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# 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 $using the Rule of 72$? 232 likes 1159 views ## Answer to a math question 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 $using the Rule of 72$?

Jon
4.6
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.

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.

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