Problem 102 - Prealgebra

Question

Perform the indicated operations. \(8 \cdot \frac{1}{4}\)

Step-by-Step Solution

Answer

The result of the operation is 2.

Step 1: Understanding the Problem

The problem requires performing a multiplication operation between the number 8 and the fraction \(\frac{1}{4}\). This means we need to calculate what 8 times one-fourth (a quarter) is.

Step 2: Multiplying the Numerator

Since we are multiplying a whole number by a fraction, we multiply the whole number by the numerator of the fraction. Here, we multiply 8 by 1, leading to 8 as the result.

Step 3: Understanding the Denominator

The denominator remains the same, which is 4 in this fraction. Since we only multiply the numerator by the whole number, the denominator is unaffected in this operation.

Step 4: Simplifying the Result

After performing the multiplication, we have \(\frac{8}{4}\). To simplify this fraction, we divide the numerator by the denominator, obtaining the simplified result: 2.

Key Concepts

Whole Number and Fraction Operations

When you multiply a whole number by a fraction, you're actually finding a portion of that whole number. In this type of operation, we focus on interacting just with the numerator of the fraction. Here, we are multiplying a whole number, which is 8, by the fraction \(\frac{1}{4}\). Here’s how it works:

• First, take the whole number 8 and multiply it by the numerator of the fraction, which is 1.
• This gives us 8.
• The denominator of the fraction, 4, remains unchanged.

Therefore, the multiplication results in the fraction \(\frac{8}{4}\). Keeping multiplication simple by only affecting the numerator is a key principle when working with whole numbers and fractions together.

Simplifying Fractions

Simplifying a fraction means reducing it to its smallest possible equivalent. This allows us to see the fraction in its simplest form, making it easier to understand and use in other calculations. In our example, we finished with the fraction \(\frac{8}{4}\). Here's how we simplify it:

• Look at the numerator, 8, and the denominator, 4, and figure out if there is a common factor. The highest number that divides both evenly is 4.
• Divide both the numerator and the denominator by 4.
• This becomes \(\frac{8 \div 4}{4 \div 4} = \frac{2}{1}\), which simplifies to just 2.

Now, instead of the fraction \(\frac{8}{4}\), we have the whole number 2, which is much simpler to visualize and further work with.

Numerator and Denominator Concepts

A fraction consists of two parts: a numerator and a denominator. Understanding these components is crucial for performing operations like multiplication or simplification.

• The numerator is the top number, indicating how many parts we are considering. In \(\frac{1}{4}\), 1 is the numerator.
• The denominator is the bottom number, which shows the total number of equal parts the whole is divided into. Here, 4 is the denominator.
• When you multiply fractions, you only multiply the numerators together and keep the original denominator, in this case, the denominator of 4 is unchanged from \(\frac{1}{4}\).

Knowing this helps you understand why multiplication affects only the numerator in operations involving whole numbers and fractions, and how we end up simplifying them after operations.