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To solve this problem, we need to calculate the amount of quarterly deposits required to reach a savings goal of $20,000 in five years, given an interest rate of 4.7% compounded semi-annually.

### Step-by-Step Solution:

1. **Identify the Variables**:
   - **Future Value (FV)**: $20,000 (the goal in 5 years).
   - **Interest Rate (annual)**: 4.7%, compounded semi-annually.
   - **Number of Years (N)**: 5 years.
   - **Compounding Frequency**: Semi-annually means the interest compounds twice per year.
   - **Deposit Frequency**: Quarterly (every 3 months), so there will be 4 deposits per year.

2. **Convert the Interest Rate for Compounding Periods**:
   - Since the interest rate is compounded semi-annually, we divide the annual rate by 2 for each semi-annual period.
   - Semi-annual interest rate $ r_{\text{semi}} = \frac{4.7\%}{2} = 2.35\% $ per period.

3. **Determine the Number of Periods**:
   - For 5 years, with quarterly deposits, there will be $ 5 \times 4 = 20 $ quarterly deposits.

4. **Set up the Future Value of an Annuity Due Formula**:
   Since deposits are made at the beginning of each quarter, we use the **Future Value of an Annuity Due** formula:
   \[
   FV = P \times \frac{(1 + r)^n - 1}{r} \times (1 + r)
   \]
   where:
   - $ P $ is the quarterly deposit amount.
   - $ r $ is the effective interest rate per quarter.
   - $ n $ is the total number of deposits.

5. **Calculate Effective Quarterly Interest Rate**:
   - First, we find the quarterly effective interest rate based on the semi-annual rate:
     \[
     r_{\text{quarter}} = (1 + r_{\text{semi}})^{\frac{1}{2}} - 1
     \]
     Plugging in 2.35%, we can calculate this rate.

6. **Solve for $ P $**:
   Rearrange the formula to solve for the quarterly deposit amount, $ P $, that will accumulate to $20,000.

Let me calculate the exact value for you.The required beginning-of-quarter deposit to reach a savings goal of $20,000 in 5 years, with a 4.7% interest rate compounded semi-annually, is approximately **$883.21**.

Would you like further details on the calculations, or do you have any additional questions?

### Related Questions:
1. How would the deposit amount change if the interest rate were compounded quarterly instead?
2. What would the deposit be if the goal was to reach $25,000 instead of $20,000?
3. How does semi-annual compounding differ from continuous compounding in terms of required deposits?
4. What if deposits were made at the end of each quarter instead of the beginning?
5. How would a higher interest rate, say 6%, impact the deposit amount?

### Tip:
For problems involving compound interest with regular deposits, understanding the difference between an **ordinary annuity** (end-of-period payments) and an **annuity due** (beginning-of-period payments) is crucial for accurate calculations.

What beginning-of-quarter deposits are needed to reach a savings goal of $20,000 in 5 years with a 4.7% interest rate compounded semi-annually?

Learn how to determine the quarterly deposit amount needed to reach a $20,000 savings goal in 5 years with a 4.7% interest rate compounded semi-annually. This problem involves understanding future value of annuities, compound interest calculations, and effective rate conversions. Ideal for finance and mathematics students.

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