Problem 19 - 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: Define Variables

Let's define the variables involved in the problem. Let \( x \) represent the amount of money Craig inherited from his uncle.

Step 2: Set Up the Equation

According to the problem, Craig combines his inheritance \( x \) with \( \\(22,000 \) he already saved, then doubles this amount through an investment. Finally, he ends up with \( \\)134,000 \). This information can be expressed with the following equation: \[ 2(x + 22,000) = 134,000 \].

Step 3: Simplify the Equation

First, divide both sides of the equation by 2 to simplify it: \[ x + 22,000 = 67,000 \].

Step 4: Solve for x

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

Step 5: Calculate the Inherited Amount

Perform the subtraction to find the value of \( x \): \[ x = 45,000 \]. Hence, Craig inherited \( \$45,000 \).

Key Concepts

Algebraic Equation

Algebraic equations are mathematical statements where two expressions are set equal to each other. These equations involve variables (like \(x\) in this case) that represent unknown values we need to find.
There are many types of algebraic equations, but they all aim to establish relationships between different quantities. In this exercise, the relationship is between Craig's inherited amount and the total amount available after his investment. An algebraic equation allows us to work backward from known outcomes to discover unknown initial values.
By defining a variable for the unknown value, an algebraic equation is a powerful tool for finding solutions to real-world problems. In Craig’s case, the equation \(2(x + 22,000) = 134,000\) was used to determine how much money he inherited.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. Linear equations are a specific type of algebraic equation where each term is either a constant or a product of a constant and a single variable, and the variable is raised to the power of one.
For the inheritance problem, we know Craig’s savings equation had a linear component: \(x + 22,000\). The whole equation \(2(x + 22,000) = 134,000\) requires steps to isolate \(x\).

• First, simplify the equation. Multiply or divide both sides to remove any constants affecting the group. In this case, divide both sides by 2: \(x + 22,000 = 67,000\).
• Once simplified, isolate \(x\) by performing inverse operations. Subtract \(22,000\) from both sides to find \(x = 45,000\).

Solving linear equations sequentially helps in understanding how each action transforms the equation and leads to the solution.

Mathematical Reasoning

Mathematical reasoning is the logical thinking process needed to make sense of a problem and figure out how to solve it. It allows you to move from a real-world outline, with numbers and known outcomes, to a step-by-step solution using mathematical principles.
In the current exercise, mathematical reasoning starts by understanding the problem: Craig saves, inherits, and invests. The findings lead to translating this story into mathematical form.
The reasoning involves:

• Identifying what is known - e.g., total savings, initial savings, and outcome after investment.
• Defining the unknown in terms of a variable - \(x\) for inheritance.
• Constructing an equation based on these relationships - symbolically summarizing the situation.
• Performing systematic operations to reveal the unknown.

A strong foundation in mathematical reasoning enhances your ability to tackle complex problems by breaking them into simple, manageable parts.