if after 6 years (in 2018), i used 20% of total amount for my 18% birthday. The rest of the amount then invest at a flat rate of 6.35% p.a. How long it takes in years to earn back the amount used for 18 th birthday ?

To solve this problem, we need to calculate the time it takes to earn back the amount used for the 18th birthday through the interest earned on the remaining amount.

### Step-by-Step Solution:

1. **Initial Setup:**
   - Let's denote the total initial amount as \( A \).
   - In 2018, you used 20% of \( A \) for your 18th birthday. So, the amount used is \( 0.2A \).
   - The remaining amount after the expenditure is \( 0.8A \).
   
2. **Interest Calculation:**
   - The remaining amount \( 0.8A \) is invested at a flat rate of 6.35% per annum.
   - The interest earned each year on the remaining amount is \( 0.0635 \times 0.8A \).

3. **Time to Earn Back the Spent Amount:**
   - We need to find the time \( t \) it takes for the interest earned to equal the amount used for the 18th birthday, which is \( 0.2A \).
   - The amount earned over time \( t \) from interest is given by \( t \times 0.0635 \times 0.8A \).
   - Set the interest earned equal to the amount used: 
     \[
     t \times 0.0635 \times 0.8A = 0.2A
     \]
   - Simplify the equation by canceling out \( A \) from both sides:
     \[
     t \times 0.0635 \times 0.8 = 0.2
     \]
   - Simplify further:
     \[
     t \times 0.0508 = 0.2
     \]
     \[
     t = \frac{0.2}{0.0508}
     \]
   - Calculate \( t \):
     \[
     t \approx \frac{0.2}{0.0508} \approx 3.937 \text{ years}
     \]

### Final Answer:
It takes approximately 3.94 years to earn back the amount used for the 18th birthday.

Would you like more details on any step? Here are some related questions:

1. How does the flat rate affect the time required to earn back the spent amount?
2. What would be the impact if the interest rate was compounded annually instead of being a flat rate?
3. How would the result change if a different percentage of the total amount was spent initially?
4. What is the difference between simple interest and compound interest in this context?
5. Can you provide a breakdown of how much interest is earned each year until the amount is earned back?

**Tip:** Always remember that the time required to earn back an amount depends on the interest rate and the remaining balance. Higher interest rates or larger remaining balances lead to quicker recovery.

Discover how long it takes to earn back the amount used for an 18th birthday after investing the remaining funds at a flat interest rate of 6.35% per annum. Detailed step-by-step solution provided.

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