An investor splits $10,000 between two accounts. One account earns 5% annual interest, and the other earns 8%. If the total interest earned after one year is $680, how much is invested in each account? - High Altitude Science
How an Investor Split $10,000 to Earn $680 in One Year: The Math Behind the Interest Breakdown
How an Investor Split $10,000 to Earn $680 in One Year: The Math Behind the Interest Breakdown
When managing investments, understanding how to allocate your capital strategically can significantly impact your returns. A common question investors face is how to split funds between two accounts earning different annual interest rates. A classic real-world example involves balancing moderate returns (5%) and higher-yield accounts (8%) to achieve a targeted income. In this case, we explore how an investor split $10,000 between two accounts—earning 5% and 8%—to generate exactly $680 in annual interest.
The Investment Split Explained
Understanding the Context
Let’s define the variables clearly:
Let \( x \) be the amount invested at 5% annual interest.
Then, \( 10,\!000 - x \) is invested at 8% annual interest.
The interest earned from each account is calculated as:
- Interest from 5% account: \( 0.05x \)
- Interest from 8% account: \( 0.08(10,\!000 - x) \)
According to the problem, the total interest earned after one year is $680:
\[
0.05x + 0.08(10,\!000 - x) = 680
\]
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Key Insights
Solving for \( x \)
Start by expanding the equation:
\[
0.05x + 800 - 0.08x = 680
\]
Combine like terms:
\[
-0.03x + 800 = 680
\]
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Subtract 800 from both sides:
\[
-0.03x = -120
\]
Divide by -0.03:
\[
x = \frac{-120}{-0.03} = 4,\!000
\]
So, $4,000 is invested at 5%, and the remainder — $10,000 – $4,000 = $6,000 — is invested at 8%.
Verification
Calculate interest from each account:
- 5% of $4,000 = \( 0.05 \ imes 4,\!000 = 200 \)
- 8% of $6,000 = \( 0.08 \ imes 6,\!000 = 480 \)
Total interest = \( 200 + 480 = 680 \), confirming the solution.
Conclusion
This scenario demonstrates a practical investment strategy: splitting funds between a conservative 5% account and a higher-yield 8% account to meet a precise financial goal. By applying basic algebra, the investor efficiently determined that $4,000 should go into the lower-yield account and $6,000 into the 8% account. Understanding this approach empowers investors to make informed decisions when balancing returns, risk, and time in their portfolio planning.
Key Takeaways:
- Use algebraic equations to model investment allocation problems.
- Splitting funds strategically between fixed-income accounts optimizes returns.
- Always verify your calculation to ensure accurate financial outcomes.
