Elena has \$18,500 to invest. She invests some of it at $8.4\%$ annual simple interest for $1$ year, and the remainder at $11.6\%$ annual simple interest for $8$ months. At the end of the year, her total interest earned is \$1,500. How much did she invest at each rate?

• A. \$11,000 at $8.4\%$ and \$7,500 at $11.6\%$
• B. \$7,500 at $8.4\%$ and \$11,000 at $11.6\%$
• C. \$10,399.92 at $8.4\%$ and \$8100.08 at $11.6\%$
• D. \$8100.08 at $8.4\%$ and \$10,399.92 at $11.6\%$

Answer: (C)

Verified step by step guidance

1. Define variables for the amounts Elena invests at each interest rate. Let \(x\) be the amount invested at 8.4% and \(y\) be the amount invested at 11.6%.
2. Write an equation representing the total amount invested: \(x + y = 18500\).
3. Calculate the interest earned from each investment using the simple interest formula \(I = P \times r \times t\), where \(P\) is the principal, \(r\) is the annual interest rate (expressed as a decimal), and \(t\) is the time in years. For the 8.4% investment, the time is 1 year, so interest is \$0.084 \times x \times 1\(. For the 11.6% investment, the time is 8 months, which is \)\frac{8}{12} = \frac{2}{3}\( years, so interest is \)0.116 \times y \times \frac{2}{3}$.
4. Write an equation for the total interest earned: \$0.084x + 0.116 \times y \times \frac{2}{3} = 1500$.
5. Solve the system of two equations: \(x + y = 18500\) and \$0.084x + 0.116 \times y \times \frac{2}{3} = 1500\( to find the values of \)x\( and \)y$, which represent the amounts invested at each rate.