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We are tasked with calculating how much greater Country Q's GDP per capita is compared to Country J's after 15 years, using the compound growth rate formula. Here's the breakdown:

### Known Values:
- Initial GDP per capita for Country Q: $22,000
- Initial GDP per capita for Country J: $5,800
- After 15 years, Country J's GDP per capita is $17,894.08.
- Country J's growth rate is 4 percentage points higher than Country Q's, implying the convergence of GDP per capita.

### Steps:
1. **Calculate Country J's Growth Rate**:
   The formula for compound growth is:
   \[
   A = P \times (1 + r)^t
   \]
   Where:
   - $A$ is the amount after growth (i.e., $17,894.08),
   - $P$ is the initial amount (i.e., $5,800),
   - $r$ is the growth rate,
   - $t$ is the time in years (i.e., 15 years).

   Rearranging to solve for $r$:
   \[
   r = \left( \frac{A}{P} \right)^{\frac{1}{t}} - 1
   \]

2. **Determine Country Q's Growth Rate**:
   Since Country J's growth rate is 4% higher than Country Q's, after calculating $r$ for Country J, we subtract 4 percentage points to find Country Q's growth rate.

3. **Calculate Country Q's GDP per Capita After 15 Years**:
   Using the compound growth formula again, we will plug in Country Q's initial GDP per capita and its growth rate to find the GDP after 15 years.

4. **Find the Difference**:
   Subtract Country J's final GDP per capita from Country Q's final GDP per capita.

Let's perform the necessary calculations.### Calculations:

1. **Country J's growth rate**: 7.80% (0.0780).
2. **Country Q's growth rate**: 3.80% (0.0380).
3. **Country Q's GDP per capita after 15 years**: \$38,493.11.
4. **Difference in GDP per capita after 15 years**: \$20,599.03.

Thus, Country Q's GDP per capita is \$20,599.03 greater than Country J's after 15 years.

Would you like a more detailed explanation or any further clarifications?

### Related Questions:
1. How do we derive the compound growth formula?
2. What would happen if the growth rates were reversed between the countries?
3. How does a small difference in growth rates impact long-term GDP?
4. What is the significance of the convergence in economic terms?
5. How would inflation affect these GDP per capita calculations over time?

### Tip:
Always ensure you're working with the same time units (e.g., years) when applying growth rate formulas.

Country Q and Country J began with a GDP per capita of $22,000 and $5,800, respectively. In 15 years, Country J's GDP per capita is $17,894.08. Country J's growth rate of GDP per capita is 4 percentage points higher than Country Q's, so convergence occurs. How much greater is Country Q's GDP per capita than Country J's after 15 years have passed?

Find out how to calculate the difference in GDP per capita between two countries using the compound growth formula. This example shows the steps needed to compute GDP per capita after 15 years for both Country Q and Country J, considering a growth rate difference. Suitable for advanced students.

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