Problem 77 - Elementary Algebra

Question

In 2012 , there were 360,000 Florida home loans in foreclosure. Find the number of Florida home loans. Round to the nearest hundred. With 14 percent of Florida home loans in foreclosure, abandoned houses are scattered across the state. (Source: www .nytimes.com, Feb. 23, 2012)

Step-by-Step Solution

Answer

There are approximately 2,571,400 Florida home loans.

Step 1 - Convert the Percentage to a Decimal

First, convert 14 percent to a decimal by dividing by 100. \( 14\text{ percent} = \frac{14}{100} = 0.14\)

Step 2 - Set Up the Equation

Let \( x \) represent the total number of Florida home loans. Given that 14 percent of these loans were in foreclosure, set up the equation: 0.14x = 360,000

Step 3 - Solve for x

To find \( x \), divide both sides of the equation by 0.14: \[ x = \frac{360,000}{0.14} \]

Step 4 - Calculate the Total Number of Loans

Perform the division to find the initial answer: \[ x = 2,571,428.5714 \]

Step 5 - Round to the Nearest Hundred

Round the number of home loans to the nearest hundred: \[ x \approx 2,571,400\]

Key Concepts

converting percentages to decimals

Understanding how to convert percentages to decimals is crucial for solving percentage problems in algebra. A percentage represents a part of 100, and to convert it to a decimal, you simply divide by 100. For instance, if you have 14%, you convert it to a decimal by performing the division: \( \frac{14}{100} = 0.14 \). This step is foundational because it transforms the percentage into a usable numerical format for algebraic equations.

Remember these quick tips:

• Move the decimal point two places to the left when converting from percentage to decimal.
• Always ensure the percentage value is divided by 100.

Practicing this conversion will help you tackle more complex percentage problems efficiently.

setting up algebraic equations

Setting up algebraic equations is like laying the groundwork for solving any algebra problem. It's about translating a word problem into a mathematical statement. In our exercise, we know that 14% of the total home loans are under foreclosure, and there are 360,000 such loans.

We let \( x \) represent the total number of Florida home loans. Since 14% of these loans are foreclosed, we set up the equation \( 0.14x = 360,000 \). This equation tells us that 14 percent of the total number of loans equals 360,000.

Clearly identify your variables and translate the given information into a structured equation. This practice makes solving the problem straightforward and logical.

solving for variables

Once you have your algebraic equation set up, the next step is solving for the variable. In our case, the variable \( x \) represents the total number of Florida home loans. We need to isolate \( x \) on one side of the equation to find its value.

From the equation \( 0.14x = 360,000 \), we solve for \( x \) by dividing both sides by 0.14:
\[ x = \frac{360,000}{0.14} \]
This division gives us \( x = 2,571,428.5714 \). Solving for variables requires practicing basic algebraic operations like multiplication, division, addition, and subtraction. Become familiar with these operations; they will help you solve for unknowns in any equation.

rounding numbers

Rounding numbers is the final refinement step in mathematical problems, ensuring results are presented in a simple and concise manner. In our case, the total number of Florida home loans calculated to be \( 2,571,428.5714 \) should be rounded to the nearest hundred.

To round this number:

• Look at the digit in the tens place. If it’s 5 or higher, round up.
• If it’s less than 5, round down.

For \( 2,571,428.5714 \), we look at the 4 in the tens place and round down:
\[ x \text{ becomes } \text{ approximately } 2,571,400 \].

Rounding helps in making long decimal numbers more readable and interpretable.