Problem 87 - Algebra for College Students

Question

Answer the question with an algebraic expression. The perimeter of a square is \(m\) meters. How long, in centimeters, is each side of the square?

Step-by-Step Solution

Answer

Each side is \( 25m \) cm long.

Step 1: Understanding the Problem

We are given the perimeter of a square as \( m \) meters and need to find the length of each side in centimeters. Recall that the perimeter \( P \) of a square is the total distance around the square.

Step 2: Formula for the Perimeter of a Square

The formula for the perimeter of a square is \( P = 4s \), where \( s \) is the length of one side of the square.

Step 3: Express Side Length in Meters

Since the perimeter is \( m \) meters, we can set up the equation \( 4s = m \). Solving for \( s \), we get \( s = \frac{m}{4} \) meters.

Step 4: Convert Side Length to Centimeters

Convert \( s \) from meters to centimeters by recognizing that 1 meter is equal to 100 centimeters. Therefore, each side in centimeters is \( s = \frac{m}{4} \times 100 = \frac{100m}{4} \) centimeters.

Step 5: Simplify the Expression

Simplify the expression to find the final length of one side in centimeters: \( \frac{100m}{4} = 25m \). So, each side of the square is \( 25m \) centimeters.

Key Concepts

Perimeter of a Square

When you think of a square, imagine a four-sided shape where each side is the same length. The perimeter is the distance around the entire square. If you were walking around its edges, starting and ending at the same point, you'd cover the perimeter.
To find the perimeter of a square, you simply multiply the length of one side by four since all four sides are equal.
Here's the mathematical formula:

• Perimeter \( P = 4s \, \) where \( s \ \) is the length of one side.

Understanding this formula is key to solving many square-related problems.
Once you know any one of these variables, you can easily find the others. For instance, if you know the perimeter, you can calculate the side length of the square. This allows you to handle complex problems by breaking them down into simpler steps.

Conversion of Units

Converting units is like changing languages to make sure everyone understands. For this exercise, we need to convert meters into centimeters since metric system units like these are often used in math and science to simplify calculations.
In the metric system, these conversions are straightforward:

• 1 meter = 100 centimeters

This means that to convert meters to centimeters, you multiply by 100. It's important to become familiar with such conversions as they often come in handy not just in math problems, but also in real-world situations. For instance, if you have a length measured in meters but your ruler is marked in centimeters, you’ll need to do a quick conversion.
Understanding these conversions ensures accuracy in calculations and helps you solve problems involving different measurements easily.

Solving Equations

Solving equations involves finding out what number a variable stands for. In algebra, equations are like scales trying to stay balanced.
When you encounter an equation such as \( 4s = m \), you are tasked with finding the value of \( s \), which represents the length of one side of the square.
Here are the basic steps to solve it:

• Isolate the variable by performing the same operation on both sides. For our problem:
  • Divide both sides by 4: \( s = \frac{m}{4} \, \)

Now, you will have \( s \) expressed in terms of \( m \), allowing further calculations or conversions, such as converting the length into centimeters afterwards. Solving equations is a foundational skill in algebra because it helps us to understand how variables relate to each other and find unknown values.
With practice, solving equations becomes a powerful tool in tackling problems efficiently.