Let's break down the problem step by step: --- ## **Step 1: Compute the Annual Interest Earned from Fund X** - Principal in Fund X: \( 10,000 \) - Nominal annual rate (quarterly): \( 9\% \) compounded quarterly ### **Quarterly Rate** \[ \text{Quarterly Rate} = \frac{9\%}{4} = 2.25\% = .0225 \] ### **Annual Accumulation Factor** \[ (1 + .0225)^4 = 1.0225^4 \approx 1.09308 \] ### **Annual Interest Earned** \[ \text{Interest} = 10,000 \times (1.09308 - 1) = 10,000 \times .09308 = 930.80 \] So, at the end of each year, Rachael withdraws **$930.80** from Fund X and deposits it into Fund Y. --- ## **Step 2: Fund Y Accumulation (5 Years, Effective Rate \(i\))** Each year for 5 years, she deposits $930.80 into Fund Y, which earns annual effective rate \(i\). After 5 deposits, the accumulated value is $5461. This is a future value of an ordinary annuity: \[ \text{Future Value} = P \frac{(1 + i)^n - 1}{i} \] Where: - \( P = 930.80 \) - \( n = 5 \) - Future Value = 5461 So: \[ 5461 = 930.80 \cdot \frac{(1 + i)^5 - 1}{i} \] --- ## **Step 3: Solve for \(i\)** \[ \frac{(1 + i)^5 - 1}{i} = \frac{5461}{930.80} = 5.866 \] \[ (1 + i)^5 - 1 = 5.866i \] \[ (1 + i)^5 = 1 + 5.866i \] This equation must be solved numerically. --- ### **Step 4: Estimate \(i\) (Trial and Error or Financial Calculator)** Let's try a few values: #### **Try \(i = 10\% = .10\):** \[ (1.10)^5 = 1.61051 \] \[ 1.61051 = 1 + 5.866 \times .10 = 1.5866 \] This is close, but a bit high. #### **Try \(i = 9.5\% = .095\):** \[ (1.095)^5 = 1.5735 \] \[ 1.5735 = 1 + 5.866 \times .095 = 1.5573 \] Still a bit high. #### **Try \(i = 9\% = .09\):** \[ (1.09)^5 = 1.53862 \] \[ 1.53862 = 1 + 5.866 \times .09 = 1.52794 \] Very close! --- ### **Interpolate for a More Accurate Value** \[ i \approx .09 + \frac{1.53862 - 1.52794}{(1.53862 - 1.52794) + (1.5735 - 1.53862)} \times (.095 - .09) \] \[ = .09 + \frac{.01068}{.01068 + .03488} \times .005 \] \[ = .09 + \frac{.01068}{.04556} \times .005 \] \[ = .09 + .2347 \times .005 \] \[ \approx .09 + .00117 = .0912 \] --- ## **Final Answer** ### **The annual effective interest rate \(i\) is approximately:** \[ \boxed{9.1\%} \]