Question

Translate to an equation and solve. The Ramos family has \(\$ 45,000\) in home equity. If this represents \(18 \%\) of the value of the home, how much is the home worth?

Short answer

Answer: The total value of the home is $250,000.

Step-by-step solution

Define variables and equation

Let x represent the total value of the home. We know that the home equity ($45,000) is 18% of the total value (x). So, we can create the following equation: $$ 0.18x = 45000 $$

Solve for x

To find the value of x, we simply need to divide both sides of the equation by 0.18: $$ x = \frac{45000}{0.18} $$

Find the total value of the home

Now, we can calculate the value of x: $$ x = \frac{45000}{0.18} = 250000 $$ So, the total value of the home is \( \$ 250,000\).

Key concepts

Percent to Decimal Conversion

When working with percentages, it's often necessary to convert them into decimal form to perform calculations. Percent means 'per hundred', so converting a percent to a decimal is as simple as dividing by 100. This can be done by moving the decimal point two places to the left.

For example, to convert 18% into a decimal, you divide 18 by 100, which is equivalent to shifting the decimal two places to the left resulting in 0.18. This conversion is essential before using the percentage in any kind of equation or formula. Keeping this in mind ensures accurate calculations, especially when dealing with financial matters like home equity.

To further simplify understanding, consider these conversions:

• 50% becomes 0.50
• 25% becomes 0.25
• 75% becomes 0.75

Remembering that percentages are just fractions of 100 can make this process intuitive and straightforward.

Equation Solving

Solving equations is a fundamental part of algebra that involves finding the value for a variable that makes the equation true. The key is to perform the same operations on both sides of the equation to maintain its balance. In the example from our exercise, you start with the equation representing the relationship between the home equity and the home's total value.

When you have an equation like \( 0.18x = 45000 \), your goal is to isolate the variable, \( x \), which represents the unknown value you're solving for. By dividing both sides by 0.18, you effectively reverse the multiplication of \( x \) by 0.18, leaving \( x \) on its own. This step, called isolating the variable, is at the heart of solving most algebraic equations.

Once isolated, you can calculate the value of \( x \), giving you the answer you need. Ensuring you correctly follow the order of operations and use proper algebraic manipulation is crucial for finding the correct solution.

Home Equity Calculation

Home equity is the share of a property's value that the owner actually owns. It is calculated by subtracting any outstanding loan balances from the property’s market value. In many common scenarios, such as the one presented in the exercise, you're given the amount of home equity and a percentage that it represents of the home's total value.

To find the total value of the home, you use the formula:

• Equity Value = (Percentage of Home Owned as a Decimal) × Total Home Value

From this relationship, if you're solving for the Total Home Value, you can rearrange the formula and divide the Equity Value by the percentage (in decimal form).

In the exercise, with an equity value of \( \(45,000 \) representing 18% of the home, you find the home’s worth by dividing \( \)45,000 \) by 0.18. This type of calculation is crucial for homeowners who need to understand their financial stake in their property, for purposes such as selling the home, applying for home equity loans, or for individual net worth assessments.