Problem 310 - Intermediate Algebra

Question

When the Szetos sold their home, the selling price was \(500 \%\) of what they had paid for the house 30 years ago. Explain what \(500 \%\) means in this context.

Step-by-Step Solution

Answer

500% means the selling price was 5 times the original price.

Step 1: Understand the Context

To understand what 500% means in this context, first recognize that percentages are a way to express a number as a fraction of 100.

Step 2: Identify the Original Price

Let the original price of the house 30 years ago be represented by the variable, say, \( P \).

Step 3: Calculate the Increase

500% means five times the original amount. Mathematically, this is represented as \( \frac{500}{100} \times P = 5P \).

Step 4: Determine the Selling Price

The selling price of the house is given as 500% of the original price. Therefore, \( 500\text{\textpercent} \text{ of } P = 5P \).

Step 5: Interpret the Meaning

In this context, 500% means that the selling price was five times the original price paid 30 years ago.

Key Concepts

percentage increase

To understand percentages, remember that they express a number as a part of 100. When the problem states a 500% increase, it means the value has multiplied by five. This multiplication is straightforward: if the initial value is represented as P, then a 500% increase translates mathematically as \( \frac{500}{100} = 5 \). Thus, the new value is five times the initial value.

original price

The original price is the starting point in any percentage calculation. In our context, let the original price of the house 30 years ago be P. This value P is essential because all percentage calculations revolve around this initial figure. If you know the original price, it's easy to derive other values, like the increased price, using percentages.

selling price

The selling price is the amount received after the original price has been increased by a certain percentage. Here, we know that the selling price is 500% of the original price. This means it is five times the original price, P. So, if the original price is P, the selling price is calculated as 5P. Always remember that selling price = original price + profit from increase.

mathematical interpretation

Mathematical interpretation involves translating word problems into mathematical equations. In this case, the statement 'selling price was 500% of the original price' converts to \( \text{Selling Price} = 5P \). This tells us that whatever the initial price P was, the selling price is five times larger due to the 500% increase. Equations like \ 5P \ help to clarify how much the price has grown by showing precise relationship between the original and new values.