Problem 68 - Algebra 1: Concepts and Skills

Question

Multiply the fractions. $$\frac{1}{4} \cdot \frac{1}{4}$$

Step-by-Step Solution

Answer

The product of the fractions \(\frac{1}{4}\) and \(\frac{1}{4}\) is \(\frac{1}{16}\).

Step 1: Multiply the Numerators

In our case, we have two fractions \(\frac{1}{4}\) and \(\frac{1}{4}\). To start with, multiply the numerators together. The numerator of the first fraction is 1 and the numerator of the second is also 1. So, \(1 \times 1 = 1\). Thus, the numerator of the product is 1.

Step 2: Multiply the Denominators

Next, multiply the denominators of the two fractions together. The denominator of the first fraction is 4 and the denominator of the second is also 4. So, \(4 \times 4 = 16\). Thus, the denominator of the product is 16.

Step 3: Write the result

The result of the multiplication of two fractions is a new fraction where the numerator is the result of the multiplication of the numerators and the denominator is the result of the multiplication of the denominators. For our fractions, the result is \(\frac{1}{16}\).

Key Concepts

Understanding the Numerator

The numerator is a fundamental part of any fraction. It is the top number in a fraction and indicates how many parts of a whole are considered. If we look at the fraction \(\frac{1}{4}\), the numerator is 1. In essence:

• The numerator tells us "how many."
• In fraction multiplication, we multiply the numerators of the involved fractions.

When multiplying fractions like \(\frac{1}{4} \cdot \frac{1}{4}\), our numerators are both 1. This means:

• Multiply 1 (numerator of the first fraction) by 1 (numerator of the second fraction).
• This results in a numerator of 1 for the product.

Decoding the Denominator

The denominator is the bottom number in a fraction and plays a key role in defining what equal parts make up the whole. For the fraction \(\frac{1}{4}\), the denominator is 4. Understanding denominators is crucial:

• The denominator tells us into how many equal parts the whole is divided.
• For multiplication, we multiply the denominators together to get the new denominator.

In our example with \(\frac{1}{4} \cdot \frac{1}{4}\), both denominators are 4:

• Multiply 4 (denominator of the first fraction) by 4 (denominator of the second fraction).
• The resultant denominator for our product then is 16.

Exploring the Product of Fractions

The product of fractions results from multiplying two fractions together to form a new fraction. This involves multiplying both the numerators and the denominators separately.

• The numerator of the product is the product of the numerators from each fraction.
• The denominator of the product is the product of the denominators from each fraction.

Using our given example \(\frac{1}{4} \cdot \frac{1}{4}\), we find:

• The numerators multiply to 1, and the denominators multiply to 16.
• Therefore, the product of the fractions is \(\frac{1}{16}\).

This illustrates that by multiplying the fractional parts, a new fraction emerges, representing a smaller portion of the whole, especially in cases like ours where both numerators are less than the denominators.