A machine with a current value of 40.000 Euro will be replaced after 10 years.

It is expected that the machine can be sold by that time for 4.000 Euro.

A new machine will cost 52.000 Euro at that time.

In order to have sufficient money available to replace the machine after 10 years, the same amount of money is transferred into a bank account at the end of each year.


The bank account gives a compound interest of 6% per year during the first 8 years. During the last two twee years the account does not give any interest.


How much money must be transferred each year into the bank account?

To calculate how much money needs to be transferred into the bank account each year to replace the machine after 10 years, we can follow these steps:

### 1. Determine the total amount required after 10 years:

- The new machine will cost €52,000.
- The current machine can be sold for €4,000 at that time.

Thus, the net amount required is:
\[
\text{Net amount required} = 52,000 - 4,000 = 48,000 \text{ Euro.}
\]

### 2. Break down the time period:

- For the first 8 years, the bank account earns **6% interest** per year.
- For the last 2 years, there is **no interest**.

### 3. Approach: 

We need to calculate the **equal yearly deposits** made at the end of each year such that the total amount in the account after 10 years equals €48,000. This includes 6% interest for the first 8 years and no interest for the last two years.

Let’s denote:
- \( P \) as the amount deposited at the end of each year.
- \( r = 0.06 \) (6% interest rate).

We’ll calculate this in two steps:

#### Step 1: For the first 8 years (compound interest at 6%)
The deposits made at the end of each year will grow according to the compound interest formula. Each year’s deposit grows for a different number of years. The total after 10 years for the deposits made in the first 8 years is given by:

\[
P \times (1 + r)^8 + P \times (1 + r)^7 + P \times (1 + r)^6 + \dots + P \times (1 + r)^1
\]

This is a geometric series, and the sum for the first 8 years can be expressed as:

\[
P \times \sum_{k=1}^{8} (1 + r)^k = P \times \frac{(1 + r)^9 - (1 + r)}{r}
\]

#### Step 2: For the last 2 years (no interest)
The deposits made in years 9 and 10 do not accrue any interest. So, their contribution is simply \( 2P \).

### 4. Combine the two steps

The total value in the account after 10 years is:

\[
P \times \left( \frac{(1 + r)^9 - (1 + r)}{r} \right) + 2P = 48,000
\]

Now, let’s compute this equation to find \( P \).The amount that must be transferred into the bank account at the end of each year is approximately **€3,842.67** in order to have enough money to replace the machine after 10 years.

Would you like more details on any of these steps or have further questions?

Here are 5 related questions:
1. How does compound interest affect savings over time compared to simple interest?
2. How would the calculation change if the interest rate were different for each of the 8 years?
3. What if the machine's resale value after 10 years was different—how would that impact the annual deposits?
4. Could an investment with a higher rate of return reduce the yearly payments significantly?
5. How would early withdrawals from the bank account affect the ability to meet the goal?

**Tip:** Compound interest grows faster the longer it’s applied, so the earlier you start saving, the less you need to contribute each period.

A machine with a current value of 40,000 Euro will be replaced after 10 years. It is expected that the machine can be sold by that time for 4,000 Euro. A new machine will cost 52,000 Euro at that time. In order to have sufficient money available to replace the machine after 10 years, the same amount of money is transferred into a bank account at the end of each year. The bank account gives a compound interest of 6% per year during the first 8 years. During the last two years, the account does not give any interest. How much money must be transferred each year into the bank account?

Find out how to calculate the annual amount to save for a machine replacement. This problem involves compound interest over 8 years, followed by 2 years without interest, with a goal of accumulating €48,000. Learn how geometric series and interest formulas apply to real-life financial planning.

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