Question

For Exercises \(1-20,\) solve. In \(1990,\) Karen bought a collection of five paintings by a relatively unknown artist. Three of the paintings cost \(\$ 250\) each, and the other two cost \(\$ 400\) each. In 2010 , she put the paintings up for auction and they sold for \(\$ 1500 .\) What was Karen's net? Was it a profit or loss?

Short answer

Answer: Karen had a net loss of $50.

Step-by-step solution

Calculate the initial cost of the paintings

To find the initial cost of the paintings in 1990, calculate the cost of three paintings at \(\$250\) each and two paintings at \(\$400\) each. Add these amounts together to get the total initial cost. Initial cost = \((3 * 250) + (2 * 400)\)

Calculate the total initial cost

Now we will calculate this value: Initial cost = \((3 * 250) + (2 * 400) = 750 + 800 = 1550\) The initial cost for Karen was \( \$ 1550.\)

Calculate the net gain or loss

We can find out her net by finding the difference between her initial cost and the selling price in 2010. Net gain/loss = Selling price - Initial cost Net gain/loss = \(1500 - 1550\)

Determine the profit or loss

Now we need to calculate the value of the net, Net gain/loss = \(1500 - 1550 = -50\) Since the net gain/loss is a negative value, Karen had a loss: Karen's net loss = \(\$50\) The answer is that Karen had a net loss of \( \$ 50.\)

Key concepts

Initial Cost Calculation

Calculating the initial cost of an item or collection is a key step in understanding your financial position. It involves adding up all the expenses incurred to acquire the items initially. In Karen's case, she needed to calculate the cost of the five paintings she bought in 1990.
To find this initial cost, Karen followed a straightforward method:

• She had three paintings that each cost \( \\(250 \), so she multiplied \( 3 \times 250 \), resulting in \( \\)750 \).
• The other two paintings cost her \( \\(400 \) each, so she multiplied \( 2 \times 400 \), which gave her \( \\)800 \).
• By adding the two amounts together, \( 750 + 800 \), Karen calculated her total initial cost as \( \$1550 \).

This total is crucial for any further financial calculations, as it sets the base against which future gains or losses are measured.

Net Gain or Loss

Calculating net gain or loss is an important element in evaluating the success of an investment. Once you have the initial cost, you can determine how much money has been gained or lost by subtracting this cost from the final selling price. For Karen:

• The paintings were sold for \( \\(1500 \) in 2010.
• She subtracted her initial cost, \( \\)1550 \), from the selling price, which is \( 1500 - 1550 \).
• The result was \(-\$50\), indicating a net loss.

Remember, a positive result here would signify a gain, whereas a negative result indicates a loss. This calculation helps assess whether an investment was profitable or not, guiding future investment decisions.

Selling Price vs. Initial Cost

Understanding the difference between the selling price and the initial cost is essential for assessing any transaction. In Karen's situation, the selling price of her paintings did not exceed their initial cost.

• Initially, the paintings' combined purchase cost \( \\(1550 \).
• When sold, they fetched \( \\)1500 \).
• Because \( 1500 < 1550 \), this means the selling price was less than the initial cost.

This difference can drastically affect whether you make a profit or incur a loss. A basic understanding of these simple comparisons helps to make informed financial judgments efficiently.

Mathematical Reasoning in Finance

Mathematical reasoning is a powerful tool in finance, helping individuals make informed decisions about investments. By applying it to Karen’s auction scenario, we see how basic arithmetic operations can illuminate financial outcomes.
The steps involved in Karen's situation included:

• First, calculating total purchase costs by multiplication and addition to find the initial cost.
• Then, using subtraction to compare that cost with the selling price.
• Finally, interpreting the result to indicate gain or loss.

Karen's case illustrates how critical these mathematical concepts are in real-world financial analysis. Such reasoning isn't just about numbers—it's about enhancing economic understanding and improving financial strategies.