Problem 27 - Precalculus Mathematics for Calculus

Question

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: Write the Equation

Let's denote the money Craig inherits as \( x \). He already has \\(22,000 saved. After inheriting, his total savings become \( x + 22,000 \). This amount is then doubled due to a lucky investment, resulting in \( 2(x + 22,000) \). He ends up with \\)134,000, so we can set up the equation: \( 2(x + 22,000) = 134,000 \).

Step 2: Simplify the Equation

First, let's simplify the equation by dividing both sides by 2 to isolate the expression for his total savings before the investment: \[ x + 22,000 = \frac{134,000}{2} \] which simplifies to \[ x + 22,000 = 67,000 \].

Step 3: Solve for x

To find out how much Craig inherited, we need to solve for \( x \). Subtract \$22,000 from both sides of the equation: \[ x = 67,000 - 22,000 \].\ This results in \( x = 45,000 \).

Step 4: Verify the Solution

To verify, if Craig inherits \\(45,000, his total savings before the investment would be \\)67,000 (\\(45,000 + \\)22,000). Doubling this amount results in \$134,000, which matches the amount he ends up with. So, our solution is correct.

Key Concepts

Linear Equations

Linear equations are a foundational concept in algebra. They involve relationships where the highest exponent of the variable is one, making them straightforward to solve. For Craig's problem, the linear equation models the situation where we have an unknown amount he inherits and known changes to this amount over time.

The equation we used is in the form \(2(x + 22,000) = 134,000\). This symbolizes:

• Craig's inheritance, represented as \(x\).
• The savings he already has, which is \(22,000\).
• The outcome from these after a crucial investment, resulting in \(134,000\).

To solve, we organized and modified the equation by simplifying both sides, eventually isolating the variable \(x\). This helped us determine exactly how much Craig inherited. Such equations showcase how algebra can help decode real-life situations into solvable steps.

Investment Problems

Investment problems in algebra often involve determining the growth or change of an amount due to certain conditions or rates. In this example, after inheriting a certain amount, Craig doubled his total savings. This symbolizes a fundamental investment principle, where an initial amount is multiplied by a certain rate (in this case, 2) to achieve a future sum.

Here's how you can think of this problem:

• Start with an initial amount, which is the inheritance plus current savings.
• Apply the investment factor—in Craig's case, a doubling effect.
• Match this outcome to a future target amount (\(134,000\)).

By setting these elements into a linear equation, algebra allows us to track and solve for unknowns like the initial inherited amount, providing clarity on investment outcomes. Such exercises help build a deeper understanding of how money can grow with strategic decisions.

Savings Calculations

Calculating savings is a real-world skill that can be cleverly tackled using algebra. In scenarios like Craig's, it's not just about the amount saved, but how those savings are affected by additions and multipliers.

Here's how to break it down:

• Begin with known savings—in Craig's case, \(22,000\).
• Add new funds—like the inheritance he's about to receive.
• Apply any growth factor—such as doubling savings through an investment.
• Use these changes to find a balance that matches your goal (\(134,000\) in Craig's story).

Through each of these steps, algebra helps us understand the movement of money, laying out a path to track how savings grow and reach specific goals. By understanding how to manipulate and solve for unknowns, you sharpen your ability to plan financial futures.