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The problem you have provided involves two stages of savings: 

1. **From age 50 to 60:** The woman deposits $1,700 at the end of each quarter in an account that earns **7% interest compounded quarterly**. This is a case of an **ordinary annuity** (payments made at the end of each period) with **quarterly compounding**.
   
2. **From age 60 to 65:** She withdraws the entire amount saved and transfers it to a new mutual fund account that earns **9% interest compounded monthly**. Afterward, she deposits **$200 at the end of each month** for 5 years (until she reaches age 65). This is a new annuity with **monthly compounding**.

We'll go step by step to calculate the final amount at age 65.

---

### Step 1: Calculate the amount in the account at age 60

For the first part, the formula for the **future value of an ordinary annuity** is:

\[
FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right)
\]

Where:
- $ P = 1700 $ (quarterly deposit)
- $ r = \frac{0.07}{4} = 0.0175 $ (quarterly interest rate, as 7% is compounded quarterly)
- $ n = 4 \times 10 = 40 $ (the number of quarters in 10 years)

Substitute these values into the formula to calculate the future value at age 60.

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### Step 2: Transfer to the new mutual fund

At age 60, she withdraws the amount and places it in a mutual fund with **9% interest compounded monthly**. For 5 years, she will deposit **$200 at the end of each month**. This is a new annuity, and we use the same **future value formula**, but with monthly compounding:

\[
FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) + PV \times (1 + r)^n
\]

Where:
- $ P = 200 $ (monthly deposit)
- $ r = \frac{0.09}{12} = 0.0075 $ (monthly interest rate)
- $ n = 12 \times 5 = 60 $ (the number of months in 5 years)
- $ PV $ is the amount transferred from the first account.

---

Let me compute the solution for you.Here are the results of the calculations:

1. **At age 60**, the amount in the retirement account after 10 years of quarterly deposits of $1,700, earning 7% compounded quarterly, is approximately **$97,298.03**.

2. **At age 65**, after transferring this amount to a mutual fund earning 9% compounded monthly, and making additional monthly deposits of $200 for 5 years, the total amount in the account is approximately **$167,422.50**.

This is the amount she will have in her account when she reaches age 65.

Would you like further clarification or details about these calculations?

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Here are 5 related questions that expand on this topic:

1. How would the result change if the interest rate during the first 10 years was compounded monthly instead of quarterly?
2. What happens to the future value if the monthly contribution of $200 is increased by 5% each year?
3. How would the future value change if she deposited $300 monthly instead of $200 during the second phase?
4. Can you calculate the total interest earned from both phases of the savings plan?
5. How would the future value change if the second mutual fund's interest rate was 10% instead of 9%?

**Tip:** Always check whether the compounding period matches the frequency of the contributions to avoid errors in future value calculations.

Starting at age 50, a woman puts $1700 at the end of each quarter into a retirement account that pays 7% interest compounded quarterly. When she reaches age 60, she withdraws the entire amount and places it in a mutual fund account that pays 9% compounded monthly. From then on she deposits $200 in the same mutual fund at the end of each month. How much is in the account when she reaches age 65?

Learn how to calculate the future value of retirement savings involving two stages: first with quarterly deposits at 7% interest compounded quarterly, followed by monthly deposits at 9% interest compounded monthly. This problem covers key concepts such as annuities and compounded interest, suitable for Grades 11-12 or college-level students.

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