Problem 87 - Intermediate Algebra

Question

The perimeter of a square is \(c\) centimeters. How long is each side of the square? 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 \(\frac{c}{4}\) cm, and in the second case, it is \(25m\) cm.

Step 1: Define the Perimeter of a Square

The perimeter of a square is the total length around the square, which is equal to 4 times the length of one side. Thus, if the length of a side is \( s \), the perimeter \( P \) is given by \( P = 4s \). We are given that \( P = c \) centimeters.

Step 2: Solve for One Side in Centimeters

To find the length of one side \( s \) in centimeters, we use the equation from Step 1. Set \( 4s = c \) and solve for \( s \) by dividing both sides by 4. This gives \( s = \frac{c}{4} \) centimeters.

Step 3: Define Another Perimeter in Meters

Similarly, if the perimeter is given as \( m \) meters, then using the formula \( P = 4s_m \), where \( s_m \) is the side length in meters, we have \( 4s_m = m \).

Step 4: Solve for One Side in Meters

Solve the equation from Step 3 for \( s_m \) by dividing both sides by 4. This gives \( s_m = \frac{m}{4} \) meters.

Step 5: Convert Meters to Centimeters

Since we need the side length in centimeters, convert \( s_m \) from meters to centimeters by using the conversion factor 1 meter = 100 centimeters. Thus, \( s_{cm} = \frac{m}{4} \times 100 \) gives us \( s_{cm} = 25m \) centimeters.

Key Concepts

Converting Units

Whenever you need to convert between units, understanding the conversion process is essential. Let's delve into a common conversion: meters to centimeters.
The relationship between meters and centimeters is straightforward and crucial for numerous mathematical problems, including geometry.

• Firstly, note that 1 meter is equivalent to 100 centimeters.
• To convert meters to centimeters, multiply the number of meters by 100.
• For example, if you have a length of 2 meters, it equals 200 centimeters because 2 meters * 100 = 200 centimeters.

In our exercise, this conversion was key to determining the side length of a square in centimeters. By using this conversion factor, we can transform any measurement from meters to centimeters effectively.

Solving Equations

Solving equations involves finding an unknown variable's value by performing operations that isolate the variable on one side. Let's discuss how to do this using a simple equation. Suppose you have the equation for the perimeter of a square: \[ P = 4s \]Here, \( P \) represents the perimeter, and \( s \) is the side length. If you know the perimeter, say \( P = c \) centimeters, solving for \( s \) requires a few steps:

• Start by setting the equation: \( 4s = c \).
• To isolate \( s \), divide both sides by 4.
• This results in: \( s = \frac{c}{4} \)

Such methods are standard in algebra and provide a systematic approach to finding unknowns. In the context of our problem, this procedure enables us to find a square's side length when given its perimeter.

Algebraic Expressions

Understanding algebraic expressions is fundamental to solving mathematical problems. Algebraic expressions are combinations of variables, numbers, and operations that represent specific relationships.In our discussion about the square's perimeter, an algebraic expression is used to relate the perimeter to the side length of the square. The formula, \[ P = 4s, \] where \( P \) is the perimeter and \( s \) is the side length, is a prime example.

• The expression \( 4s \) indicates multiplying the side length by 4. This multiplication embodies the concept that a square's perimeter is the sum of its four equal sides.
• By using such expressions, you can translate real-world problems into mathematical language, allowing for precise and accurate problem-solving.

Through familiarizing with algebraic expressions, you gain a powerful tool for deciphering and solving various mathematical problems.