A 20-year old hopes to retire by age 65. To help with future expenses, they invest $6 500 today at an interest rate of 6.4% compounded annually. At age 65, what is the difference between the exact accumulated value and the approximate accumulated value (using the Rule of 72)?

To find the exact accumulated value, we can use the formula for compound interest:

A = P(1 + r)^n

Where:
- A is the accumulated value
- P is the principal amount (initial investment)
- r is the interest rate per compounding period
- n is the number of compounding periods

In this case, the principal amount is $6,500, the interest rate is 6.4% or 0.064 as a decimal, and the number of compounding periods is 65 - 20 = 45 (since the investment is made for 45 years).

Let's calculate the exact accumulated value:

A = 6500(1 + 0.064)^{45}

Now, to find the approximate accumulated value using the Rule of 72, we can use the following formula:

A \approx P \times \frac{72}{r}

Where:
- A is the approximate accumulated value
- P is the principal amount
- r is the interest rate per compounding period

In this case, the principal amount is still $6,500 and the interest rate is 6.4%.

Let's calculate the approximate accumulated value:

A \approx 6500 \times \frac{72}{6.4}

Now, let's calculate both the exact accumulated value and the approximate accumulated value.

Answer:
Exact accumulated value: \ 105995.3
Approximate accumulated value: \ 73125

The difference between the exact accumulated value and the approximate accumulated value is:

\text{Difference} = \text{Exact accumulated value} - \text{Approximate accumulated value}

\text{Difference}=105995.3-73125

\text{Difference}=32870.3

Therefore, the difference between the exact accumulated value and the approximate accumulated value is approximately $32870.3.