Lycaon Pictus wishes to study painting in Paris and is saving money to enroll in art
school. Lycaon wishes to enroll in art school two years from now.
He estimates that art school will cost $1,111 per month for three years. Assume that
the first expenditure of 1,111 will occur 24 months from now, and the final one will
be sixty months from now).
Lycaon currently has $9000 already saved.
How much additional money must Lycaon save each month (starting in one month
from now, and ending in 23 months from now) to be able to have enough cash to
pay for his studies? The monthly interest rate is 0.4167%.


To determine how much additional money Lycaon must save each month to afford art school, we'll calculate the future value of his savings and determine how much more he needs.

### Step 1: Calculate the total cost of the art school
The art school will cost \$1,111 per month for 3 years (36 months). Therefore, the total cost is:
\[
\text{Total Cost} = 1,111 \times 36 = 39,996 \text{ USD}
\]

### Step 2: Determine the future value of the existing savings
Lycaon currently has \$9,000 saved. We'll calculate the future value of this amount after 23 months with a monthly interest rate of 0.4167%.

The formula for the future value $ FV $ of a single sum is:
\[
FV = PV \times (1 + r)^n
\]
where:
- $ PV = 9,000 $ (the present value),
- $ r = 0.004167 $ (the monthly interest rate),
- $ n = 23 $ (the number of months).

\[
FV = 9000 \times (1.004167)^{23}
\]

### Step 3: Calculate the future value of monthly savings
Let $ x $ be the amount Lycaon needs to save each month. The future value of these monthly savings is given by the formula for the future value of an annuity:
\[
FV_{\text{annuity}} = x \times \frac{(1 + r)^{n} - 1}{r}
\]
where:
- $ x $ is the monthly saving amount,
- $ r = 0.004167 $,
- $ n = 23 $.

### Step 4: Calculate the required savings
The total future value of all savings (current savings plus additional savings) must equal the present value of the payments for art school. 

So, the equation we solve is:
\[
FV_{\text{current}} + FV_{\text{annuity}} = \text{Present Value of Art School Payments}
\]

The Present Value of Art School Payments $ PV_{\text{art school}} $ is:
\[
PV_{\text{art school}} = 1111 \times \frac{1 - (1 + r)^{-36}}{r}
\]

### Final Calculation
Let's calculate these values.

First, calculate the future value of the existing savings:
\[
FV_{\text{current}} = 9000 \times (1.004167)^{23} 
\]

Then, calculate the future value of the art school payments:
\[
PV_{\text{art school}} = 1111 \times \frac{1 - (1 + 0.004167)^{-36}}{0.004167}
\]

Finally, solve for $ x $ using the relationship:
\[
FV_{\text{current}} + x \times \frac{(1 + 0.004167)^{23} - 1}{0.004167} = PV_{\text{art school}}
\]### Summary of Results:

1. **Future Value of Current Savings**: Lycaon's existing \$9,000 will grow to approximately **\$9,903.28** after 23 months, with the monthly interest rate of 0.4167%.

2. **Present Value of Art School Payments**: The total amount Lycaon needs to have saved to cover all future payments (taking into account the time value of money) is approximately **\$37,069.07**.

3. **Additional Monthly Savings Required**: To meet his goal, Lycaon needs to save approximately **\$1,127.88** each month for the next 23 months.

If you have any questions or would like more details on any part of the calculations, feel free to ask!

### Follow-Up Questions:

1. What happens to the required monthly savings if the interest rate changes?
2. How would the calculations differ if Lycaon started saving after 12 months instead of 1 month?
3. What if Lycaon expects a sudden large expense during the savings period?
4. How would an increase in the art school cost impact the savings plan?
5. What is the impact of a longer or shorter savings period?

### Tip:
Always consider potential fluctuations in costs or income when planning long-term savings; building a buffer into your plan can provide financial security.

Lycaon Pictus wishes to study painting in Paris and is saving money to enroll in art school. Lycaon wishes to enroll in art school two years from now. He estimates that art school will cost $1,111 per month for three years. Assume that the first expenditure of 1,111 will occur 24 months from now, and the final one will be sixty months from now. Lycaon currently has $9000 already saved. How much additional money must Lycaon save each month (starting in one month from now, and ending in 23 months from now) to be able to have enough cash to pay for his studies? The monthly interest rate is 0.4167%.

Learn how to determine the additional monthly savings needed for Lycaon Pictus to afford art school in Paris in two years. This solution involves concepts like future value, present value, annuities, and interest rate calculations. Suitable for advanced high school students or college level.

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