Problem 23 - Prealgebra

Question

Define a variable. Then write an equation and solve. A number times 3 is 45

Step-by-Step Solution

Answer

The number is 15.

Step 1: Define the Variable

Let's define the unknown number as the variable \( x \). This variable will represent the number we need to find.

Step 2: Write the Equation

According to the problem, a number times 3 is 45. This can be written mathematically as \( 3x = 45 \).

Step 3: Solve the Equation

To find the value of \( x \), we need to isolate it on one side of the equation. This can be done by dividing both sides of the equation by 3: \[ x = \frac{45}{3} \]

Step 4: Simplify the Expression

Perform the division to simplify the expression: \( x = 15 \). Thus, the number is 15.

Key Concepts

Understanding Variables in Math

Variables are a fundamental part of prealgebra and mathematics in general. They act as placeholders that can represent unknown values or quantities in a mathematical expression. By assigning a letter, such as \( x \), to a variable, we can simplify and solve real-world problems.

• Representation: Variables are usually represented by letters like \( x \), \( y \), or \( z \). These letters stand in for numbers that we may not know yet.
• Usage: In the context of our exercise, the unknown number we need to find is represented by the variable \( x \).
• Benefits: Using variables helps us create general solutions that can be applied to different scenarios, beyond just the specific numbers given in the problem.

Variables allow us to translate a word problem into an algebraic expression or equation, which brings us to the next important concept: equations.

Crafting and Solving Equations

Equations are statements of equality where two expressions are equal to each other, often involving variables. In our scenario, the equation \( 3x = 45 \) stems from the phrase "a number times 3 is 45."

• Structure: An equation typically includes a variable, coefficients, and constants. In \( 3x = 45 \), \( 3 \) is the coefficient multiplying the variable \( x \), and 45 is the constant.
• Purpose: The goal in solving an equation is to find the value of the variable. This means making the equation true, where both sides are equal, by determining the value of \( x \).

To solve the equation \( 3x = 45 \), we need to isolate \( x \) by performing operations that reverse the multiplication by 3. By dividing both sides of the equation by 3, we come to find that \( x = 15 \). This means the unknown number is 15.

Effective Problem Solving Steps

Problem-solving in mathematics often follows a set of intuitive steps. These steps help organize our approach, making complex problems more manageable.

• Define the Variable: Determine what you want to find. In our example, the unknown number was defined as \( x \).
• Formulate the Equation: Translate the word problem into a mathematical equation. We turned "a number times 3 is 45" into the equation \( 3x = 45 \).
• Solve the Equation: Use algebraic methods to isolate the variable, often by performing inverse operations to simplify both sides of the equation. Dividing both sides by 3 gave us \( x = 15 \).
• Verify the Solution: Check your work by substituting the value back into the original equation to ensure both sides are equal. In our case, substituting \( 15 \) for \( x \) in \( 3x \) does equal 45, confirming our solution is correct.

These structured steps help tackle even the trickiest of math problems with ease, building up both problem-solving skills and confidence.