Problem 53 - Elementary Algebra Within Reach

Question

Geometry The perimeter of the Jamaican flag is 120 inches. Its length is twice its width. Find the dimensions of the flag.

Step-by-Step Solution

Answer

The dimensions of the Jamaican flag are a length of 40 inches and a width of 20 inches.

Step 1: Express the length in terms of the width

Given that the length is twice the width, we can express the length as L = 2W

Step 2: Substitute the length into the formula for the perimeter

We substitute the expression for the length from step 1 into the formula for the perimeter. That gives us: P = 2L + 2W = 2*(2W) + 2W = 4W + 2W = 6W

Step 3: Solve for the width

We know from the problem that the perimeter P is 120 inches. Set 120 equal to 6W, and then solve for W: 120 = 6W implies W = 20 inches

Step 4: Find the length

Substitute W = 20 inches back into the equation L = 2W we get L = 2*20 = 40 inches

Key Concepts

Geometry and measurement

Geometry plays a crucial role in understanding different shapes and their properties. In geometry, measurement helps us determine dimensions like length, width, and height. In our example, understanding the dimensions of the Jamaican flag involves geometry concepts such as length and width. Given that the flag's length is twice its width, we express this relationship in terms of geometry.
This exercise also uses measurement to determine the perimeter, which is a linear concept measuring the boundary of a shape. Measurement units such as inches are used to express these dimensions and ensure calculations are accurate. By mastering geometry and measurement principles, you can successfully solve problems related to shape and size.

Perimeter calculation

Calculating the perimeter is a fundamental aspect of geometry. The perimeter is the total distance around a particular shape.For rectangles, which is the likely shape of the flag, the perimeter formula is \[ P = 2L + 2W \] where \( L \) is the length and \( W \) is the width.
In the problem, we know the perimeter of the Jamaican flag is 120 inches. To calculate the width and length of the flag, we used this formula by substituting the given relationship \( L = 2W \). This leads to a new simplified equation: \[ P = 4W + 2W = 6W \] which we used to find the width.
Once you understand this formula, calculating the perimeter for many shapes, especially rectangles, becomes straightforward.

Variables and expressions

Variables and expressions are key elements in solving algebraic word problems. A variable represents an unknown value, often depicted with letters such as \( W \) for width and \( L \) for length.Understanding these expressions helps in forming equations that represent relationships mentioned in a problem.
In our exercise, we defined our variables based on the problem's description: \( L = 2W \). This equation demonstrates a relationship that helps us solve for one variable when another variable is known.Variables are often combined in expressions, which are phrases that include numbers and operation signs.
By mastering variables and expressions, you gain the ability to translate real-world problems into solvable mathematical equations.