Problem 109 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ 5 t \cdot 60 $$

Step-by-Step Solution

Answer

The simplified expression is \(300t\).

Step 1: Understand the Expression

The expression \(5t \cdot 60\) involves a variable \(t\) and constants. The expression denotes the product of these numbers.

Step 2: Simplify the Numerical Part

First, simplify the numerical part of the expression by multiplying \(5\) and \(60\). Calculating gives \(5 \times 60 = 300\).

Step 3: Express the Simplified Form

Multiply the numerical result by \(t\) to express the simplified form of the original expression: \(300t\).

Key Concepts

Variable Multiplication

When multiplying variables in algebraic expressions, we deal with quantities that may not yet have a known value. In our exercise, the variable involved is denoted by \( t \). Variable multiplication involves using a placeholder to represent unknown or changing quantities. For example, if we have \( 5t \), this implies that \( t \) is multiplied by \( 5 \), even though \( t \) doesn't have a specific value yet.
In equations or expressions, variables can stand for real-world things like time, distance, or any other value you might quantify. If there were more variables involved, we'd perform multiplication similarly. However, since there is only one variable in our example, the process is simply multiplying it by the number we arrive at through numerical multiplication.

Numerical Multiplication

The term "numerical multiplication" refers to multiplying numbers without variables. In our example, before dealing with the variable \( t \), we first simplify the numerical part by calculating \( 5 \times 60 \).

• The number \( 5 \) represents a constant. It's a fixed and determined value.
• The number \( 60 \) is also a constant here, needing to be multiplied by \( 5 \).

Performing this multiplication gives us \( 300 \). Handling numerical multiplication separately helps in breaking down complex expressions and dealing with them in parts. This strategy helps prevent mistakes when variables come into play.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form, making them easier to interpret or use in more complex equations. Let's simplify the given expression \( 5t \cdot 60 \).After multiplying the numbers \( 5 \) and \( 60 \), we arrive at \( 300 \). Now, incorporating the variable back, the simplified expression becomes \( 300t \).
Key points to remember:

• First, handle any numerical parts separately to reduce errors.
• Next, multiply the result with any variables involved.
• Always aim for the most compact form of the expression.

Simplifying might seem straightforward with numbers and variables, but it's an essential skill in algebra, especially as you tackle more complicated equations later on in your studies.