A firm transforms its business from one model to another. For this its product and service development investments rise from 20 000 to 32 000 euros.
a) By how much as a percentage did these grow?
b) If the investments then drop by 20 % in the following year, how much are these then?
c) If these then keep on decreasing a steady percent rate from this amount to 28 000 euros a year in the following 3 years,
how much is the yearly rate of change?

Let's tackle each part of the problem step by step:

### a) By how much as a percentage did these investments grow?
We know the initial investment was 20,000 euros, and it increased to 32,000 euros.

To calculate the percentage increase:
\[
\text{Percentage Increase} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100
\]
Substituting the values:
\[
\text{Percentage Increase} = \frac{32,000 - 20,000}{20,000} \times 100 = \frac{12,000}{20,000} \times 100 = 60\%
\]

So, the investments grew by **60%**.

---

### b) If the investments then drop by 20%, how much are these then?
Now, we know that the investment after the growth is 32,000 euros. If it drops by 20%, we can calculate the new value as:

\[
\text{New Investment} = \text{Current Investment} - \left(\frac{20}{100} \times \text{Current Investment}\right)
\]
\[
\text{New Investment} = 32,000 - \left(0.20 \times 32,000\right)
\]
\[
\text{New Investment} = 32,000 - 6,400 = 25,600 \, \text{euros}
\]

So, after a 20% drop, the investments are **25,600 euros**.

---

### c) What is the yearly rate of change if the investments decrease steadily to 28,000 euros in 3 years?

We start from 25,600 euros and after 3 years, the investment decreases to 28,000 euros. Since we are asked to find the yearly percentage rate of change (\(r\)), we can use the formula for exponential decay:
\[
A = P \times (1 + r)^n
\]
Where:
- \(A = 28,000\) (final amount),
- \(P = 25,600\) (initial amount),
- \(n = 3\) (years),
- \(r\) is the yearly rate of change (we will solve for this).

Substituting the values:
\[
28,000 = 25,600 \times (1 + r)^3
\]
Dividing both sides by 25,600:
\[
\frac{28,000}{25,600} = (1 + r)^3
\]
\[
1.09375 = (1 + r)^3
\]
Taking the cube root of both sides:
\[
1 + r = \sqrt[3]{1.09375}
\]
Using a calculator:
\[
1 + r = 1.03004
\]
Thus:
\[
r = 1.03004 - 1 = 0.03004 \, \text{or} \, 3.004\%
\]

The yearly rate of change is approximately **3.004%**.

---

### Summary:
a) The investments grew by **60%**.  
b) After a 20% drop, the investments are **25,600 euros**.  
c) The yearly rate of change is approximately **3.004%**.

---

Would you like further clarification on any of these steps?

Here are 5 related questions you might find helpful:
1. How is exponential growth or decay applied in business investments?
2. What happens if the yearly decrease rate is higher than 3.004%?
3. How would you calculate the investment after 5 years at this rate?
4. What if the initial increase had been to 40,000 euros instead of 32,000?
5. How does the formula for exponential decay relate to real-world business scenarios?

**Tip:** To approximate the yearly rate of change in multi-year growth or decay problems, the exponential formula \(A = P(1 + r)^n\) is key for finding consistent rate trends.

A firm transforms its business from one model to another. For this its product and service development investments rise from 20,000 to 32,000 euros.
a) By how much as a percentage did these grow?
b) If the investments then drop by 20% in the following year, how much are these then?
c) If these then keep on decreasing a steady percent rate from this amount to 28,000 euros a year in the following 3 years, how much is the yearly rate of change?

Explore how to calculate business investment growth by 60%, handle a 20% drop, and find the yearly exponential decay rate to 28,000 euros over 3 years. This detailed step-by-step guide covers percentage increase and exponential decay for Grades 9-12.

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