Problem 17 - Prealgebra

Question

Solve each of these problems using the method developed in this section. In 2006 the average price of a home began to fall in most real estate markets across the country. The median price of a single family home in the U.S. was \(\$ 227,000\) in \(2006 .\) The median price is now \(\$ 195,500 .\) By what percent did the median price of a single family home drop? Round your answer to the nearest tenth of a percent.

Step-by-Step Solution

Answer

The median price dropped by 13.9%.

Step 1: Understand the Problem

We need to find the percentage decrease in the median price of a single-family home from 2006 to now. The formula for percentage decrease is:\[ \text{Percentage Decrease} = \left(\frac{\text{Old Price} - \text{New Price}}{\text{Old Price}}\right) \times 100 \]

Step 2: Substitute the Given Values

Substitute the old price and new price into the percentage decrease formula. Here, the old price is \(\\( 227,000\) and the new price is \(\\) 195,500\):\[ \text{Percentage Decrease} = \left(\frac{227,000 - 195,500}{227,000}\right) \times 100 \]

Step 3: Calculate the Price Difference

Subtract the new price from the old price to find the drop in price:\[ 227,000 - 195,500 = 31,500 \]

Step 4: Solve for Percentage Decrease

Use the price difference to find the percentage decrease:\[ \text{Percentage Decrease} = \left(\frac{31,500}{227,000}\right) \times 100 \]

Step 5: Perform the Division and Multiplication

Calculate the division and multiply by 100:\[ \frac{31,500}{227,000} \approx 0.1387 \]\[ 0.1387 \times 100 = 13.87 \]

Step 6: Round to the Nearest Tenth

Round 13.87 to the nearest tenth of a percent, which gives us:\[ 13.9\% \]

Key Concepts

Understanding Percent Change

Percent change is a way to express the difference between two values as a percentage. This concept is very useful when comparing quantities over time, like prices or population growth. The formula to calculate percent change is:

• Percent Change = \(\frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\)
• If the result is positive, it indicates a percent increase.
• If negative, it shows a percent decrease.

In real estate, checking the percent change in home prices can inform buyers and sellers about market trends. The calculation helps them understand if the market is gaining or losing value over time. Knowing how to calculate this percent change is essential.
It allows stakeholders to make informed decisions based on how much prices have fluctuated compared to previous periods.

Explaining Median Price

The median price of homes provides a clearer picture of the typical value of properties than the average price might. Here's why:

• Definition: The median is the middle value in a list of numbers. In the context of home prices, it represents the price point at which half the homes are sold for less and half for more.
• Avoids Skewing: Unlike an average, the median isn't skewed by extremely high or low prices. It gives a better indication of what a typical buyer might expect to pay.

Knowing the median price can help buyers and sellers understand market conditions, such as whether the market is dominated by high-end or lower-end properties. Regularly updated median prices can also offer insights into broader economic conditions, reflecting trends like inflation or deflation in the housing market.

Following a Step-by-Step Solution

A step-by-step solution is an excellent approach for thoroughly understanding problem-solving methods. It breaks the problem into manageable pieces, making it easier to follow and less overwhelming.

• First, read and understand the problem. Identify what you want to find and what information is given.
• Second, substitute the known values into the appropriate formula, like the percent change formula in our exercise.
• Third, perform the calculations systematically. Handle one operation at a time to avoid errors.

In the context of our exercise, we moved from understanding the drop in home prices to calculating the exact percentage of decrease. By following a detailed, logical sequence of steps, you not only solve the problem but also understand the reasoning behind each step. This method builds confidence and skill in solving similar problems in the future.