The per capita income of a developing country is $10,000, while that of an industrialized country is $40,000. If the developing country's income per capita grows at 6% per year while the industrialized income per capita grows at 3%, approximately how long will it take for the developing country's income per capita to equal that of the industrialized country's (rounded to the nearest tenth of a year)?
Question
The per capita income of a developing country is 40,000. If the developing country's income per capita grows at 6% per year while the industrialized income per capita grows at 3%, approximately how long will it take for the developing country's income per capita to equal that of the industrialized country's (rounded to the nearest tenth of a year)?
Solution
To solve this problem, we can use the formula for compound interest, which is A = P(1 + r/n)^(nt), where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (in decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for, in years.
However, in this case, we are not looking for the final amount (A), but the time (t) it will take for the developing country's per capita income to equal that of the industrialized country's. Therefore, we need to rearrange the formula to solve for t.
First, let's set up the equation with the given values:
10,000(1 + 0.06/1)^(1*t)
We can simplify this to:
4 = (1.06)^t
To solve for t, we can use the logarithm base 1.06 on both sides of the equation:
log1.06(4) = t
Using a calculator, we find that log1.06(4) ≈ 23.45.
Therefore, it will take approximately 23.5 years (rounded to the nearest tenth of a year) for the developing country's income per capita to equal that of the industrialized country's, assuming the annual growth rates remain constant.
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