A state lotto has a prize that pays $600 each week for 15 years. If the state can earn 3% interest on investments, how much money will they need to put into an account now to cover the weekly prize payments?


The problem can be solved by calculating the present value of an annuity, where the prize of $600 is paid weekly for 15 years at a 3% annual interest rate. We need to adjust for weekly payments and then calculate the present value.

### Step-by-step solution:

1. **Convert the annual interest rate to a weekly rate:**
   Since interest is compounded weekly, we convert the annual rate of 3% into a weekly interest rate.
   \[
   \text{Weekly interest rate} = \frac{3\%}{52} = \frac{0.03}{52} \approx 0.000576923 \text{ per week.}
   \]

2. **Calculate the number of weekly payments:**
   Since the payments last for 15 years, the total number of weeks in 15 years is:
   \[
   \text{Number of payments} = 15 \times 52 = 780 \text{ weeks.}
   \]

3. **Use the Present Value of Annuity formula:**
   The formula for the present value $PV$ of an annuity is:
   \[
   PV = P \times \left(1 - (1 + r)^{-n}\right) / r
   \]
   Where:
   - $P = 600$ (weekly payment),
   - $r = 0.000576923$ (weekly interest rate),
   - $n = 780$ (number of payments).

4. **Substitute the values into the formula:**
   \[
   PV = 600 \times \left(1 - (1 + 0.000576923)^{-780}\right) / 0.000576923
   \]

Now, let's calculate this.The state will need to put approximately **$376,780.68** into an account now to cover the weekly prize payments of $600 for 15 years, assuming a 3% annual interest rate.

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the required amount change if the interest rate were 4%?
2. What if the prize payments lasted for 20 years instead of 15?
3. How much would the state have to put in if the payments were monthly instead of weekly?
4. What would be the future value of the payments after 15 years?
5. How much interest would the state earn over the entire period?

**Tip:** The higher the interest rate, the lower the present value needed to cover future payments.

A state lotto has a prize that pays $600 each week for 15 years. If the state can earn 3% interest on investments, how much money will they need to put into an account now to cover the weekly prize payments?

Calculate the present value of a $600 weekly lotto prize over 15 years at a 3% annual interest rate. This detailed guide includes step-by-step calculations using the Present Value of Annuity formula. Ideal for students studying finance or advanced math.

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