Problem 93 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ \frac{4}{3}\left(\frac{6}{5}\right) $$

Step-by-Step Solution

Answer

The product is \( \frac{8}{5} \).

Step 1: Understand the Multiplication Task

In this problem, we need to multiply the two fractions: \( \frac{4}{3} \) and \( \frac{6}{5} \). The method involves multiplying the numerators together and the denominators together.

Step 2: Multiply the Numerators

Multiply the numerators of both fractions: \( 4 \) and \( 6 \).

Step 3: Multiply the Denominators

Multiply the denominators of both fractions: \( 3 \) and \( 5 \).

Step 4: Write the Resulting Fraction

Combining the results from Steps 2 and 3, we get a new fraction: \( \frac{24}{15} \).

Step 5: Simplify the Fraction

To simplify \( \frac{24}{15} \), find the greatest common divisor (GCD) of 24 and 15, which is 3. Divide both the numerator and the denominator by 3.

Step 6: Write the Simplified Fraction

After dividing both the numerator and the denominator by 3, the simplified fraction is \( \frac{8}{5} \).

Key Concepts

Understanding the Numerator

In a fraction, the numerator is the number on the top. It tells you how many parts you have. For the fraction \( \frac{4}{3} \), the numerator is 4. When you're multiplying fractions, you multiply the numerators together. Here, we multiply 4 from the first fraction by 6 from the second fraction. You get 24, which becomes the numerator of your new fraction. This operation combines the parts of the two fractions into a single value. It's essential to remember that the numerator shows the actual "count" of units in a fraction.
Keep in mind that multiplying numerators is a straightforward process. Just line up the top numbers and multiply them together. This step is simple, yet crucial for understanding how the "pieces" of each fraction fit together in the solution.

Understanding the Denominator

The denominator is the number that appears on the bottom of a fraction. It indicates into how many equal parts the whole is divided. In our example, the fraction \( \frac{4}{3} \) has 3 as its denominator. When multiplying fractions, you multiply the denominators as well. In this case, 3 and 5 are multiplied to get 15, which becomes the denominator of the new fraction \( \frac{24}{15} \).
The denominator is essential because it tells you about the size of the pieces you are dealing with. By multiplying denominators, you're effectively determining how the pieces from each individual fraction are compatible, thus forming a unified whole. This concept will help take you a long way in understanding how each part (numerator and denominator) works in harmony to express the value of the fraction.

Greatest Common Divisor (GCD)

The Greatest Common Divisor, often abbreviated as GCD, is the largest number that can evenly divide two or more numbers. It plays a key role in simplifying fractions. In the fraction \( \frac{24}{15} \), the GCD of 24 and 15 is 3. To find this, look for the largest number that divides both 24 and 15 without leaving a remainder.

• List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• List the factors of 15: 1, 3, 5, 15
• The largest common factor is 3.

Using the GCD helps you reduce fractions to their simplest form by dividing both the numerator and the denominator by this number. This process is helpful for making calculations easier and for obtaining a fraction that is easier to understand and compare with other fractions.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator share no common factors other than 1. This process makes fractions easier to compare, add, and subtract. To simplify the fraction \( \frac{24}{15} \), we use the GCD, which is 3, to divide both the numerator and the denominator.

• Divide the numerator: \( 24 \div 3 = 8 \)
• Divide the denominator: \( 15 \div 3 = 5 \)

Thus, \( \frac{24}{15} \) simplifies to \( \frac{8}{5} \). Simplification preserves the value of the fraction while changing its appearance to a form that is often more practical or easier to work with. Understanding this technique is crucial for solving problems that involve fractions in any mathematical operation.