Problem 60 - Elementary Algebra

Question

Set up an algebraic equation and then solve. James invested \(\$ 1,600\) in two accounts. One account earns \(4.25 \%\) simple interest and the other earns \(8.5 \%\). If the interest after 1 year was \(\$ 85,\) how much did he invest in each account?

Step-by-Step Solution

Answer

James invested $1200 in the 4.25% account and $400 in the 8.5% account.

Step 1: Define Variables

Let the amount invested in the account with 4.25% interest be \( x \). The amount invested in the account with 8.5% interest will then be \( 1600 - x \), since the total investment is $1600.

Step 2: Set Up the Equation for Interest

The total interest earned from both accounts in one year is $85. The interest from the first account is \( 0.0425x \) and from the second account is \( 0.085(1600-x) \). Therefore, the equation is \( 0.0425x + 0.085(1600-x) = 85 \).

Step 3: Simplify the Equation

Distribute the 0.085 in the equation: \( 0.0425x + 136 - 0.085x = 85 \). To simplify, combine like terms: \( -0.0425x + 136 = 85 \).

Step 4: Solve for x

Subtract 136 from both sides of the equation: \( -0.0425x = 85 - 136 \). Simplify to get \( -0.0425x = -51 \). Divide both sides by -0.0425 to solve for \( x \): \( x = \frac{-51}{-0.0425} \approx 1200 \).

Step 5: Calculate the Investment in Second Account

If \( x = 1200 \), then the amount invested in the second account is \( 1600 - x = 1600 - 1200 = 400 \).

Key Concepts

Investment Problems

Investment problems are a common topic in algebra that involve determining the allocation of a specific amount of money across different investment options.
In these scenarios, each option often has its own unique interest rate, influencing the total returns. For example, investing in a low-risk account generally offers a lower interest rate, while riskier options might offer higher returns.
The key to solving investment problems is to carefully distribute the total investment so that all conditions given in the problem are met.

• Understand the total amount: Clearly define the total initial investment.
• Identify different investment options: Each with their respective rates or conditions.
• Use equations to represent allocations: Set up algebraic equations based on the problem's conditions.

By systematically solving these equations, you'll determine precisely how much money goes into each investment option to achieve a desired result.

Simple Interest

Simple interest is a way of calculating the interest earned on an investment based on the original amount, or principal, without compounding.
The formula for simple interest is straightforward: \[ I = P \cdot r \cdot t\]Where:

• I is the interest earned.
• P is the principal amount invested.
• r is the annual interest rate (expressed as a decimal).
• t is the time the money is invested for, typically in years.

In the given problem, both investments generate simple interest, with different rates applied to each account for a one-year term.
Understanding how to apply this formula is crucial for tackling simple interest problems.

Problem-Solving Steps

When tackling algebraic equations in investment problems, following a structured set of steps can greatly enhance your ability to find accurate solutions.
Here’s a simple problem-solving guide to follow:

• Define Variables: Decide what each variable represents. For instance, if you're dealing with two accounts, let one variable represent the amount in one account.
• Set Up the Equation: Use the information given to form an equation that represents the total interest or conditions described in the problem.
• Simplify the Equation: Distribute and combine like terms to make solving easier.
• Solve for the Unknown: Isolate the variable to find its value, factoring in all given data.
• Calculate Other Values: Once the main variable is solved, use it to find additional required values, such as the amount invested in other accounts.

By methodically applying these steps, you can effectively solve most algebraic investment problems, ensuring both accuracy and understanding.