# Calculate the approximate number of years it will take for real GDP per person to double if an economy maintains an economic growth rate of 12 percent a year and a population growth rate of 10...

Calculate the approximate number of years it will take for real GDP per person to double if an economy maintains an economic growth rate of 12 percent a year and a population growth rate of 10 percent a year.

It will take how many years for real GDP per person to double?

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Hello!

GDP per person is total GDP divided by a population size. Let's denote the initial GDP as `A` and the initial population size as `B.` Then the initial GDP per person is `A/B.`

After a year, GDP will grow by 12 percent and becomes `A*1.12.` A population will grow by 10 percent and becomes `B*1.10.` So GDP per person becomes `A/B*1.12/1.1.`

After each next year GDP per person will be multiplied by the same factor, `1.12/1.1.` Thus after `n` years it becomes `A/B*(1.12/1.1)^n.`

And the problem is to find such `n` that `A/B*(1.12/1.1)^n=A/B*2,` or `(1.12/1.1)^n=2.`

To solve this equation it is necessary to use logarithms. Take natural logarithm on both sides:

`n*ln(1.12/1.1)=ln(2), or n=ln(2)/(ln(1.12/1.1)).`

This is equal to approximately 38.5. Therefore it will take **39** full years for real GDP per person to double.