Problem 27 - Algebra and Trigonometry

Question

Inheritance Craig is saving to buy a vacation home. He inherits some money from a wealthy uncle, then combines this with the \(\$ 22,000\) he has already saved and doubles the total in a lucky investment. He ends up with \(\$ 134,000-\) just enough to buy a cabin on the lake. How much did he inherit?

Step-by-Step Solution

Answer

Craig inherited $45,000.

Step 1: Set up the Equation

Let's denote the amount Craig inherited as \( x \). The total amount of money Craig had after combining his savings and inheritance is \( x + 22,000 \). This amount was then doubled.

Step 2: Express the Doubled Amount

After doubling the combined amount of savings and inheritance, Craig had \( 2(x + 22,000) \). This total amount equals \( 134,000 \) dollars, so we can set up the equation: \[ 2(x + 22,000) = 134,000 \]

Step 3: Simplify the Equation

Divide both sides of the equation by 2 to solve for \( x + 22,000 \): \[ x + 22,000 = \frac{134,000}{2} \]Which simplifies to:\[ x + 22,000 = 67,000 \]

Step 4: Solve for Inheritance

Subtract \( 22,000 \) from both sides to solve for \( x \):\[ x = 67,000 - 22,000 \]\[ x = 45,000 \]

Key Concepts

Solving Equations

Equations are wonderful tools in mathematics that allow us to represent real-world problems in a structured way. They consist of expressions set equal to each other. In essence, when we solve an equation, we are finding the value of the variable that makes this equality true. For example, in Craig's inheritance problem, the variables and equations help us to calculate exactly how much he inherited.To solve these types of problems, think of the equation as a balanced scale.

• You perform the same operation on both sides to maintain balance.
• Operations can include addition, subtraction, multiplication, or division.

Breaking down Craig's problem, the first step is setting up an equation: \[ 2(x + 22,000) = 134,000 \]This equation represents the entire scenario, including inheritance and savings. Moving forward, simplifying the equation helps isolate the variable, making it easier to find its value. Each step requires precision, as any miscalculation can lead to incorrect results. Finally, solving for the variable requires logical progression of operations to determine that Craig inherited \(45,000\), showcasing the utility of solving equations in real applications.

Inheritance

Inheritance in this context refers to the money Craig received from his uncle. It's an example of external financial sources contributing to personal wealth. In Craig's situation, this inheritance was key to reaching his monetary goal of buying a vacation home.Let's delve into how inheritance might simplify financial concerns:

• It's often a one-time financial boost, not regularly planned income.
• Inheritance can dramatically alter financial plans and needs.

In the problem, Craig combined this inheritance with his own savings of \(22,000\), thereby increasing his total available funds. This combination is common in financial planning, where external funds are included in personal investments and savings strategies. Understanding how inheritance impacts goals helps in planning future financial actions—just like Craig did, using it to make a sound investment choice.

Investment Doubling

Investment doubling refers to an investment strategy where the returns from an investment amount to twice the original sum. It highlights both the potential for growth and the risks involved in market investments. For Craig, after combining his inheritance and savings, the decision to invest luckily doubled his total amount, taking advantage of market conditions to significantly boost his purchasing power. Here are some insights into investment doubling:

• It requires strategic investment choices with an understanding of risks and returns.
• While the term 'doubling' seems simple, achieving this requires patience or a favorable market.

Craig's scenario suggests that his doubling result was serendipitous, indicating high return on investment. The lesson here is that while doubling investments is possible, it's important to remember that market investments come with inherent risks and are not always predictable. Understanding this concept is crucial for anyone looking to invest and potentially double their money, just as Craig did.