A university would like to establish a perpetuity, the interest from which will fund a scholarship for a talented student each year. The value of this prize should be $3500 per year.

a) If the interest on this investment is 3.6% per annum, how much will the university need to invest?

The university only has $60000 to invest.

b) What annual interest rate is required to pay the scholarship in perpetuity? Round your answer to two decimal places.

To solve these problems, we need to use the concept of a perpetuity. A perpetuity is a type of annuity that lasts forever, where the same amount is paid out each year indefinitely. The present value $ P $ of a perpetuity can be calculated using the formula:

\[ P = \frac{A}{r} \]

where:
- $ A $ is the annual payment,
- $ r $ is the annual interest rate (expressed as a decimal).

### Part (a)

Given:
- $ A = \$3500 $ (annual scholarship amount)
- $ r = 3.6\% = 0.036 $ (annual interest rate)

We need to find $ P $ (the principal amount needed to invest):

\[ P = \frac{3500}{0.036} \]

Let's calculate:

\[ P = \frac{3500}{0.036} = 97,222.22 \]

The university will need to invest **$97,222.22** to fund a $3500 per year scholarship at an interest rate of 3.6% per annum.

### Part (b)

Given:
- $ A = \$3500 $ (annual scholarship amount)
- $ P = \$60000 $ (amount available to invest)

We need to find $ r $ (the required annual interest rate):

Using the formula for the present value of a perpetuity:

\[ P = \frac{A}{r} \]

Rearranging to solve for $ r $:

\[ r = \frac{A}{P} \]

Substitute the given values:

\[ r = \frac{3500}{60000} \]

Let's calculate:

\[ r = \frac{3500}{60000} = 0.0583333 \]

To express $ r $ as a percentage, we multiply by 100:

\[ r = 0.0583333 \times 100 = 5.83\% \]

The required annual interest rate is **5.83%** to pay the scholarship in perpetuity with a $60000 investment.

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Would you like further details or have any questions about these calculations? Here are some additional related questions for further practice:

1. If the interest rate were 4%, how much would the university need to invest for the $3500 annual scholarship?
2. How much would the university need to invest if the scholarship amount was increased to $5000 per year at an interest rate of 3.6%?
3. What would be the annual scholarship amount if the university invested $80000 at an interest rate of 3.6%?
4. If the university wanted to provide a $4500 annual scholarship, what interest rate would be needed if they could only invest $60000?
5. How much would the university need to invest to provide a $3500 scholarship if the interest rate dropped to 3%?
6. If the interest rate increases to 4.5%, how much would the university need to invest for the same $3500 scholarship?
7. What would be the principal amount needed to fund a perpetuity that pays out $2500 annually at an interest rate of 3.6%?
8. If the university invests $75000 at an interest rate of 4%, what annual scholarship amount can it fund in perpetuity?

**Tip:** When dealing with financial calculations involving perpetuities, always ensure the interest rate is expressed as a decimal in formulas.

Learn how to calculate the initial investment needed for a perpetuity to fund a $3500 scholarship annually at a 3.6% interest rate, and determine the required interest rate for perpetuity funding with a $60000 investment. Practice financial calculations with detailed solutions.

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