The problem given in the image is about calculating the total amount of money accumulated after 16 years with an annual interest rate of 3%, where $300,000$ yen is deposited at the beginning of each year. The problem states that the compound interest formula factor $(1.03)^{16}$ is approximated as $1.6$.

Step-by-Step Solution

1. Given Data:

   • Annual deposit: $300,000$ yen
   • Interest rate: $3\%$ per year
   • Number of years: $16$
   • Compound interest factor: $(1.03)^{16} = 1.6$

2. Formula for Future Value of Annuity: The future value of an annuity can be calculated using the formula: $FV = P \times \frac{(1 + r)^n - 1}{r}$ where:

   • $P$ is the annual deposit
   • $r$ is the annual interest rate
   • $n$ is the number of years

3. Substitute the Given Values: $P = 300,000 \, \text{yen}$ $r = 0.03$ $n = 16$ $(1 + r)^n = 1.6$

4. Calculate the Future Value: $FV = 300,000 \times \frac{1.6 - 1}{0.03}$ Simplify the expression inside the fraction: $FV = 300,000 \times \frac{0.6}{0.03}$

5. Perform the Division: $\frac{0.6}{0.03} = 20$

6. Multiply by the Annual Deposit: $FV = 300,000 \times 20$ $FV = 6,000,000 \, \text{yen}$

Conclusion

The total amount accumulated after 16 years is $\boxed{6,000,000}$ yen.

Would you like more details or have any questions?

Here are 5 questions you might consider asking:

1. How do you derive the formula for the future value of an annuity?
2. Can you explain another example of compound interest calculations?
3. How does the interest rate affect the future value of an annuity?
4. What are the differences between simple and compound interest?
5. How can I calculate the future value for different compounding periods?

Tip: When dealing with annuities or compound interest problems, always ensure the interest rate and the number of periods match the compounding frequency (e.g., annually, semi-annually).