Problem 14 - Prealgebra

Question

Find each of the following products. (Multiply.) $$\frac{2}{3} \cdot \frac{4}{5} \cdot \frac{1}{3}$$

Step-by-Step Solution

Answer

The product is \(\frac{8}{45}\).

Step 1: Understand the problem

We need to find the product of three fractions: \( \frac{2}{3} \), \( \frac{4}{5} \), and \( \frac{1}{3} \). This requires multiplying all these fractions together to get a single fraction in its simplest form.

Step 2: Multiply the numerators

To multiply fractions, we start by multiplying the numerators. Multiply \(2\), \(4\), and \(1\):\[2 \times 4 \times 1 = 8\]So the product of the numerators is 8.

Step 3: Multiply the denominators

Next, multiply the denominators: \(3\), \(5\), and \(3\):\[3 \times 5 \times 3 = 45\]So the product of the denominators is 45.

Step 4: Form the new fraction

Now rewrite the product of the fractions as a single fraction using the product of the numerators and denominators:\[\frac{8}{45}\]The multiplied result is \(\frac{8}{45}\).

Step 5: Simplify the fraction

Check if \(\frac{8}{45}\) can be simplified. Since 8 and 45 do not have any common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Numerators and Denominators

In any fraction, the **numerator** is the top part, and it indicates the number of parts you have. The **denominator** is the bottom part, and it shows the total number of equal parts the whole is divided into. When you're working with fractions, especially involving multiplication, it's important to understand these terms.

For example, in the fraction \( \frac{2}{3} \), "2" is the numerator and "3" is the denominator. When multiplying fractions, you'll be multiplying all the numerators together and all the denominators together.

Understanding these two components is crucial because they help you perform operations like multiplication and make sense of the result. By doing so, you achieve a product which is again a fraction, expressed with a new numerator and a new denominator.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means expressing the fraction such that the numerator and denominator are as small as possible, without changing the fraction's value. For example, the fraction \( \frac{8}{24} \) can be simplified by dividing both the numerator and the denominator by their greatest common factor (GCF), which in this case is 8. This results in \( \frac{1}{3} \).

It's important to check if the fraction can be simplified after performing any arithmetic operation like multiplication. In our exercise, the product of the fractions \( \frac{2}{3}, \frac{4}{5}, \text{ and } \frac{1}{3} \) results in \( \frac{8}{45} \).

Since 8 and 45 have no common factors greater than 1, you can't simplify it further, thus \( \frac{8}{45} \) is already in its simplest form. Regular practice in simplifying fractions enhances accuracy and efficiency in solving more complex fraction problems.

Fraction Multiplication Process

The process of multiplying fractions involves a straightforward method that makes it easier than it might initially seem. Here are the steps you'll follow:

• Multiply the numerators of all the fractions together. This gives you the numerator of the result.
• Multiply the denominators of all the fractions together. This gives you the denominator of the result.
• Form a new fraction from these two products.
• Simplify the fraction, if possible, to achieve the simplest form.

Let's see the process in action with the example \( \frac{2}{3} \), \( \frac{4}{5} \), and \( \frac{1}{3} \):
- Multiply the numerators: \( 2 \times 4 \times 1 = 8 \)
- Multiply the denominators: \( 3 \times 5 \times 3 = 45 \)
- Combine these results into a new fraction: \( \frac{8}{45} \)
This method is simple and logical, allowing you to multiply fractions effectively and is applicable to any number of fractions, whether they are two or more.