Problem 10 - Intermediate Algebra

Question

a. The number of decades, is, the number of years, divided by 10. b. The cost of dining out, equals, the cost of the meal, plus \(\$ 15\) for parking.

Step-by-Step Solution

Answer

To find decades, divide years by 10. Total dining cost is meal cost plus $15 for parking.

Step 1: Understanding the Concept of Decades

A decade is a period of 10 years. Therefore, to find the number of decades from a given number of years, you simply divide the number of years by 10.

Step 2: Formula for Decades

Mathematically, the number of decades can be expressed as: \[ \text{Decades} = \frac{\text{Years}}{10} \]

Step 3: Understanding Dining Out Costs

When calculating the total cost of dining out, you need to include both the cost of the meal and an additional fixed cost for parking, which is $15.

Step 4: Formula for Total Cost of Dining

The total cost can be expressed as: \[ \text{Total Cost} = \text{Cost of Meal} + 15 \]

Key Concepts

Decades Calculation

Understanding how to calculate decades is straightforward and can be a handy skill in various situations. A decade, by definition, is a span of 10 years. This concept is grounded in the idea of breaking down large numbers into smaller, more manageable pieces.

When you have a number representing years and you want to know how many complete decades are included in that time span, you divide the years by 10. For example, if you have 45 years, dividing by 10 gives you 4.5 decades. This suggests there are 4 full decades and another half decade (5 years).

The mathematical expression for this is:

• \[ ext{Decades} = \frac{\text{Years}}{10} \]

This simple calculation helps in various fields such as history, project planning, and even personal life reflections, allowing you to quickly convert and understand time spans in a decade format.

Cost Calculation

Calculating costs is an essential part of managing finances, whether for personal budgets, business expenses, or simply planning a day out. This concept can involve a basic operation where you sum up various smaller expenses to find the total.

For instance, when you dine out, expenses usually include more than just the meal's price. A common additional cost is parking, which can be a fixed amount, such as $15. Therefore, to calculate the total expenditure for dining out, you would add the cost of the meal to the parking fee.

The formula is as follows:

• \[ ext{Total Cost} = \text{Cost of Meal} + 15 \]

This formula provides a quick and easy way to ensure you have accounted for all costs involved. By understanding these basic cost calculations, you can avoid surprises in your expenses and manage your finances more effectively.

Algebraic Expressions

Algebraic expressions form the building blocks of algebra, a branch of mathematics dealing with symbols and the rules for manipulating those symbols. Expressions can represent numbers, variables, operations, and operators.

An algebraic expression might involve variables that stand for numbers. These expressions can be as simple as \(x + 5\) or more complex like \(3x^2 + 2x - 7\). The key is understanding how these components work together to represent quantities and solve problems.

In practical applications like those in our exercise, algebraic expressions help structure calculations logically. For example, the expressions used to calculate decades and costs represent numerical relationships that allow for straightforward evaluation. Learning to work with these expressions is crucial, as they form the foundation for more advanced mathematical analysis, aiding in both problem-solving and strategic planning.