Problem 63 - Prealgebra

Question

A rectangle has a perimeter of 9.5 inches. If the length is 2.75 inches, find the width.

Step-by-Step Solution

Answer

The width of the rectangle is 2 inches.

Step 1: Understand the Formula for Perimeter

The perimeter of a rectangle is calculated using the formula: \( P = 2L + 2W \), where \( P \) is the perimeter, \( L \) is the length, and \( W \) is the width.

Step 2: Plug in the Known Values

Substitute the given values for the perimeter and the length into the formula: \( 9.5 = 2(2.75) + 2W \).

Step 3: Simplify the Equation

Calculate \( 2(2.75) \), which gives you \( 5.5 \). This simplifies the equation to: \( 9.5 = 5.5 + 2W \).

Step 4: Solve for the Width

Subtract \( 5.5 \) from both sides of the equation to isolate the term with \( W \): \( 9.5 - 5.5 = 2W \), leading to \( 4 = 2W \).

Step 5: Divide to Find the Width

Divide both sides by 2 to find \( W \): \( W = \frac{4}{2} \). This gives \( W = 2 \).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics that helps you find unknown values. In our exercise, we have the equation derived from the perimeter formula of a rectangle. Start by isolating the unknown, which is the width (\( W \)).
The equation given is simplified to \( 9.5 = 5.5 + 2W \).

• First, perform the inverse operation to eliminate the known number from the side with the unknown. This means subtracting \( 5.5 \) from both sides.
• Once simplified, you have \( 4 = 2W \).
• To get \( W \) by itself, divide both sides by 2, which gives \( W = 2 \).

These steps show how equations are manipulated to solve for an unknown quantity.

Geometry Concepts

In geometry, understanding shapes and their properties is critical. A rectangle is a quadrilateral with opposite sides being equal and parallel. The rectangle's perimeter, which is the total distance around it, is given by the formula \( P = 2L + 2W \).

• This formula comes from the sum of all sides: two lengths and two widths.
• Recognizing these properties helps us apply our knowledge of formulas to find missing dimensions.

In our problem, knowing one dimension (length) and the perimeter, we apply geometric principles to find the width.

Prealgebra Problems

Prealgebra is a critical step in mathematics education where understanding the basic concepts of algebra starts. This includes working with simple equations and formulas.
The given problem is prealgebra in nature because it involves substituting values into a formula, simplifying, and solving for a variable.

• Start with understanding the problem, breaking it down into smaller steps such as establishing what is known (the length and perimeter) and what is unknown (the width).
• Using basic arithmetic operations, solve for the unknown by following a logical sequence.

Prealgebra problems like this prepare students for more advanced mathematical concepts by establishing a strong foundation in problem-solving and logical thinking.