Problem 22 - Introductory Algebra for College Students

Question

According to the American Bureau of Labor Statistics, you will devote 37 years to sleeping and watching TV. The number of years sleeping will exceed the number of years watching TV by 19. Over your lifetime, how many years will you spend on each of these activities?According to the American Bureau of Labor Statistics, you will devote 32 years to sleeping and eating. The number of years sleeping will exceed the number of years eating by 24 Over your lifetime, how many years will you spend on each of these activities?

Step-by-Step Solution

Answer

For the first problem, you will spend 28 years sleeping and 9 years watching TV. For the second problem, you will spend 28 years sleeping and 4 years eating.

Step 1: Establish the system of equations for the first problem

Given that the total time spent sleeping and watching TV = 37 years and the time spent sleeping exceeds the time spent watching TV by 19 years, the system of equations is established as follows: \[ x + y = 37 \] (where x represents sleeping and y represents watching TV) and \[ x = y + 19 \] (since the time sleeping is more than the time watching TV).

Step 2: Solve the system of equations for the first problem

Substitute equation 2 into equation 1 to get: \[ (y + 19) + y = 37 \]. Solving this equation results in y = 9. By substituting y = 9 into equation 2, x = 9 + 19 = 28.

Step 3: Establish the system of equations for the second problem

Given that the total time spent sleeping and eating = 32 years and the time spent sleeping exceeds the time spent eating by 24 years, the system of equations is established as follows: \[ a + b = 32 \] (where a represents sleeping and b represents eating) and \[ a = b + 24 \] (since the time sleeping is more than the time eating).

Step 4: Solve the system of equations for the second problem

Substitute equation 4 into equation 3 to get: \[ (b + 24) + b = 32 \]. Solving this equation results in b = 4. By substituting b = 4 into equation 4, a = 4 + 24 = 28.

Key Concepts

System of Equations

A system of equations is essentially a set of two or more equations that are related through shared variables. In this context, it helps us solve problems that involve multiple conditions. Let's walk through this process using the system of equations from the presented problems:

1. For the first problem, we were given two pieces of information: the total time spent on both sleeping and watching TV is 37 years, and sleeping exceeds watching TV by 19 years. With these conditions, the system of equations was formed:

• \[ x + y = 37 \]
• \[ x = y + 19 \]

2. Similarly, for the second problem, the system of equations was created using the statements that the combined years for sleeping and eating is 32, and sleeping and eating varies by 24 years.

• \[ a + b = 32 \]
• \[ a = b + 24 \]

The system of equations is a powerful tool in problem-solving as it allows us to find values of the unknowns (like years spent on activities) that meet all given conditions simultaneously.

Problem-Solving

The essence of problem-solving in mathematics is to understand the situation, use logical reasoning, and apply the right strategies to find a solution. Each step should contribute to cracking the underlying problem. In our case study, we approached it methodically.

1. **Understanding the Problem:** First, we identified the conditions given in the exercise—relationships and totals between different life activities like sleeping, watching TV, and eating.
2. **Developing a Strategy:** Next, we translated these relationships into a system of equations as it offered clarity by providing a structured way to handle the information.
3. **Executing the Plan:** Solving the equations step-by-step involved logical deductions and substitutions.

• For instance, substituting one equation into another was crucial in narrowing down the exact number of years for each activity.

4. **Evaluating Solutions:** Finally, we verified if the solutions found meet all the conditions initially set. If they do, the problem is solved. Otherwise, we may need to revise our interpretation or calculations.

Mathematical Modeling

Mathematical modeling involves transforming real-world problems into mathematical language, enabling us to analyze and solve them. In your exercise, we modeled the situations of daily life activities as equations.

1. **Identifying Variables:** Before diving into equations, we chose appropriate variables to represent the unknowns—like the number of years devoting to each activity. In this scenario, variables such as \( x, y, a, \) and \( b \) depicted years spent sleeping, watching TV, and eating, respectively.
2. **Formulating Equations:** Using the descriptions given in the problem, such as the difference in years and sum totals, we constructed algebraic equations that mirrored these real-life situations.
3. **Analyzing and Solving:** The mathematics behind interpreting and solving these equations helps ensure that the model provides realistic and accurate results for the initial query, aiding better decision-making or predictions about time allocation across various activities in one’s life. After the solutions were found, the model would then serve its purpose of inferring how our time might be distributed.