To solve the problem in the image regarding the monthly annuity system, let's break it down:

Given Data:

• Principal Loan (P): Rp 10,000,000
• Interest Rate (i): 2% per month = 0.02
• Duration (n): 1 year = 12 months

Formula for Monthly Annuity Payment (A):

$A = \frac{P \cdot i}{1 - (1 + i)^{-n}}$

Step-by-Step Solution:

a. Monthly Installment (A)

1. Substituting the values: $A = \frac{10,000,000 \cdot 0.02}{1 - (1 + 0.02)^{-12}}$

2. Simplify the denominator: $(1 + 0.02)^{-12} = 1.02^{-12} \approx 0.7854$ $1 - 0.7854 = 0.2146$

3. Solve for $A$: $A = \frac{10,000,000 \cdot 0.02}{0.2146} = \frac{200,000}{0.2146} \approx 931,925.74$

   Monthly Installment (A): Rp 931,926

b. Principal Portion in the 5th Month

1. The interest portion for any month is calculated as: $\text{Interest Payment} = \text{Outstanding Balance} \cdot i$

2. Principal portion is: $\text{Principal Payment} = A - \text{Interest Payment}$

3. Outstanding balance after each month can be calculated using: $\text{Balance}_{k} = A \cdot \frac{(1 + i)^n - (1 + i)^k}{i}$ For the 4th month ($k = 4$), calculate the balance before the 5th month: $\text{Balance}_{4} = 931,926 \cdot \frac{(1.02)^{12} - (1.02)^4}{0.02}$

   • $(1.02)^{12} \approx 1.2682$
   • $(1.02)^4 \approx 1.0824$

   Substituting: $\text{Balance}_{4} = 931,926 \cdot \frac{1.2682 - 1.0824}{0.02} \approx 931,926 \cdot 9.789 \approx 9,118,961$

   Interest for the 5th month: $\text{Interest}_{5} = \text{Balance}_{4} \cdot 0.02 = 9,118,961 \cdot 0.02 = 182,379$

   Principal for the 5th month: $\text{Principal}_{5} = A - \text{Interest}_{5} = 931,926 - 182,379 = 749,547$

   Principal Portion (5th Month): Rp 749,547

c. Interest Portion in the 4th Month

Using the same approach, calculate the balance before the 4th month ($k = 3$): $\text{Balance}_{3} = 931,926 \cdot \frac{1.2682 - (1.02)^3}{0.02}$

• $(1.02)^3 \approx 1.0612$

Substituting: $\text{Balance}_{3} = 931,926 \cdot \frac{1.2682 - 1.0612}{0.02} \approx 931,926 \cdot 10.35 \approx 9,641,937$

Interest for the 4th month: $\text{Interest}_{4} = \text{Balance}_{3} \cdot 0.02 = 9,641,937 \cdot 0.02 = 192,839$

Interest Portion (4th Month): Rp 192,839

Final Answers:

a. Monthly Installment: Rp 931,926
b. Principal in the 5th Month: Rp 749,547
c. Interest in the 4th Month: Rp 192,839

Do you want a breakdown of any part, or should I calculate additional terms? Here are 5 related questions:

1. How is the outstanding balance calculated at each month?
2. How do changes in the interest rate affect the monthly installment?
3. What is the total amount of interest paid over the loan term?
4. How can an early repayment be factored into this calculation?
5. What if the loan is extended to 18 months instead of 12?

Tip: Use a financial calculator or spreadsheet for repetitive annuity calculations to save time and reduce errors!