Problem 13 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The product is \( \frac{1}{24} \).

Step 1: Identify the Fractions

The problem involves multiplying three fractions: \( \frac{1}{2} \), \( \frac{1}{3} \), and \( \frac{1}{4} \).

Step 2: Multiply the Numerators

Multiply the numerators of the fractions together. For this problem, the numerators are all 1, so: \( 1 \times 1 \times 1 = 1 \).

Step 3: Multiply the Denominators

Multiply the denominators of the fractions together: \( 2 \times 3 \times 4 \). First multiply 2 by 3 to get 6, then multiply 6 by 4 to get 24. So, \( 2 \times 3 \times 4 = 24 \).

Step 4: Form the Resulting Fraction

Combine the results from steps 2 and 3 to form the resulting fraction: \( \frac{1}{24} \).

Key Concepts

Understanding Fraction Multiplication

Fraction multiplication is a fundamental mathematical operation that is both straightforward and essential. When multiplying fractions, you don't need to worry about finding a common denominator, unlike in addition or subtraction. Instead, you simply multiply straight across. This means you multiply the numerators together and then multiply the denominators together.
Let's consider multiplying three fractions: \( \frac{1}{2} \), \( \frac{1}{3} \), and \( \frac{1}{4} \). To solve this, you would multiply:

• The numerators: \( 1 \times 1 \times 1 \)
• The denominators: \( 2 \times 3 \times 4 \)

It's important to remember this step-by-step approach for systematic calculation.

Role of Numerators in Fraction Multiplication

The numerator is the top part of a fraction. In fraction multiplication, it's what you count first. Simply multiply all the numerators together.
In our example with \( \frac{1}{2} \), \( \frac{1}{3} \), and \( \frac{1}{4} \), you multiply the numerators (which are each 1) to get:

• \( 1 \times 1 \times 1 = 1 \)

By understanding the numerator's role, you ensure that the fraction representation starts correctly. Multiplying numerators is essential to determining the top part of your resulting fraction.

Importance of Denominators in Fraction Multiplication

The denominator, the bottom part of a fraction, tells us into how many parts the whole is divided. In multiplication, you multiply each denominator together.
For the fractions \( \frac{1}{2} \), \( \frac{1}{3} \), and \( \frac{1}{4} \), happen the following:

• Multiply 2 by 3: \( 2 \times 3 = 6 \)
• Then, multiply that result by 4: \( 6 \times 4 = 24 \)

Thus, understanding denominators in this way helps you calculate the size of the parts in the final answer. Start with one, and multiply it down the list.

Forming the Resulting Fraction

After you have multiplied the numerators and denominators separately, the next step is to put them together to form a resulting fraction.
In our case study, after calculating, we have a numerator of 1 and a denominator of 24. Therefore, the resulting fraction becomes:

• \( \frac{1}{24} \)

This resulting fraction is the simplified product of your multiplication task. Ensuring the numerators and denominators are correctly multiplied makes it an accurate representation of the calculation. The product, \( \frac{1}{24} \), reflects the small piece we calculated from our original fractions.