Suppose that real GDP per capita in Italy is $32,000. If real GDP per capita is growing at a rate of 2.5% per year, how many years will it take for real GDP per capita to reach $64,000? Instructions: Round your answer to 1 decimal place. years
Question
Suppose that real GDP per capita in Italy is 64,000? Instructions: Round your answer to 1 decimal place. years
Solution
This is a compound interest problem. The formula for compound interest is:
A = P (1 + r/n)^(nt)
Where: A = the amount of money accumulated after n years, including interest. P = the principal amount (the initial amount of money) r = annual interest rate (in decimal) n = number of times that interest is compounded per year t = the number of years the money is invested for
In this case, we want to find out how many years (t) it will take for the initial GDP per capita (P) of 64,000 (A) with an annual growth rate (r) of 2.5% or 0.025 (in decimal form). The growth is compounded once per year, so n = 1.
So, we need to solve the following equation for t:
64,000 = 32,000 (1 + 0.025/1)^(1*t)
Simplify the equation:
2 = (1 + 0.025)^t
Take the natural logarithm (ln) of both sides:
ln(2) = t * ln(1.025)
Solve for t:
t = ln(2) / ln(1.025)
Using a calculator, we find that t ≈ 27.7 years.
So, it will take approximately 27.7 years for real GDP per capita to reach $64,000, if it grows at a rate of 2.5% per year.
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