Problem 51 - Algebra for College Students

Question

Perform the indicated operations. When possible write down only the answer. $$\text { One-half of } \frac{4 x}{3}$$

Step-by-Step Solution

Answer

\frac{2x}{3}

Step 1: Understand the Problem

To solve this problem, you need to find one-half of the given fraction \(\frac{4x}{3}\). This means you will multiply \(\frac{4x}{3} \) by \(\frac{1}{2}\).

Step 2: Set Up the Multiplication

Write the multiplication expression: \(\frac{1}{2} \times \frac{4x}{3}\).

Step 3: Multiply the Fractions

Multiply the numerators together and the denominators together. This will give you: \(\frac{1 \times 4x}{2 \times 3} = \frac{4x}{6}\).

Step 4: Simplify the Fraction

Simplify \(\frac{4x}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives you: \(\frac{4x \div 2}{6 \div 2} = \frac{2x}{3}\).

Key Concepts

simplifying fractions

Learning to simplify fractions is a key skill in math. Simplifying a fraction means making the numerator (top number) and the denominator (bottom number) as small as possible. But they still have to show the same value. Simplify by finding a number that both parts of the fraction can be divided by, called the greatest common divisor (GCD).

For example, in the final step of our exercise, we simplified \(\frac{4x}{6}\) to \(\frac{2x}{3}\).
We divided both 4 and 6 by their GCD, which is 2. This gave us the fraction in its simplest form. Simplifying helps make the fraction easier to understand and use.

multiplying fractions

When you multiply fractions, it’s quite different from adding or subtracting them. You don’t need a common denominator. Instead, you multiply the numerators together and the denominators together.

In our exercise, we multiplied \(\frac{1}{2}\)\text{x} \( \frac{4x}{3} \). This gave us a new fraction: \(\frac{1 \text{x} 4x}{2 \text{x} 3} \). We then got \(\frac{4x}{6} \). Remember to multiply straight across the top and bottom. This step creates a new fraction which might need simplification.
Multiplying fractions makes many math problems easier and more manageable.

greatest common divisor

Finding the greatest common divisor (GCD) is often crucial in math, especially for simplifying fractions. The GCD of two numbers is the largest number that can divide both without leaving a remainder.

To simplify \(\frac{4x}{6} \) in our exercise, we identified 2 as the GCD of 4 and 6. Dividing both by 2, we got \(\frac{2x}{3} \).

Here’s how to find the GCD:

• List the factors of both numbers.
• Identify the greatest number that appears in both lists.
• Use it to simplify your fraction by dividing both the numerator and the denominator.

Using the GCD helps keep your fractions neat and easier to work with.