Problem 61 - Algebra 1

Question

Simplify the expression. $$\frac{1}{8} y \cdot 8$$

Step-by-Step Solution

Answer

The simplified expression is \(y\)

Step 1: Identify the Numerator

The expression \(\frac{1}{8} y\) can be interpreted as \(y\), divided by 8.

Step 2: Multiply by the given number

Multiply the fraction (\(\frac{1}{8} y\)) by 8, which is multiplying \(y\) by \(1\) (since \(8 \div 8 = 1\)).

Step 3: Simplify the expression

Now, simplify the expression to get the final answer.

Key Concepts

Fractions in Algebra

Fractions are a fundamental concept in algebra, representing division between two numbers. When working with algebraic fractions, we often see variables involved, such as \(\frac{1}{8} y\). This expression can read as "\(y\) is divided by 8." Understanding this division is crucial since it allows us to see how a fraction represents a part of a whole or the result of a division operation.

Algebraic fractions follow the same rules as numerical fractions. So, the fraction \(\frac{1}{8}\) implies one out of eight parts. When a variable is introduced, it indicates that this same fractioning process applies to whatever value \(y\) takes. Thus, knowing how to interpret and manipulate algebraic fractions is essential for simplifying expressions and solving equations.

Multiplying Fractions

Multiplying fractions is a straightforward process once you understand the mechanics. To multiply two fractions, you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. For example, multiplying \(\frac{1}{8}\) by \(8\) involves writing \(8\) as a fraction: \(\frac{8}{1}\). Then, it becomes a matter of straight multiplication:

• Multiply the numerators: \(1 \times 8\)
• Multiply the denominators: \(8 \times 1\)

This gives you \(\frac{8}{8}\). Since the numerator and the denominator are the same, this fraction equals 1. Therefore, \(\frac{1}{8} \cdot 8\) simplifies to 1. This is crucial for simplifying expressions involving fractions in algebra.

Simplification Process

The simplification process is about reducing expressions to their simplest form. In our example, the expression \(\frac{1}{8} y \cdot 8\) relies on understanding both fractions and multiplication. Let's break it down:

• First, recognize the multiplication \(\frac{1}{8} y \cdot 8\) as two parts: \(\frac{1}{8} \cdot 8\) is a fraction multiplied by its denominator, simplifying to 1.
• This leaves you with \(y\) multiplied by 1, which is simply \(y\). Multiplying anything by 1 leaves it unchanged.

As a result, the simplified expression is just \(y\). This straightforward simplification shows how breaking down an expression into manageable parts can simplify the overall problem. By understanding each component, fraction, multiplication, and simplification, you'll navigate algebra with greater ease.