Question

For her first 3 years with a company, Marissa earned $$\$ 45,200$$ each year. For the next 5 years she earned $$\$ 55,400$$ per year. What was the average amount that Marissa earned per year over the 8-year period? A. $$\$ 49,075$$ B. $$\$ 50,300$$ C. $$\$ 51,575$$ D. $$\$ 52,100$$

Short answer

\(Total \ earnings \ for \ first \ 3 \ years = (45,200) \times (3) = 135,600\) \(Total \ earnings \ for \ next \ 5 \ years = (55,400) \times (5) = 277,000\) \(Total \ earnings \ over \ 8 \ years = (135,600) + (277,000) = 412,600\) \(Average \ annual \ earnings = \frac{412,600}{8} = 51,575\) The correct answer is C. \( \$ 51,575\)

Step-by-step solution

Calculate earnings for the first 3 years

First, we need to find the total earnings for Marissa during her first 3 years with the company. Since she earned $$\$ 45,200$$ each year, we can calculate her total earnings by multiplying her annual earnings by 3: Total earnings for first 3 years = (Annual earnings) × (Number of years) = ($$45,200$$) × (3)

Calculate earnings for the next 5 years

Next, we need to find the total earnings for Marissa during the next 5 years with the company. Since she earned $$\$ 55,400$$ each year, we can calculate her total earnings by multiplying her annual earnings by 5: Total earnings for next 5 years = (Annual earnings) × (Number of years) = ($$55,400$$) × (5)

Calculate total earnings over 8 years

Now that we have the total earnings for both periods, we can find the overall total earnings for Marissa over the 8-year period by adding the earnings from the two periods: Total earnings over 8 years = (Total earnings for first 3 years) + (Total earnings for next 5 years)

Calculate the average annual earnings

To find the average amount Marissa earned per year, we will divide her total earnings over the 8-year period by the number of years, 8: Average annual earnings = (Total earnings over 8 years) / (Number of years) = (Total earnings over 8 years) / (8)

Compare the result with the given options

Finally, we will compare our calculated average annual earnings with the given options (A, B, C, and D) to identify the correct answer.

Key concepts

Average Annual Earnings Calculation

Understanding how to calculate average annual earnings is crucial for various financial assessments, including salary negotiations, job evaluations, and personal financial planning. In the context of a GED Mathematics test, mastering this calculation can often solve questions related to income over different periods.

To calculate average annual earnings, you must total all the earnings received over a given period and then divide by the number of years in that period. This concept is rooted in the mean, which is a basic statistical measure representing the central value of a data set.

Using Marissa's example, we combine her earnings from different periods of her employment to compute her total earnings. Then, dividing by the total number of years she's been working, we get her average annual earnings. This method shows students the practical use of averages and provides them with skills they can use beyond the classroom.

GED Prep Math Problems

Preparing for the GED Mathematics test involves tackling a variety of math problems that test your quantitative reasoning and problem-solving abilities. Problems like calculating average annual earnings are typical on the GED, as they mix real-world scenarios with the need to apply basic mathematical operations.

To effectively prepare for these problems, it's important to understand not just the basic math operations—addition, subtraction, multiplication, and division—but also how to apply them in practical contexts. Practice problems and textbooks often frame these operations within everyday situations to make them more relatable and to improve understanding.

For GED prep, focusing on step-by-step solutions encourages a thorough understanding of the process behind each answer, which is more beneficial than simply memorizing formulas or numbers. It's about cultivating a mindset of analytical thinking and logical reasoning, which is crucial for success on the GED and future mathematical challenges.

Mathematical Problem-Solving

Mathematical problem-solving entails more than just performing calculations; it's about understanding the situation, figuring out which mathematical operations to use, and applying these operations in the right order. It's a core competency for educational success and an essential skill in many career paths.

In solving the example problem, the process begins with a clear comprehension of the problem statement. Next, we break down the problem into smaller, manageable parts—calculating Marissa's earnings in different intervals and then combining these to find a total. The final step involves averaging this total over the years worked.

Key strategies in mathematical problem-solving include identifying relevant information, selecting the right tools and formulas, and checking whether the solution makes sense in the context of the problem. Encouraging students to explain their reasoning, as done in a step-by-step approach, solidifies their understanding and equips them with the ability to tackle a wide range of mathematical problems.