Question

Two square rugs have a combined area of 20 square meters. If the area of the large rug is four times as large as the area of the small rug, what is the perimeter of the smaller rug? (A) 2 meters (B) 4 meters (C) 8 meters (D) 16 meters

Short answer

The perimeter of the smaller rug is 8 meters (Option C).

Step-by-step solution

Translate the problem into algebraic expressions and equations.

Let's denote the area of the large rug as L and the area of the small rug as S. The problem states that the combined area of the two rugs is 20 square meters. So, we can write the following equation: \( L + S = 20 \) The problem also mentions that the area of the large rug is four times the area of the small rug. So, we can write another equation as: \( L = 4S \)

Solve the system of equations for the area of the smaller rug.

We can substitute the expression for L from the second equation into the first equation to solve for S: \( 4S + S = 20 \) Combine the terms on the left side: \( 5S = 20 \) Now, divide both sides by 5 to find the value of S: \( S = 4 \) Thus, the area of the smaller rug is 4 square meters.

Find the perimeter of the smaller rug.

Recall that the rug is a square, so each side of the rug has the same length. Let's denote the length of each side of the smaller rug as x. We know that the area of a square is the side length squared, so we have: \[ x^2 = S \] We already know the area of the smaller rug (S = 4). Now, we can find the side length (x): \[ x^2 = 4 \] To find the value of x, take the square root of both sides: \[ x = \sqrt{4} \] \[ x = 2 \] Since the rug is a square, the perimeter is equal to the sum of all 4 sides: \[ Perimeter = 4x \] Now, substitute the value of x we found: \[ Perimeter = 4(2) \] \[ Perimeter = 8 \] Therefore, the perimeter of the smaller rug is 8 meters (Option C).

Key concepts

Understanding Geometry in the Context of Rugs

Geometry is all about the shapes and their properties. In this exercise, the main shape of interest is a square. Squares are very special because all four of their sides are equal in length, and their opposite sides are parallel. This means we can use simple formulas such as area and perimeter to solve problems related to squares.

Here, we are dealing with two square rugs. One is larger, and the other is smaller. Understanding that each rug is a square helps us to apply the correct geometric formulas that involve side lengths, areas, and perimeters. With squares, once we know one side length, we can easily find other properties like the area by squaring the side length, or the perimeter by multiplying the side length by four.

Using Algebra to Translate the Problem

Algebra helps us to translate word problems into mathematical equations. This is very useful in breaking down problems step by step. Let's start by defining some variables.

In this problem, we define:

• The area of the larger rug as \( L \).
• The area of the smaller rug as \( S \).

From the information given in the problem, we can create equations such as \( L + S = 20 \) to represent the combined area of the rugs.

To simplify further, since the larger rug's area is four times that of the smaller rug, we write \( L = 4S \). This system of equations can then be solved to find the specific areas of each rug. Algebra makes it easier to track relationships and solve for unknowns efficiently.

Approaching Problem Solving Strategically

Problem solving requires breaking down the given details and using logical reasoning to find a solution.

To solve this exercise, we first translated the word problem into algebraic equations. This helped us understand the relationships between the different quantities involved. Next, we solved the equations step by step to find the area of the smaller rug.

We identified that the final step in problem solving was to find the perimeter of the smaller rug, using the square's properties. By taking a systematic approach, we ensured nothing was overlooked, and all given information was used effectively. This structured way of solving problems can be useful across different types of math challenges.

Calculating the Perimeter Effectively

Perimeter calculation is straightforward for squares. The perimeter is the measure of the total length around the shape.

For a square, you can find the perimeter by multiplying the side length by four, since all sides are equal. Given the area of a square rug, you can find the side length by taking the square root of the area.

• If the area \( S \) of the smaller rug is \( 4 \text{ m}^2 \), the side length \( x \) is noted as \( x = \sqrt{4} \), which results in \( x = 2 \text{ m} \).
• Therefore, the perimeter \( P \) is \( 4 \times x = 4 \times 2 = 8 \text{ m} \).

So, whenever you're calculating the perimeter of a square, remember it's just about squaring the side lengths then multiplying by four. This method makes it easy to derive the perimeter when you know the area.