Question

Dictionopolis has a real GDP per capita of $5,000, which is growing at 10% per year. Sitara has a real GDP per capita of $10,000, which is growing at 5% per year. Real GDP per capita for the two nations will converge in about _____ years.

Answer 1

By solving the equation (1.10/1.05)^t = 2, we find that the real GDP per capita for Dictionopolis and Sitara will converge in approximately 14 years.

Step-by-step explanation:

To find out how many years it will take for the real GDP per capita of Dictionopolis and Sitara to converge, we need to use the formula for compound growth. The calculation can be done by setting the future value of Dictionopolis's GDP per capita equal to the future value of Sitara's GDP per capita and solving for the number of years, t.

The formula for future value is:
FV = PV (1 + r)^t,

where FV is the future value, PV is the present value, r is the rate of growth, and t is the number of years.

For Dictionopolis:

FV = $5,000 (1 + 0.10)^t

For Sitara:

FV = $10,000 (1+ 0.05)^t

Now, equating both future values and solving for t, we get:

$5,000 (1 + 0.10)^t = $10,000 (1 + 0.05)^t

To find the number of years, we need to solve this equation:

(1 + 0.10)^t / (1 + 0.05)^t = 2

(1.10/1.05)^t = 2

(1.047619)^t = 2

Taking the natural log of both sides gives us:

t * ln(1.047619) = ln(2)

t = ln(2) / ln(1.047619)

t = 14.21

The answer is that the GDP per capita for the two nations will converge in about 14 years.