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)?

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Jon

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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.

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:

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

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:

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:

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

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