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Chapter 6: Problem 3

Write the formula that relates the cost \(C\) of purchasing \(q\) items that cost d$ each.

Short Answer

Expert verified
The formula for the cost of purchasing \(q\) items, each costing 'd' dollars, is given by \(C = q \times d\)

Step by step solution

01

Identify known quantities

We know that \(q\) represents the quantity or number of items and 'd' represents the cost of each item.
02

Understand the relationship

The cost of purchasing items can be determined by multiplying the quantity of the items by the cost of each item.
03

Write the Formula

Following from above, we get the formula: \(C = q \times d\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations form the backbone of algebra and are crucial for understanding how different variables relate to one another. In our exercise, we explore a simple yet powerful linear equation: \(C = q \times d\). Here, the cost \(C\) is a function of two variables: the number of items \(q\) and the cost per item \(d\). The term "linear" comes into play because when plotted on a graph, this equation forms a straight line. This simplicity allows us to predict values of \(C\) for various \(q\) and \(d\).

Understanding linear equations helps us to identify:
  • Change in one variable affects the other鈥攖he more items \(q\), the higher the total cost \(C\).
  • The rate of change, given by \(d\), tells us how much \(C\) changes as \(q\) changes.
By mastering these, you can not only solve problems like this one but also apply these concepts to various real-life scenarios.
Cost Calculation
Understanding cost calculation is essential both in mathematics and in everyday life. In this context, calculating the total cost of goods involves determining how much money is needed to purchase a given number of items. To do this, we use the formula \(C = q \times d\), where:
  • \(q\) represents the number of items you want to buy.
  • \(d\) is the price per each individual item.
This formula straightforwardly tells us how to calculate the total spending based on the quantity and price per unit. For example, if you want to buy 10 pencils at $0.50 each, your total cost is \(C = 10 \times 0.50 = 5\). This simple arithmetic, when understood deeply, can help you make informed financial decisions.
Mathematical Relationships
Mathematical relationships help us make sense of how different quantities interact with each other. The relationship described by the equation \(C = q \times d\) is a classic example of direct proportion. This means that as one variable changes, the other changes in a consistent way.

In our scenario:
  • If \(q\), the quantity of items, doubles, then the cost \(C\) will also double, assuming \(d\) remains constant.
  • Similarly, if the cost \(d\) per item increases, total cost \(C\) increases given a fixed \(q\).
These types of linear, proportional relationships are foundational in algebra. Understanding them helps in making predictions and seeing potential outcomes in a clear, systematic way. Mathematically, we express these relationships not just theoretically, but practically, influencing decisions from budgeting groceries to handling complex business finances.

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Most popular questions from this chapter

Unit cost of grass seed A 50 -pound bag of grass seed costs \(\$ 222.50 .\) Find the unit cost of grass seed.

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{6 a-6 b+6 c}{9 a-9 b+9 c}$$

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{3 x+3 y}{x^{2}+x y}$$

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are 0. SEE EXAMPLE 10. (OBJECTIVE 4) $$\frac{x-y}{y-x}$$

Write each expression in simplest form. If it is already in simplest form, so indicate. Assume that no denominators are \(0 .\) $$\frac{2 x^{2}-8}{x^{2}-3 x+2}$$
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