Problem 71 - Algebra 1: Concepts and Skills

Question

Multiply the fractions $$ \frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} $$

Step-by-Step Solution

Answer

The result of multiplying the fractions \(\frac{1}{3} \cdot \frac{1}{3} \cdot \frac{1}{3}\) is \(\frac{1}{27}\).

Step 1: Multiplication of the first two fractions

To begin with, multiply the first two fractions together. This is done by simply multiplying the numerator (top number) of the first fraction by the numerator of the second fraction and writing the result as the numerator of the new fraction. Then, do the same with the denominators (bottom numbers). For \(\frac{1}{3} \cdot \frac{1}{3}\) , multiplying the numerators 1 and 1 gives 1, and multiplying the denominators 3 and 3 gives 9. So, \(\frac{1}{3} \cdot \frac{1}{3} = \frac{1}{9}\).

Step 2: Multiplication of the resulting fraction by the third fraction

Now, multiply the resulting fraction, \(\frac{1}{9}\), by the third fraction, \(\frac{1}{3}\). Again, multiply the numerators together and the denominators together to get the new fraction. The numerator is 1 × 1 = 1 and the denominator is 9 × 3 = 27. So, \(\frac{1}{9} \cdot \frac{1}{3} = \frac{1}{27}\).

Key Concepts

Understanding Fractions

Fractions are a type of mathematical expression used to represent a part of a whole. They consist of two main parts: the numerator and the denominator.
Fractions are often used in everyday life to describe quantities that are less than a whole. For instance, if you eat one slice of a pizza that has eight slices, you have eaten 1/8 of the pizza. In mathematical terms, fractions allow us to express numbers that lie between whole numbers. This is crucial in various applications like measurements, cooking, and sharing resources evenly. To deeply understand fractions, it's useful to think about them as division problems, where the numerator is divided by the denominator.

Numerator and Denominator

In every fraction, the top number is called the numerator, and the bottom number is called the denominator. These two components tell you two things:

• The numerator indicates how many parts are being considered or taken.
• The denominator represents how many equal parts the whole is divided into.

For example, in the fraction \( \frac{3}{4} \):

• The numerator is 3, indicating three parts.
• The denominator is 4, showing that these parts are from a whole divided into four pieces.

Understanding the roles of the numerator and denominator is vital when performing operations like addition and multiplication of fractions. They help determine the size of the fraction relative to a whole.

Multiplication of Fractions

Multiplying fractions is straightforward once you understand the roles of the numerator and denominator. The process involves:

• Multiplying the numerators of the fractions together to get the new numerator.
• Multiplying the denominators together to get the new denominator.

For instance, let's multiply \( \frac{1}{3} \times \frac{1}{3} \):

• Multiply the numerators: 1 × 1 = 1.
• Multiply the denominators: 3 × 3 = 9.

This results in \( \frac{1}{9} \). When multiplying more than two fractions, continue this process. For example, \( \frac{1}{9} \times \frac{1}{3} \) gives:

• Numerators: 1 × 1 = 1.
• Denominators: 9 × 3 = 27.

Thus, the result is \( \frac{1}{27} \). Remember, fraction multiplication is all about simplifying the problem into smaller, repeatable steps!