Question

For Exercises \(1-20,\) solve.For Exercises \(1-20,\) solve. In \(1998,\) Mario purchased a collection of seven pieces of antique furniture. Four of the pieces cost \(\$ 1200\) each, and the other three cost \(\$ 1500\) each. In 2010 , he sold the entire collection at an auction for \(\$ 8600 .\) What was Mario's net? Was it a profit or loss?

Short answer

Answer: Mario faced a loss of $700.

Step-by-step solution

Calculate the total cost of the furniture in 1998

Four pieces cost \(\$1200\) each, so their total cost is \(4 \times 1200 = \$4800\). Three other pieces cost \(\$1500\) each, so their total cost is \(3 \times 1500 = \$4500\). Now, add these two amounts to get the total cost: \(4800 + 4500 = \$9300\).

Calculate the net profit or loss

The selling price of the furniture in 2010 was \(\$8600\). To find the net profit or loss, we subtract the total cost from the selling price: \(8600 - 9300 = -\$700\).

Determine if it is a profit or loss

Since the net value calculated is negative, this means Mario faced a loss of \(\$700\).

Key concepts

Basic Algebra

Basic algebra is a fundamental aspect of mathematics that involves the use of symbols and letters to represent numbers and quantities in formulas and equations. The beauty of algebra lies in its ability to solve problems by manipulating these symbols following specific rules and operations such as addition, subtraction, multiplication, and division.

In the context of our exercise with Mario's furniture, basic algebra comes into play when combining the costs of individual items to calculate the total cost. For instance, when we see that four pieces cost \(\$1200\) each, we perform a multiplication:\( 4 \times 1200 \). This usage of algebra simplifies the process of working with several quantities at once, rather than adding up \(\$1200\) four separate times.

Profit and Loss Concepts

Understanding the profit and loss concepts is crucial for anyone involved in financial transactions. Profit occurs when the selling price of an item exceeds the cost price, while a loss happens when the cost price is higher than the selling price.

To calculate Mario’s net result, we compare the original cost of purchasing the furniture to the final selling price. A profit or loss is determined by the difference between these two figures. This calculation is made simple using the formula:\[ \text{Net Profit (or Loss)} = \text{Selling Price} - \text{Total Cost} \]. If the result is positive, it's a profit; if it's negative, that indicates a loss. As it was in Mario's case, where he suffered a loss since his selling price was lower than the purchase price.

Mathematical Problem Solving

Mathematical problem solving is a systematic process that involves understanding the problem, devising a plan, carrying out that plan, and then revisiting the problem to check the results. The exercise with Mario’s antique furniture transaction showcases how this approach is applied.

The understanding phase involved recognizing that we need to find the cost of each group of furniture and their sum to get the total cost. The planning phase included deciding to use multiplication and addition for these calculations. Execution of the plan was performing the calculations, and finally, verifying the result meant comparing the selling price with the total cost to determine if there was a profit or loss. Such a methodical approach ensures thoroughness and accuracy in solving mathematical problems, and particularly so in real-world financial situations such as this.