# 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? 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...

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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.

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