: Identify the operation

We need to multiply two fractions: \( \frac{2}{3} \) by \( \frac{3}{4} \).

Step 2: Multiply the numerators

Multiply the numerators together: \( 2 \times 3 = 6 \).

Step 3: Multiply the denominators

Multiply the denominators together: \( 3 \times 4 = 12 \).

Step 4: Construct the new fraction

Combine the results from Steps 2 and 3. The resulting fraction is \( \frac{6}{12} \).

Step 5: Simplify the fraction

Simplify \( \frac{6}{12} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 6. So, \( \frac{6}{12} = \frac{1}{2} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential part of working with fractions. It involves reducing a fraction to its simplest form, where the numerator and the denominator have no common divisors other than 1. By simplifying fractions, we make them easier to understand and compare.

To simplify a fraction, you can follow these steps:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• Write the resulting fraction, which is now in its simplest form.

For example, if you start with the fraction \( \frac{6}{12} \), you can simplify it by determining that the GCD of 6 and 12 is 6. Then, divide both numbers by 6 to get \( \frac{1}{2} \). This means \( \frac{1}{2} \) is the simplest form of \( \frac{6}{12} \).

Simplifying fractions helps ensure that answers are presented in their most concise form, making them tidy and straightforward.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can evenly divide two or more numbers. It plays a crucial role in simplifying fractions, as it helps reduce the fraction to its simplest form.

To find the GCD of two numbers, you can use either of these methods:

• **Listing Factors**: Write down all the factors of each number and identify the largest factor that appears in both lists.
• **Euclidean Algorithm**: Use division to reduce the numbers step-by-step until you find the GCD. In essence, keep finding remainders until a remainder of 0 is reached—then, the divisor at this step is the GCD.

For instance, to find the GCD of 6 and 12, we note the factors are:
- **6**: 1, 2, 3, 6
- **12**: 1, 2, 3, 4, 6, 12

The common factors are 1, 2, 3, and 6, with 6 being the greatest, so the GCD is 6. Knowing the GCD allows you to simplify fractions like \( \frac{6}{12} = \frac{1}{2} \) after dividing both terms by 6.

Numerators and Denominators

In fractions, the terms "numerator" and "denominator" refer to the two numbers that make up the fraction. Understanding these parts is vital for fraction operations like multiplication or simplification.

The **numerator** is the top number in a fraction, expressing how many parts of a whole are being considered. For example, in the fraction \( \frac{2}{3} \), 2 is the numerator.

The **denominator** is the bottom number, indicating the total number of equal parts into which the whole is divided. In \( \frac{2}{3} \), 3 is the denominator.

When multiplying fractions, both numerators are multiplied together, and both denominators are multiplied together. For instance:

• In \( \frac{2}{3} \times \frac{3}{4} \), multiply the numerators: 2 and 3 to get 6.
• Multiply the denominators: 3 and 4 to get 12.

After multiplication, the resulting fraction \( \frac{6}{12} \) can be simplified using the GCD of 6 and 12, achieving the simplest form \( \frac{1}{2} \). Understanding numerators and denominators helps navigate fractions more easily during operations and simplification.