Aidan invests €5000 in a special savings account for two years. The rate is 3% for the first year with a higher rate for the second year. Tax at 40% will be deducted each year from the interest earned. a) How much will Aidan's investment be worth    at the end of the first year, after tax is paid? b)  Aidan calculates  that, after tax has been deducted, his investment will be worth €5196.89 at the end  of the second year. Calculate the rate of interest for the second year. 

Let's break the problem into two parts and solve each part step by step.

### a) Investment value at the end of the first year (after tax)

Aidan starts with an investment of €5000, and the interest rate for the first year is 3%. However, after earning interest, 40% of the interest will be deducted as tax.

1. **Interest for the first year**:  
   The interest before tax for the first year is calculated as:
   \[
   \text{Interest before tax} = 5000 \times 0.03 = 150 \text{ euros}
   \]

2. **Tax on the interest**:  
   The tax rate is 40%, so the tax amount is:
   \[
   \text{Tax amount} = 150 \times 0.40 = 60 \text{ euros}
   \]

3. **Interest after tax**:  
   The interest remaining after tax is:
   \[
   \text{Interest after tax} = 150 - 60 = 90 \text{ euros}
   \]

4. **Investment value at the end of the first year**:  
   Aidan's total investment value at the end of the first year, after tax, is:
   \[
   \text{Investment value after year 1} = 5000 + 90 = 5090 \text{ euros}
   \]

### b) Calculate the rate of interest for the second year

Aidan's investment at the end of the second year is €5196.89, and the value after the first year is €5090. We need to calculate the interest rate for the second year, knowing that 40% of the interest is deducted as tax.

1. **Interest earned in the second year (before tax)**:  
   Let the interest rate for the second year be \( r \). The interest before tax can be calculated as:
   \[
   \text{Interest before tax} = 5090 \times r
   \]

2. **Tax on the interest for the second year**:  
   The tax is 40%, so the tax amount for the second year is:
   \[
   \text{Tax amount} = 5090 \times r \times 0.40
   \]

3. **Interest after tax**:  
   The interest after tax is:
   \[
   \text{Interest after tax} = 5090 \times r \times (1 - 0.40) = 5090 \times r \times 0.60
   \]

4. **Total investment value at the end of the second year**:  
   According to the problem, the total value at the end of the second year is €5196.89. Therefore:
   \[
   5090 + (5090 \times r \times 0.60) = 5196.89
   \]

5. **Solve for \( r \)**:
   Subtract 5090 from both sides:
   \[
   5090 \times r \times 0.60 = 5196.89 - 5090 = 106.89
   \]
   Now solve for \( r \):
   \[
   r = \frac{106.89}{5090 \times 0.60} = \frac{106.89}{3054} \approx 0.035 \text{ or } 3.5\%
   \]

So, the interest rate for the second year is approximately **3.5%**.

---

### Summary:
- **a)** Aidan's investment at the end of the first year (after tax) is **€5090**.
- **b)** The interest rate for the second year is **3.5%**.

Would you like further clarification or more details? Here are some related questions:

1. How would the outcome change if the tax rate were different?
2. What is the compound interest formula without taxes involved?
3. How would the total amount be affected by a different second-year interest rate?
4. What would happen if the interest were compounded monthly instead of yearly?
5. Can the same approach be used if the investment period were longer, such as five years?

**Tip**: When dealing with percentages and taxes, always calculate the total interest first, then apply the tax deduction.

Aidan invests €5000 in a special savings account for two years. The rate is 3% for the first year with a higher rate for the second year. Tax at 40% will be deducted each year from the interest earned. a) How much will Aidan's investment be worth at the end of the first year, after tax is paid? b) Aidan calculates that, after tax has been deducted, his investment will be worth €5196.89 at the end of the second year. Calculate the rate of interest for the second year.

Learn how to calculate the value of Aidan's €5000 investment after one year, factoring in a 3% interest rate and 40% tax. Also, calculate the interest rate for the second year, with a final value of €5196.89. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation