Problem 32 - Algebra for College Students

Question

The cost of publishing pamphlets varies directly as the number of pamphlets produced. If it costs \(\$ 96\) to publish 600 pamphlets, how much does it cost to publish 800 pamphlets? \(\$ 128\)

Step-by-Step Solution

Answer

It costs $128 to publish 800 pamphlets.

Step 1: Understand Direct Variation

The problem states that the cost of publishing pamphlets varies directly as the number of pamphlets. This means that the cost (C) is directly proportional to the number of pamphlets (P). Mathematically, this can be expressed as \( C = k \cdot P \), where \( k \) is the constant of proportionality.

Step 2: Determine the Constant of Proportionality

Using the given information that it costs \( \$96 \) to publish 600 pamphlets, substitute these values into the equation \( C = k \cdot P \) to find \( k \): \( 96 = k \cdot 600 \). Solve for \( k \) to get \( k = \frac{96}{600} = 0.16 \).

Step 3: Calculate the Cost for 800 Pamphlets

Now that we know \( k = 0.16 \), we can find the cost to publish 800 pamphlets by substituting \( P = 800 \) back into the equation: \( C = 0.16 \cdot 800 \). Calculate \( C \) to get \( C = 128 \).

Step 4: Verify the Solution

Our calculation shows that the cost for 800 pamphlets is \( \$128 \). The proportional relationship holds as the constant \( k \) remains the same. Therefore, the solution is verified.

Key Concepts

Constant of Proportionality

In many real-world scenarios, two quantities may change in relation to each other at a consistent rate. This is where the idea of a "constant of proportionality," often represented as \( k \), comes into play. When we say that a variable, like cost, varies directly as another, such as the number of items like pamphlets, we imply a straightforward linear relationship between the two. Mathematically, this direct relationship is expressed by the formula: \[C = k \cdot P\] where \( C \) is the cost, \( P \) is the number of pamphlets, and \( k \) is the constant of proportionality.

• This constant \( k \) remains unchanged regardless of the quantity, provided the relationship is purely proportional.
• In our exercise, if it costs \( \\(96 \) to publish 600 pamphlets, then \( k \) can be determined by rearranging the formula to get \( k = \frac{C}{P} \), giving \( k = 0.16 \).

This means, for every single pamphlet produced, the cost increases by \\)0.16, hence defining the consistent rate of variation between the cost and the number of pamphlets produced.

Proportions in Mathematics

Proportions in mathematics define relationships between quantities where they increase or decrease relative to one another at a fixed ratio. In the context of our problem on pamphlets, proportions help us understand how costs scale with the number of items produced.

• When two quantities, like cost and number of pamphlets, are proportional, their ratio remains constant. This is the essence of direct variation.
• For instance, in our exercise, the cost per pamphlet is a fixed amount, as shown by the constant of proportionality \( k = 0.16 \).

Proportional relationships are invaluable in various fields such as economics, physics, and engineering. Recognizing these relationships allows us to predict the change in one quantity when another varies, similar to how we predicted the cost of 800 pamphlets from the given data on 600 pamphlets.

Problem-Solving Steps

Successful problem-solving in mathematics often involves logical and methodical steps, as demonstrated in calculating the pamphlet costs in our exercise. Here's a simplified breakdown:

• Understand the Problem: Recognize that the question involves direct variation between costs and the number of pamphlets. This sets the stage for utilizing a direct proportionality formula.
• Determine the Constant: By substituting given values into the formula \( C = k \cdot P \), solve for \( k \), ensuring you find the constant rate of change between the two variables.
• Apply the Constant: Use the determined \( k \) to compute unknown values, like calculating the cost for a new number of pamphlets, ensuring the logic holds by solving \( C = 0.16 \cdot 800 \).
• Verify Your Solution: Double-check computations and ensure the solution is consistent with the problem's context and logic. In our case, both theoretical understanding and the arithmetic led to a cost of \$128 for 800 pamphlets.

By following these structured steps, approaching complex problems becomes more manageable, aiding both learning and application.