Problem 37 - Prealgebra

Question

Use the associative property to rewrite each of the following expressions, and then simplify as much as possible. $$\frac{1}{5}(5 y)$$

Step-by-Step Solution

Answer

The expression simplifies to \(y\).

Step 1: Apply the Associative Property

The associative property states that the way in which numbers are grouped does not change their sum or product. For multiplication, this property can be written as \((a \cdot b) \cdot c = a \cdot (b \cdot c)\). In our expression, we start by regrouping: \[ \frac{1}{5} \cdot (5 \cdot y) = \left(\frac{1}{5} \cdot 5\right) \cdot y \]

Step 2: Simplify the Expression

Simplify the expression inside the parentheses. Multiplying \(\frac{1}{5} \cdot 5\) gives you \(1\): \[ \left(\frac{1}{5} \cdot 5\right) \cdot y = 1 \cdot y \]

Step 3: Further Simplify

Any number multiplied by 1 remains unchanged. Thus, we simplify the expression further: \[ 1 \cdot y = y \]

Key Concepts

Understanding Multiplication

Multiplication is a fundamental arithmetic operation that involves combining groups of equal sizes. When you multiply two numbers, you find out how many objects you would have if you had that number of groups. For example, in our exercise with the expression \( \frac{1}{5}(5y) \), we are dealing with multiplying a fraction and a variable that includes multiplication too.

When it comes to working with fractions, such as \( \frac{1}{5} \), multiplication becomes a tool to scale or divide a quantity into smaller parts. Here is how multiplication steps in:

• Think of multiplication as repeated addition or a way to quickly combine groups of things.
• With fractions, multiplying means finding a fraction of a number, like 1/5 of 5 is 1.
• If you multiply 5 groups of \( y \), it's the same as adding \( y \) five times.

Understanding these basic concepts of multiplication, especially with fractions and variables, is essential since they form the basis of many algebraic manipulations.

Simplification Made Simple

Simplification is all about making an expression as simple as it can be. In the exercise, the expression \( \left(\frac{1}{5} \cdot 5\right) \cdot y \) was simplified to \( y \). This process involves reducing expressions by performing arithmetic operations or using properties like the associative property.

Some tips for simplification include:

• Always look for opportunities to eliminate fractions by multiplying terms together.
• Use properties of numbers, such as the fact that multiplying anything by 1 leaves it unchanged.
• Combine like terms, when possible, to reduce expressions to their simplest forms.

In our example, the fraction \( \frac{1}{5} \) is multiplied by 5 to get 1, thanks to the associative property helping with grouping terms. Simplification often means making complex calculations easier to understand and work with.

The Role of Prealgebra

Prealgebra is often the first step in high school mathematics. It provides the foundation for algebra, focusing on arithmetic operations and properties that make tackling more advanced math possible. In the given exercise, concepts from prealgebra play a critical role.

When learning prealgebra, students are introduced to properties of numbers like the associative property. Here's why it's important:

• The associative property allows you to change the grouping of numbers without changing the result, which simplifies calculations.
• Understanding how to manipulate variables, such as \( y \) in \( 5y \), prepares you for solving algebraic equations.
• These skills provide the basis for tackling more complex mathematical problems later on.

By mastering prealgebra, students study the basics of expressions, enabling them to work with variables, numbers, and operations efficiently.