Problem 264 - Elementary Algebra

Question

Carol invested \(\$ 2,560\) into two accounts. One account paid \(8 \%\) interest and the other paid \(6 \%\) interest. She earned \(7.25 \%\) interest on the total investment. How much money did she put in each account?

Step-by-Step Solution

Answer

Carol invested \(\text{\textdollar}1600\) at 8% and \(\text{\textdollar}960\) at 6%.

Step 1 - Define Variables

Let the amount invested at 8% be denoted as \(x\). Therefore, the amount invested at 6% will be \(2560 - x\).

Step 2 - Set Up Interest Equations

The interest from the account earning 8% is \(0.08x\). The interest from the account earning 6% is \(0.06(2560 - x)\).

Step 3 - Express Total Interest

According to the problem, the total interest earned is 7.25% of the total investment. Therefore, the total interest is \(0.0725 \times 2560\), which amounts to 185.6.

Step 4 - Create an Equation

Set up the equation for the total interest: \[0.08x + 0.06(2560 - x) = 185.6\]

Step 5 - Solve for x

Simplify and solve the equation: \[0.08x + 153.6 - 0.06x = 185.6\] Combine like terms: \[0.02x + 153.6 = 185.6\] Subtract 153.6 from both sides: \[0.02x = 32\] Divide both sides by 0.02: \[x = 1600\]

Step 6 - Find the Amount in the Other Account

Since \(x = 1600\), the amount invested at 6% is \(2560 - 1600 = 960\).

Step 7: Conclusion

Carol invested \(\text{\textdollar}1600\) in the account that paid 8% interest and \(\text{\textdollar}960\) in the account that paid 6% interest.

Key Concepts

Linear Equations

Linear equations represent mathematical statements of equality that involve constants and variables. These equations are essential in solving investment problems because they allow us to find unknown values. In Carol's exercise, we set up a linear equation to find out how much she invested in each account. By defining variables, like letting the amount at 8% be denoted as \(x\), we can further simplify and solve these problems. The equation we use is created based on the interests from different accounts.

Percentage Calculation

Percentage calculation is crucial when dealing with interest rates in an investment problem. It measures parts per hundred and helps us understand portions of the whole. In Carol's problem, the accounts paid 8% and 6% interest, which required calculating these percentages on the amounts invested. Calculations like \(0.08x\) for the interest from the 8% account and \(0.06(2560 - x)\) for the 6% account tell us how much interest is earned from each. Also, understanding the given 7.25% total interest earned helps set up the necessary equations.

Investment Distribution

Investment distribution refers to how Carol spread her money across two accounts with different interest rates. To find the amounts, we started by letting \(x\) be the amount invested at 8%, hence the remaining \(2560 - x\) would be invested at 6%. This strategy simplifies the problem and helps set up the interest equations. Finally, solving these equations helps us figure out that Carol invested \(\text{\$1600}\) at 8% and \(\text{\$960}\) at 6%. Understanding how to distribute investments efficiently can greatly affect the overall returns.

Problem-Solving Steps

Solving Carol's investment problem requires a series of well-defined steps:

• Define Variables: Start by denoting the amount invested at 8% with \(x\), making the 6% invested amount \(2560 - x\).
• Set Up Interest Equations: Calculate the interest for each account using the percentages given.
• Express Total Interest: Determine the total interest earned according to the problem, which is 7.25% of the total investment.
• Create an Equation: Set up an equation based on the total interest earned from both accounts.
• Simplify and Solve the Equation: Solve for \(x\) to find how much was invested at 8%.
• Find the Remaining Amount: Subtract the amount invested at 8% from the total to find the amount invested at 6%.
• Conclusion: Conclude by stating the amounts invested in both accounts.