Calculating Marcus's Earnings An Algebraic Expression Problem

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In this article, we'll delve into a mathematical problem involving calculating earnings and expenses. The scenario revolves around Marcus, who works on commission selling electrical appliances and incurs daily lunch expenses. We will break down the problem step-by-step, identify the key components, and construct an algebraic expression to represent Marcus's daily earnings. Understanding how to formulate such expressions is crucial for various real-world applications, such as personal finance, business calculations, and economic analysis. This exercise will help you sharpen your analytical and problem-solving skills, particularly in the realm of algebraic representation.

Marcus earns a commission of 110\frac{1}{10} of the price of every electrical appliance he sells. He also spends $10 on lunch each day. If he sells goods worth xx dollars in a day, which expression represents the amount of money Marcus has left after lunch?

To solve this problem, we need to identify the different components that contribute to Marcus's daily financial situation. These components include:

  • Commission Earnings: Marcus earns a commission based on the total value of the electrical appliances he sells.
  • Commission Rate: The commission rate is the fraction of the sales value that Marcus earns as commission.
  • Total Sales Value: This is the total dollar amount of the goods Marcus sells in a day, which is represented by the variable xx.
  • Lunch Expenses: Marcus incurs a fixed daily expense for lunch.

By identifying these components, we can break the problem down into smaller, more manageable parts. This approach will make it easier to construct an algebraic expression that accurately represents Marcus's daily earnings after accounting for lunch expenses.

1. Commission Earnings Calculation

The first step is to calculate Marcus's commission earnings. We know that he earns a commission of 110\frac{1}{10} of the price of every electrical appliance he sells. The total value of the goods he sells in a day is represented by xx dollars. Therefore, to find his commission earnings, we need to multiply the commission rate by the total sales value.

Mathematically, this can be expressed as:

Commission Earnings=Commission Rate×Total Sales Value\text{Commission Earnings} = \text{Commission Rate} \times \text{Total Sales Value}

In this case, the commission rate is 110\frac{1}{10}, and the total sales value is xx dollars. Substituting these values into the equation, we get:

Commission Earnings=110×x\text{Commission Earnings} = \frac{1}{10} \times x

This can be simplified as:

Commission Earnings=x10\text{Commission Earnings} = \frac{x}{10}

So, Marcus's commission earnings for the day are x10\frac{x}{10} dollars. This is the amount he earns before accounting for his lunch expenses.

2. Lunch Expenses

The second component we need to consider is Marcus's lunch expenses. The problem states that he spends $10 on lunch each day. This is a fixed expense, meaning it does not depend on the value of the goods he sells. Therefore, we can directly subtract this amount from his commission earnings to find his net earnings for the day.

3. Expression for Net Earnings

Now that we have calculated Marcus's commission earnings and identified his lunch expenses, we can construct an algebraic expression to represent the amount of money Marcus has left after lunch. To do this, we subtract his lunch expenses from his commission earnings.

We know that his commission earnings are x10\frac{x}{10} dollars, and his lunch expenses are $10. Therefore, the expression for his net earnings (the amount of money he has left after lunch) is:

Net Earnings=Commission Earnings−Lunch Expenses\text{Net Earnings} = \text{Commission Earnings} - \text{Lunch Expenses}

Substituting the values we found earlier, we get:

Net Earnings=x10−10\text{Net Earnings} = \frac{x}{10} - 10

This expression represents the amount of money Marcus has left after lunch on any given day, where xx is the total value of the goods he sells in dollars.

To determine the expression that represents the amount of money Marcus has left after lunch, we need to combine his commission earnings and his lunch expenses. We know that his commission earnings are 110\frac{1}{10} of the total sales value, which is represented by xx. This can be written as 110x\frac{1}{10}x or x10\frac{x}{10}.

Marcus also spends $10 on lunch each day. This is a fixed expense that needs to be subtracted from his commission earnings. Therefore, to find the amount of money Marcus has left after lunch, we subtract $10 from his commission earnings.

Combining these two components, we can write the expression as:

x10−10\frac{x}{10} - 10

This expression represents Marcus's net earnings for the day, taking into account his commission and his lunch expenses. The term x10\frac{x}{10} represents his commission earnings, and the term −10-10 represents his lunch expenses. The entire expression gives us the amount of money Marcus has left after deducting his lunch expenses from his commission earnings.

In conclusion, the expression that represents the amount of money Marcus has left after lunch is x10−10\frac{x}{10} - 10. This expression accurately captures Marcus's financial situation for the day, considering both his commission earnings and his lunch expenses. By breaking down the problem into smaller components and using algebraic representation, we were able to construct a clear and concise expression that answers the question. Understanding how to formulate such expressions is a valuable skill in mathematics and has practical applications in various real-world scenarios. This exercise demonstrates the importance of carefully analyzing the given information, identifying the key variables, and combining them in a logical manner to arrive at the correct solution.