Problem 25 - Prealgebra

Question

Jordan is paving a rectangular patio with bricks. If the patio contains a total of 308 bricks and there are 22 bricks running along the length of the patio, how many bricks run along the width of the patio?

Step-by-Step Solution

Answer

14 bricks run along the width of the patio.

Step 1: Set up the equation

We need to determine the number of bricks along the width of the patio. Let's call this number \( w \). The problem states there are 22 bricks along the length of the patio and the total number of bricks is 308. We can set up the equation relating these quantities as \( 22 \times w = 308 \).

Step 2: Solve for the width

To find the width, we need to solve the equation \( 22 \times w = 308 \) for \( w \). We do this by dividing both sides of the equation by 22: \[ w = \frac{308}{22} \].

Step 3: Calculate the value

Calculate \( \frac{308}{22} \): dividing 308 by 22 gives 14. Therefore, \( w = 14 \).

Step 4: Conclusion

The solution to the problem indicates there are 14 bricks running along the width of the patio.

Key Concepts

Rectangular Patio

Imagine a rectangular patio as a simple flat surface with four sides, where opposite sides are equal in length. It’s like having two longer sides called the length and two shorter sides known as the width. Jordan is paving this kind of surface with bricks, arranging them neatly along both the length and width.
This basic setup allows us to express the patio's structure mathematically. In these terms, the total number of bricks used is the product of the number of bricks along the length and the number along the width. This pattern is typical for rectangles and helps in easily calculating dimensions when one side and the total area, as measured in bricks, are known.

Multiplication and Division

To solve problems involving rectangles, multiplication and division are key operations. In this scenario, we know the total number of bricks, which acts like the 'area' in a broader sense, and the number along one side, referred to as the length. Thus, we multiply the number of bricks in each row by the number of rows to get the total.
Using multiplication, the formula for total bricks becomes:

• Total Bricks = Length (in bricks) × Width (in bricks)

To find one unknown, like the width, division comes into play. By rearranging the multiplication formula, we can solve for the width by dividing the total number of bricks by the length. This reverse operation is crucial when solving real-life word problems.

Solving Equations

When given a problem like Jordan’s patio, setting up and solving equations becomes a valuable strategy. You start by translating the word problem into an equation. For Jordan, we know:

• 22 bricks line the length of the patio
• Total bricks = 308

With this, the equation becomes:\(22 \times w = 308\), where \(w\) represents the width in bricks. To find \(w\), isolate it by dividing both sides of the equation by 22. This step helps unravel the mystery number of bricks along the width, simplifying the equation to:

• Width = Total Bricks / Length

Practice makes these processes second nature, especially when you can connect the numbers to real objects like bricks.

Area and Perimeter

In geometry, understanding the area and perimeter of rectangles helps in solving real-world problems, like paving patios. Area measures the space covered by the rectangle, calculated by multiplying length by width. While Jordan's patio uses bricks to show area concretely, the concept holds universally as:

• Area = Length × Width

Perimeter, on the other hand, measures the total distance around the rectangle. While this wasn’t needed for Jordan's exact problem, understanding both concepts enriches your ability to tackle related and more complex tasks. Calculating perimeter involves adding all sides:

• Perimeter = 2 × (Length + Width)

Grasping these ideas early on sets you up for more advanced mathematics, as they apply in both practical and theoretical scenarios.