Greg bought a gold coin for $9,000. If the value of the coin increases at a constant rate of 12% every 5 years, how many years will it take for the coin to be worth $20,000?

Question

Nathan Axel · Accepted Answer

Approximately 35 years (35.23).

If the price increases by 12% every five years, to get the value after five years we multiply by #1.12#. To get the value after another five years we multiply by #1.12# again, ie we multiply the original value by #1.12^2#. So the value after #5n# years will be given by #9000*(1.12)^n# So we now want to find out when the value will be $20000. So #20000 = 9000*1.12^n# indicates that 20000/9000 = 20/9 = 1.12 Utilizing the logs from both sides, apply the LHS log rules to reduce the n: #log(20/9) = n*log(1.12)# #therefore n = log(20/9)/log(1.12) ~= 7.046# The coin reaches $20000 value after roughly 35 years.

Savannah Anderson · Answer

undefined To find out how many years it will take for the coin to be worth $20,000, we can use the formula for compound interest:

$$ A = P \times (1 + r)^n $$

Where:
- $ A $ is the final amount (which is $20,000 in this case),
- $ P $ is the initial amount (which is $9,000),
- $ r $ is the interest rate (which is 12% or 0.12),
- $ n $ is the number of periods.

Rearranging the formula to solve for $ n $:

$$ n = \frac{ \log \left( \frac{A}{P} \right) }{ \log(1 + r) } $$

Substituting the given values:

$$ n = \frac{ \log \left( \frac{20,000}{9,000} \right) }{ \log(1 + 0.12) } $$

$$ n ≈ \frac{ \log(2.222) }{ \log(1.12) } $$

$$ n ≈ \frac{ 0.3465 }{ 0.0492 } $$

$$ n ≈ 7.05 $$

So, it will take approximately 7 years for the coin to be worth $20,000.

HIX Tutor · Answer

undefined To find out how many years it will take for the coin to be worth $20,000, follow these steps:

1. **Identify Initial Value**: The coin is worth $9,000.

2. **Identify Target Value**: We want it to be worth $20,000.

3. **Identify Growth Rate**: The value increases by 12% every 5 years.

4. **Calculate Future Value Formula**:
   - Use the formula:  
   $$ \text{Future Value} = \text{Present Value} \times (1 + r)^n $$
   where $ r $ is the rate (0.12 for 12%) and $ n $ is the number of 5-year periods.

5. **Set Up the Equation**:  
   $$ 20,000 = 9,000 \times (1 + 0.12)^n $$

6. **Simplify the Equation**:
   $$ 20,000 = 9,000 \times (1.12)^n $$

7. **Divide Both Sides by 9,000**:
   $$ \frac{20,000}{9,000} = (1.12)^n $$
   $$ 2.22 \approx (1.12)^n $$

8. **Find n**:  
   - Use a calculator or logs:
   $$ n = \frac{\log(2.22)}{\log(1.12)} $$

9. **Calculate n**:  
   - Let's assume $ \log(2.22) \approx 0.346 $
   - And $ \log(1.12) \approx 0.049 $
   $$ n \approx \frac{0.346}{0.049} \approx 7.06 $$

10. **Find Total Years**:
   - Since $ n $ is the number of 5-year periods, multiply by 5:
   $$ \text{Total Years} = n \times 5 \approx 7.06 \times 5 \approx 35.3 \text{ years} $$

11. **Conclusion**:  
   It will take about **35 years** for the coin to be worth $20,000.