Problem 65 - A Graphical Approach to Precalculus with Limits

Question

Investment problems such as those in Exercises \(61-66\) can be solved by using a method similar to the one explained in Example \(2,\) along with the simple- interest formula \(I=P R T\). where I is the interest earned, \(P\) is the initial amount of money deposited, \(R\) is the annual interest rate as a decimal, and \(T\) is the time the money is deposited in years. Solve each problem. Let \(T=1\) year for each exercise. Lottery Winnings A woman won \(\$ 200,000\) in a state lottery. She first paid income tax of \(30 \%\) on the winnings. Of the rest, she invested some at \(1.5 \%\) and some at \(4 \%\) earning \(\$ 4350\) interest per year. How much did she invest at each rate?

Step-by-Step Solution

Answer

$50,000 was invested at 1.5% and $90,000 at 4%.

Step 1: Calculate Amount After Tax

First, determine the amount the woman is left with after paying the income tax. She pays 30% tax on $200,000, which is calculated as \(200,000 \times 0.30 = 60,000\). Hence, the amount left after tax is \(200,000 - 60,000 = 140,000\).

Step 2: Set Up Variables for Investments

Let \(x\) be the amount invested at 1.5% and \(140,000 - x\) be the amount invested at 4%. The reason for \(140,000 - x\) is that total available money is 140,000 after tax.

Step 3: Apply Simple Interest Formula for Each Investment

For the 1.5% investment: \(I_1 = x \times 0.015 \times 1 = 0.015x\). For the 4% investment: \(I_2 = (140,000 - x) \times 0.04 \times 1 = 0.04(140,000 - x)\).

Step 4: Set Up the Equation for Total Interest

The total interest earned from both investments is given as $4350. Therefore, we set up the equation: \(0.015x + 0.04(140,000 - x) = 4350\).

Step 5: Solve the Equation

Expand the equation: \(0.015x + 0.04 \times 140,000 - 0.04x = 4350\)Simplify: \(0.015x - 0.04x + 5600 = 4350\)Combine like terms: \(-0.025x = 4350 - 5600\)\(-0.025x = -1250\)Divide by -0.025:\(x = \frac{-1250}{-0.025} = 50,000\).Thus, \(x = 50,000\) is the amount invested at 1.5%.

Step 6: Calculate the Second Investment Amount

Use \(x = 50,000\) to find the amount invested at 4%. \(140,000 - x = 140,000 - 50,000 = 90,000\). As such, $90,000 out of the total amount was invested at 4% interest.

Key Concepts

Investment Problems

Investment problems can seem daunting at first, but they are a structured way to apply mathematical principles to real-world financial scenarios. These problems typically involve decisions about where to allocate money to receive the highest possible returns. One common method is through investments with various interest rates.

In such problems, you need to do the following:

• Account for external factors like taxes or expenses first, as these reduce the initial investment amount.
• Identify different investment options and their interest rates.
• Set up and solve equations to find out how much to invest in each option for maximum gain.

Always remember, clearly defining variables and relationships at the start saves you from confusion later on. It also guides you through the setup of your equations.

Interest Calculation

Interest calculation is essential in investment problems, helping you understand how much return you can expect on your deposited money. Simple interest is one of the most straightforward ways to calculate interest. It uses the formula \( I = P \times R \times T \), where:

• \(I\) is the interest earned.
• \(P\) is the principal amount, or the initial sum invested.
• \(R\) is the annual interest rate, expressed as a decimal.
• \(T\) is the time period for the investment, usually in years.

For our exercise, since \(T = 1\) year, this simplifies calculations significantly.

By applying this formula to each segment of an investment, you determine the interest earned individually and can sum them to find total earnings. It's a great way to gauge how different interest rates impact the overall returns on your investments.

Algebraic Equations

Algebraic equations play a critical role when solving investment problems. They enable you to translate the problem into a mathematical framework that can be solved efficiently. Setting up equations involves:

• Identifying the unknowns, such as the amounts invested at different rates.
• Expressing relationships and constraints mathematically, like the total interest earned or total amount invested post-taxes.
• Combining these expressions into one coherent equation.

In our sample problem, we have two unknowns and set up an equation by expressing them in terms of the total interest. Once the equation \(0.015x + 0.04(140,000 - x) = 4350\) is established, solving for \(x\) provides the answer to how much was invested at each rate. This process simplifies solving complex investment scenarios by filtering through unnecessary information and focusing on mathematical truths.