Problem 118 - Prealgebra

Question

Multiply or divide as indicated. $$4 \cdot \frac{1}{2}$$

Step-by-Step Solution

Answer

The result is 2.

Step 1: Identify Operation

The problem asks us to multiply two numbers: 4 and \( \frac{1}{2} \). The multiplication of a whole number by a fraction can be approached by multiplying the whole number by the numerator of the fraction and maintaining the denominator.

Step 2: Multiply the Numerator

Multiply the whole number 4 by the numerator of the fraction, which is 1. \[ 4 imes 1 = 4 \]

Step 3: Combine with the Denominator

Place the result from Step 2 over the denominator of the fraction. Since the denominator of \( \frac{1}{2} \) is 2, we write: \[ \frac{4}{2} \]

Step 4: Simplify the Fraction

To simplify \( \frac{4}{2} \), divide the numerator by the denominator. \[ \frac{4}{2} = 2 \] Thus, 4 multiplied by \( \frac{1}{2} \) equals 2.

Key Concepts

Whole Numbers

Whole numbers are the basic building blocks of mathematics that we use daily. These are non-negative numbers without any fractional or decimal component. Examples of whole numbers include 0, 1, 2, 3, and so forth. Understanding whole numbers is fundamental when dealing with operations involving fractions.

When multiplying a whole number like 4 by a fraction, the process involves the whole number interacting with the numerator of the fraction. In our example, when you see the fraction multiplication problem \(4 \cdot \frac{1}{2}\), the 4 is a whole number.

• Whole numbers do not have a denominator, but when we multiply them with fractions, we can imagine them having a denominator of 1 for easier calculation.
• Whole numbers maintain their essence in multiplication problems when multiplied by fractions because you multiply by the top of the fraction first, simplifying your operations.

Being comfortable with whole numbers helps make fraction operations straightforward and manageable.

Simplifying Fractions

Simplifying fractions is an essential skill to make your answers clearer and more concise. When you simplify a fraction, you are reducing it to its simplest form without changing its value. This is done by dividing both the numerator and denominator by their greatest common divisor (GCD).

In our example, after multiplying, we got \(\frac{4}{2}\). Here’s how to simplify it:

• Identify the GCD of 4 and 2, which is 2 in this case.
• Divide both the numerator, 4, and the denominator, 2, by 2 to get 2 and 1, respectively.
• So, \(\frac{4}{2}\) simplifies to 2.

Simplifying fractions makes it easier to understand and compare amounts since you work with whole numbers whenever possible. You always aim to represent fractions in their simplest form for clarity and ease of further calculations.

Numerator and Denominator

Understanding numerators and denominators is key to mastering fraction operations. A fraction consists of two parts: the numerator and the denominator. The numerator is the top number of a fraction, indicating how many parts are considered. The denominator is the bottom number, showing the total number of equal parts the whole is divided into.

In the fraction \(\frac{1}{2}\), 1 is the numerator, representing one part of a whole that is divided into two equal parts, as indicated by the denominator 2. Now, when we multiply a fraction by a whole number, we focus on the numerator. For the operation \(4 \cdot \frac{1}{2}\):

• We multiply the whole number (4) by the numerator (1).
• This results in a new fraction, \(\frac{4}{2}\), where the denominator remains the same.

Understanding these roles will clarify how fractions operate and ensure you multiply and simplify effectively.