Question

The cost of Brand V paper is proportional to the weight. If 18 ounces of Brand $V$ paper cost $\mathrm{\$}3.06,$ what is the cost of 24 ounces of Brand V paper? $$\begin{array}{l}\mathrm{\$}4.08\\ \mathrm{\$}5.58\\ \mathrm{\$}6.12\\ \mathrm{\$}7.31\\ \mathrm{\$}7.59\end{array}$$

Short answer

The cost of 24 ounces of Brand V paper is $4.08.

Step-by-step solution

Identify the relationship

Since the cost of Brand V paper is proportional to the weight, the relationship can be represented as: $$\text{Cost}=k\times \text{Weight}$$ where $k$ is the constant of proportionality.

Find the constant of proportionality

From the given information, 18 ounces of Brand V paper costs $\mathrm{\$}3.06$. Substitute these values into the equation to solve for $k$: $$3.06=k\times 18$$ Solve for $k$: $$k=\frac{3.06}{18}\approx 0.17$$

Calculate the cost for 24 ounces

Now, use the value of $k$ to find the cost of 24 ounces of Brand V paper: $$\text{Cost}=k\times 24$$ Substitute $k$: $$\text{Cost}=0.17\times 24\approx 4.08$$

Key concepts

Constant of Proportionality

In algebra, a proportional relationship between two quantities means that as one quantity changes, the other changes at a constant rate. This constant rate is called the 'constant of proportionality'. For instance, if the cost of Brand V paper changes directly with its weight, we can denote this relationship as $\text{Cost}=k\times \text{Weight}$, where $\text{k}$ is our constant of proportionality. In the given problem, we needed to find $k$ by using the known values: 18 ounces costing $3.06\. We derived $k$ by solving $\frac{3.06}{18}\approx 0.17$. This k value will help determine costs for different weights.

Linear Equations

Linear equations form the backbone of proportional relationships. They're equations that graph as straight lines, representing how two quantities vary together. In our problem, the equation $\text{Cost}=k\times \text{Weight}$ is linear. The graph of this equation would show a straight line illustrating how the weight of Brand V paper influences the cost. To solve linear equations, you typically isolate the variable you want to solve for by performing the same mathematical operations on each side of the equation. This principle was used to find the cost of 24 ounces of paper.

Unit Rate Calculation

Calculating the unit rate is crucial for understanding proportional relationships. A unit rate tells us how much of something there is per one unit of another thing. For Brand V paper, our unit rate is the cost per ounce. We found it by dividing the total cost by the number of ounces: $\frac{3.06}{18}\approx 0.17$. This unit rate means each ounce of paper costs approximately $0.17\. Knowing the unit rate allows us to make quick calculations for different amounts easily. For example, to find the cost of 24 ounces, we simply multiply the unit rate by 24.

Problem-Solving

Problem-solving in algebra often involves breaking down a complex issue into manageable steps. Let's revisit our steps: First, identify the relationship (proportional in this case). Next, find the constant of proportionality using given values. Lastly, use this constant to solve for unknown quantities. Following these steps systematically ensures accuracy. For our exercise, we identified the relationship (\text{Cost} = k \times \text{Weight}), calculated $k$ (0.17), and solved for the cost of 24 ounces by multiplying 0.17 by 24, resulting in $4.08\. Breaking down problems step-by-step helps simplify and solve them effectively.