Suatu pinjaman akan dilunasi dengan anuitas tahunan. Tentukan besarnya anuitas jika besarnya angsuran ke-4 dan bungan ke-4 berturut turut adalah Rp 518.000 dan Rp 127.000

To solve this problem, we need to determine the annual annuity (A) given the following information:

- The 4th installment (angsuran ke-4) is \( Rp \ 518,000 \)
- The 4th interest payment (bunga ke-4) is \( Rp \ 127,000 \)

### Step 1: Understanding the Annuity Formula

The annuity payments consist of two parts:
1. The interest on the remaining principal
2. The principal repayment

The formula for an installment in an annuity payment is:

\[
A_n = (P - (n-1) \cdot R) \times r
\]

Where:
- \( A_n \) is the total installment (Rp 518,000 for the 4th installment)
- \( r \) is the interest rate
- \( P \) is the initial principal
- \( R \) is the principal repayment (which is the same for each installment if the annuity is level)

### Step 2: Expressing the 4th Installment

The 4th installment consists of:
- Interest on the remaining principal after the first 3 installments
- The 4th principal repayment

This can be expressed as:

\[
A = \text{Principal Repayment (R)} + \text{Interest on Remaining Principal (bunga ke-4)}
\]

### Step 3: Calculation

Given:
- \( A_4 = Rp \ 518,000 \)
- \( \text{bunga ke-4} = Rp \ 127,000 \)

Let \( R \) be the principal repayment:

\[
R = A_4 - \text{bunga ke-4} = 518,000 - 127,000 = Rp \ 391,000
\]

### Step 4: Determine the Annuity (A)

The annuity payment (A) is the sum of \( R \) and the interest payment:

\[
A = R + \text{Interest Payment}
\]

Here, the annuity \( A \) is equivalent to the 4th installment since the installments are constant in a fixed annuity system:

\[
A = Rp \ 518,000
\]

### Final Answer:
The amount of the annual annuity (besarnya anuitas) is **Rp 518,000**.

Would you like to explore more on this, or have any further questions?

Here are 5 related questions to consider:
1. How would the calculation change if the interest rate changed?
2. What is the total interest paid over the life of the loan?
3. How many total payments are typically made in such an annuity?
4. What would the annuity be if the principal was higher?
5. How does the principal repayment change over time in an annuity?

**Tip:** Understanding the breakdown between principal repayment and interest in each installment is crucial for accurately determining the total cost of a loan.

Learn how to calculate the annual annuity payment using specific installment and interest data. This problem involves determining the annuity amount based on given installment details and interest payments. Suitable for advanced high school students studying finance and mathematical applications.

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