Problem 10 - Prealgebra

Question

Find each of the following products. (Multiply.) $$\frac{1}{y} \cdot 8$$

Step-by-Step Solution

Answer

\( \frac{8}{y} \)

Step 1: Understanding the Problem

The problem requires us to multiply a fraction \( \frac{1}{y} \) by 8. This is an operation involving a fraction and a whole number.

Step 2: Write the Whole Number as a Fraction

To simplify the multiplication, express 8 as a fraction: \( \frac{8}{1} \). This makes it easier to understand the multiplication process between fractions.

Step 3: Multiply the Fractions

Multiply the numerators and multiply the denominators: \( \frac{1}{y} \times \frac{8}{1} = \frac{1 \times 8}{y \times 1} = \frac{8}{y} \). This follows the rules for multiplying fractions.

Step 4: Simplify the Result

The fraction \( \frac{8}{y} \) is already in its simplest form. There are no common factors between 8 and \( y \) that could allow for further simplification.

Key Concepts

Whole Numbers

Whole numbers are the set of numbers that include 0, 1, 2, 3, and so on without any fractions or decimals. They are the fundamental building blocks of mathematics, often used in counting and basic arithmetic. Whole numbers are important because they are easy to understand and provide a simple way to perform basic operations like addition, subtraction, multiplication, and division. In the context of multiplication involving fractions, it's useful to remember these points:

• Whole numbers can be represented as fractions by writing them with the denominator 1. For example, 8 can be expressed as \( \frac{8}{1} \).
• This conversion allows whole numbers to be easily used in operations with fractions. It simplifies the multiplication process because both numbers involved are in fraction form, which makes it easier to follow the rules of fraction operations.

Fractions

Fractions represent parts of a whole and consist of two numbers: the numerator and the denominator. The numerator is the top number, indicating how many parts you have. The denominator is the bottom number, showing how many equal parts the whole is divided into. Understanding fractions is vital for learning to multiply them, especially when combined with whole numbers.

• Fractions can represent quantities less than one, exactly one, or greater than one, depending on the relationship between the numerator and the denominator.
• To multiply fractions, you simply multiply the numerators together to get the new numerator, and multiply the denominators to get the new denominator.
• This makes operations with fractions straightforward once you are comfortable with the concept of numerators and denominators.

Simplifying Fractions

Simplifying fractions involves reducing them to their smallest possible form, where the numerator and denominator share no common factors except 1. In the provided exercise, the multiplication results in a fraction \( \frac{8}{y} \), which may already be in its simplest form if 8 and \( y \) have no common factors.

• The simplification process involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by that number.
• However, if no common factors other than 1 exist, the fraction is already simplified, as demonstrated in our example.
• Simplifying makes the fraction easier to read and use in further calculations, ensuring it's in its most efficient form.