Richland’s real GDP per person is $20,000, and Poorland’s real GDP per person is $10,000. However, Richland’s real GDP per person is growing at 1 percent per year, and Poorland’s is growing at 3 percent per year. Compare real GDP per person in the two countries after 10 years and after 20 years. Approximately how many years will it take Poorland to catch up to Richland?
Instructions: Enter your responses as whole numbers.
 
GDP per person	GDP per person after 10 years	GDP per person after 20 years
Richland	$  	$  
Poorland	$  	$  
 
Poorland will catch up to Richland in   years.

To solve this problem, we will use the formula for future GDP per person with continuous growth:

\[
\text{Future GDP} = \text{Current GDP} \times (1 + \text{growth rate})^{\text{number of years}}
\]

### Step 1: Calculate the real GDP per person for Richland and Poorland after 10 and 20 years

#### Richland's GDP after 10 years:
Richland's current GDP per person = $20,000  
Richland's growth rate = 1% = 0.01 per year

Using the formula:

\[
\text{GDP after 10 years (Richland)} = 20,000 \times (1 + 0.01)^{10}
\]

#### Poorland's GDP after 10 years:
Poorland's current GDP per person = $10,000  
Poorland's growth rate = 3% = 0.03 per year

Using the formula:

\[
\text{GDP after 10 years (Poorland)} = 10,000 \times (1 + 0.03)^{10}
\]

#### Richland's GDP after 20 years:
Using the formula for Richland after 20 years:

\[
\text{GDP after 20 years (Richland)} = 20,000 \times (1 + 0.01)^{20}
\]

#### Poorland's GDP after 20 years:
Using the formula for Poorland after 20 years:

\[
\text{GDP after 20 years (Poorland)} = 10,000 \times (1 + 0.03)^{20}
\]

### Step 2: Determine when Poorland will catch up to Richland

To calculate when Poorland will catch up, we set up the equation:

\[
10,000 \times (1 + 0.03)^t = 20,000 \times (1 + 0.01)^t
\]

We will solve this equation to find $ t $, the number of years it will take for Poorland's GDP per person to equal Richland's GDP per person.

Let me compute these values.Here are the results:

- **Richland's GDP per person after 10 years**: $22,092
- **Poorland's GDP per person after 10 years**: $13,439
- **Richland's GDP per person after 20 years**: $24,404
- **Poorland's GDP per person after 20 years**: $18,061

It will take approximately **35 years** for Poorland to catch up to Richland in terms of real GDP per person.

Would you like further explanation or have any questions?

### Related Questions:
1. How would the catch-up time change if Richland's growth rate increased to 2%?
2. What if Poorland's growth rate decreased to 2%—how long would it take to catch up?
3. What happens if both countries grow at the same rate of 2% per year?
4. How much of a difference in GDP per person would there be after 50 years?
5. What is the formula to calculate future GDP using compound interest?

### Tip:
A small difference in growth rates can significantly affect long-term outcomes due to the power of compounding over time.

Richland’s real GDP per person is $20,000, and Poorland’s real GDP per person is $10,000. However, Richland’s real GDP per person is growing at 1 percent per year, and Poorland’s is growing at 3 percent per year. Compare real GDP per person in the two countries after 10 years and after 20 years. Approximately how many years will it take Poorland to catch up to Richland?

This problem compares the real GDP per person growth rates of Richland and Poorland over 10 and 20 years. Richland's GDP grows at 1% per year, while Poorland's grows at 3%. Learn how to calculate future GDP values and estimate the time it takes for Poorland to catch up. Suitable for Grades 9-12, this problem covers exponential growth and compounding concepts.

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