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In mathematics, proportional relationships are a cornerstone concept that links two quantities in a way that their ratio remains constant. When we say that one quantity is proportional to another, it means that if one quantity changes, the other changes at a consistent rate. This consistency is what we call the 'constant of proportionality', denoted typically by the letter 'k'.

For instance, let's say you're buying fruits, and the price of fruits is directly proportional to their weight. If 2 pounds of fruit cost \(6, then 4 pounds (which is double the weight) would cost \)12, and so on. The price per weight unit (pound) stays the same, illustrating a proportional relationship. In our GED Math Practice problem, the price of a billboard sign is proportional to its area. Therefore, we can write the equation for the price as: \( p = k \times A \) where \( p \) is the price, \( k \) is the constant of proportionality, and \( A \) is the area of the sign.

In real-life problems like these, understanding proportional relationships helps in predicting costs, quantities, and other factors with simplicity and accuracy.

Speaking of area, it's a measure indicating the size of a two-dimensional surface. For a rectangle, the area is calculated by multiplying the length and the width of the rectangle. The formula for the area (\( A \)) of a rectangle is: \[ A = \text{length} \times \text{width} \]
Using the dimensions provided, if we take a rectangle with a length of 5 feet and a width of 8 feet, we simply multiply these two numbers together to find the area: \( A = 5 \text{ ft} \times 8 \text{ ft} \).

Knowing how to calculate the area is essential for various real-life scenarios, such as determining the amount of paint needed for a wall, the size of a carpet for your room, or, like our exercise, the cost of a billboard sign by its size.

In algebra, solving for variables is the process of finding the values of the unknowns in equations. To solve for a variable, we often use algebraic manipulation, which involves methods like addition, subtraction, division, and multiplication, alongside the properties of equality to isolate the variable and find its value.

In our practice problem, after finding the proportionality constant, we need to solve for the length of the new sign (\( L_\text{new} \)). By using the formula for the area of a rectangle (\( A = \text{length} \times \text{width} \)), and knowing that in this case, the area is equal to the cost divided by the constant of proportionality (\( A_\text{new} = \frac{p}{k} \)), we can rearrange the equation to isolate and solve for the length (\( L_\text{new} \)). The steps of such algebraic manipulation are fundamental and highly applicable in various disciplines within and beyond mathematics.