To determine how much Ryan should pay at the end of 2 ½ years to settle all his obligations, we need to calculate the future value of each loan at the end of 2 ½ years considering the interest rate at which the money is compounded. Let's break down each loan and calculate its future value, then sum these future values to find the total amount Ryan needs to pay.

Given:

• Interest rate for the cooperative: 18% compounded monthly
• Time period: 2.5 years (which is 30 months)

1. Short Term Loan

• Principal: ₱15,000
• Time: 6 months
• Interest: 15% compounded semi-annually

Future Value (FV) formula for compounding semi-annually: $FV = P \left(1 + \frac{r}{n}\right)^{nt}$ where:

• $P = 15,000$
• $r = 0.15$
• $n = 2$ (semi-annual compounding)
• $t = \frac{6}{12} = 0.5$ years

$FV_1 = 15,000 \left(1 + \frac{0.15}{2}\right)^{2 \cdot 0.5}$

2. Cellphone Loan

• Principal: ₱20,000
• Time: 1 year
• No interest (assuming it's simple and not compounded)

Future Value (FV) formula: $FV = P \left(1 + rt\right)$ where:

• $P = 20,000$
• $r = 0$
• $t = 1$

$FV_2 = 20,000$

3. Salary Loan

• Principal: ₱50,000
• Time: 2 years
• Interest: 15% compounded monthly

Future Value (FV) formula for compounding monthly: $FV = P \left(1 + \frac{r}{n}\right)^{nt}$ where:

• $P = 50,000$
• $r = 0.15$
• $n = 12$ (monthly compounding)
• $t = 2$

$FV_3 = 50,000 \left(1 + \frac{0.15}{12}\right)^{12 \cdot 2}$

Converting each loan to a future value at 2.5 years with the cooperative's interest rate

Using the future value (FV) formula again, we need to convert the FV values obtained above to the equivalent value at 2.5 years at 18% compounded monthly.

Step 1: Calculate each loan's FV using the given interest rates

1. Short Term Loan: $FV_1 = 15,000 \left(1 + \frac{0.15}{2}\right)^{1}$ $FV_1 = 15,000 \left(1 + 0.075\right)$ $FV_1 = 15,000 \times 1.075$ $FV_1 = 16,125$

2. Cellphone Loan: $FV_2 = 20,000$

3. Salary Loan: $FV_3 = 50,000 \left(1 + \frac{0.15}{12}\right)^{24}$ $FV_3 = 50,000 \left(1 + 0.0125\right)^{24}$ $FV_3 = 50,000 \left(1.0125\right)^{24}$

Step 2: Calculate future values at 2.5 years using the cooperative's rate (18% compounded monthly)

$FV_{1,2.5y} = FV_1 \left(1 + \frac{0.18}{12}\right)^{30}$ $FV_{2,2.5y} = FV_2 \left(1 + \frac{0.18}{12}\right)^{18}$ $FV_{3,2.5y} = FV_3 \left(1 + \frac{0.18}{12}\right)^{6}$

Let's compute these step-by-step.

1. Short Term Loan FV at 2.5 years: $FV_{1,2.5y} = 16,125 \left(1 + \frac{0.18}{12}\right)^{30}$ $FV_{1,2.5y} = 16,125 \left(1 + 0.015\right)^{30}$ $FV_{1,2.5y} = 16,125 \left(1.015\right)^{30}$

2. Cellphone Loan FV at 2.5 years: $FV_{2,2.5y} = 20,000 \left(1 + \frac{0.18}{12}\right)^{18}$ $FV_{2,2.5y} = 20,000 \left(1 + 0.015\right)^{18}$ $FV_{2,2.5y} = 20,000 \left(1.015\right)^{18}$

3. Salary Loan FV at 2.5 years: $FV_{3,2.5y} = 50,000 \left(1 + \frac{0.18}{12}\right)^{6}$ $FV_{3,2.5y} = 50,000 \left(1 + 0.015\right)^{6}$ $FV_{3,2.5y} = 50,000 \left(1.015\right)^{6}$

Let's perform the final calculations.

Calculation

Let's plug these values into a Python script for accurate results.