Problem 30 - Precalculus

Question

Skippy has a total of \(\$ 10,000\) to split between two investments, One account offers \(3 \%\) simple interest, and the other account offers \(8 \%\) simple interest. For tax reasons, he can only earn \(\$ 500\) in interest the entire year. How much money should Skippy invest in each account to earn \(\$ 500\) in interest for the year?

Step-by-Step Solution

Answer

Skippy should invest \(\$6,000\) at 3% and \(\$4,000\) at 8%.

Step 1: Define Variables

Let \( x \) be the amount of money invested in the 3% interest account, and \( y \) be the amount of money invested in the 8% interest account.

Step 2: Set Up Equations

From the problem statement, we know two equations: \( x + y = 10,000 \) (total investment equation) and \( 0.03x + 0.08y = 500 \) (interest earned equation).

Step 3: Solve for One Variable

From the equation \( x + y = 10,000 \), we can express \( y \) in terms of \( x \): \( y = 10,000 - x \).

Step 4: Substitute and Simplify

Substitute \( y = 10,000 - x \) into the interest equation: \( 0.03x + 0.08(10,000 - x) = 500 \). Simplify this to \( 0.03x + 800 - 0.08x = 500 \).

Step 5: Combine Like Terms

Combine the \( x \) terms: \( -0.05x + 800 = 500 \).

Step 6: Isolate Variable

Subtract 800 from both sides to get \( -0.05x = -300 \), then divide by \(-0.05\) to find \( x \): \( x = 6,000 \).

Step 7: Find Second Variable

Using \( y = 10,000 - x \), substitute \( x = 6,000 \) to find \( y = 10,000 - 6,000 = 4,000 \).

Step 8: Verify Solution

Check if \( x = 6,000 \) and \( y = 4,000 \) satisfy the interest equation: \( 0.03(6,000) + 0.08(4,000) = 180 + 320 = 500 \), which confirms the solution.

Key Concepts

Investment Allocation

When it comes to financial planning, understanding investment allocation is crucial. In simple terms, investment allocation is about deciding how much money to put into different types of investments or accounts.
In Skippy's case, he has a total of $10,000 to invest, and he needs to decide how to divide this amount between two bank accounts with different interest rates. One account offers an interest rate of 3%, and the other offers an 8% interest rate. The goal here is to balance the amount of money in these two accounts so that the total interest for the year is exactly $500.
Effective investment allocation helps maximize returns while meeting certain constraints, like Skippy's tax condition. Considering different interest rates and figuring out how much to put in each account can optimize the gains, adhere to financial rules, or, as in Skippy's case, meet tax restrictions without exceeding the interest earnings limit.

System of Equations

To solve Skippy's investment problem, we use a mathematical method called a system of equations. A system of equations is a set of equations with multiple variables that you solve together.
In this problem, we have two equations based on the information given:

• \( x + y = 10,000 \)
• \( 0.03x + 0.08y = 500 \)

The first equation ensures that the total investment is the entire \(10,000. The second equation ensures that the total simple interest earned is exactly \)500.
Solving these equations together allows us to find exactly how much Skippy should invest in each account to satisfy both conditions. A system of equations helps simplify complex decisions into actionable numerical outcomes when dealing with such allocation problems.

Linear Equations

Linear equations are the backbone of many mathematical calculations, including solving investment problems like Skippy's. A linear equation is an equation whose graph forms a straight line in a coordinate plane. These equations involve variables raised only to the first power.
In Skippy's problem, both:

• \( x + y = 10,000 \)
• \( 0.03x + 0.08y = 500 \)

represent linear relationships.
By manipulating and solving these linear equations, we can find explicit values for the variables \(x\) and \(y\), which represent the amounts to deposit in the 3% and 8% accounts, respectively. Linear equations make it possible to determine the exact distribution of funds across accounts.

Financial Mathematics

Financial mathematics provides the tools to solve real-world investment problems. In this context, it's all about using mathematical principles to analyze financial data and make informed investment decisions.
Skippy's scenario is a classic example of employing financial mathematics to determine the optimal distribution of funds across different investment types. By computing simple interest and setting equations based on financial constraints, we use math to solve practical allocation issues.

• The simple interest formula \( I = P \times r \times t \) helps calculate the interest generated from a certain principal amount, rate, and time.
• The strategic use of equations makes it easier to figure out the right mix of investments required to fulfill specific financial goals.

Through these methods, financial mathematics transforms abstract numbers into actionable solutions, aiding people like Skippy in making sound investment decisions. Understanding these concepts empowers individuals to manage personal finances with confidence.