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**Hint**: In the above problem, the population of Andhra Pradesh in 1988 is given and the population growth rate per annum is also given. And we have to find the population after 20 years which we are going to calculate by using the following formula $ {{P}_{2008}}={{P}_{1988}}{{\left( 1+\dfrac{r}{100} \right)}^{n}} $ . In this formula, $ {{P}_{2008}} $ is the population in 2008 and $ {{P}_{1988}} $ is the population in $ 1988 $ , “r” is the growth rate per annum and “n” is the number of years from 1988 to 2008. Now, substitute the population in 1988, growth rate as $ 2.4\% $ and n as 20 and then solve this equation to get the population in 2008.

**:**

__Complete step-by-step answer__The population of Andhra Pradesh in 1988 is given as $ 5.4\times {{10}^{7}} $ .

The population growth rate per annum is given as $ 2.4\% $ . We were asked to find the population in 2008.

The number of years passed from 1988 to 2008 is 20 years. We are going to use the following formula to calculate the population in 2008 as follows:

$ {{P}_{2008}}={{P}_{1988}}{{\left( 1+\dfrac{r}{100} \right)}^{n}} $ ………. Eq. (1)

In the above formula, $ {{P}_{2008}} $ is the population in 2008 and $ {{P}_{1988}} $ is the population in $ 1988 $ , “r” is the growth rate per annum and “n” is the number of years from 1988 to 2008.

Substituting $ {{P}_{1998}}=5.4\times {{10}^{7}} $ , “r” as $ 2.4\% $ and n as 20 in eq. (1) we get,

$ \begin{align}

& {{P}_{2008}}=5.4\times {{10}^{7}}{{\left( 1+\dfrac{2.4}{100} \right)}^{20}} \\

& \Rightarrow {{P}_{2008}}=5.4\times {{10}^{7}}{{\left( 1+0.024 \right)}^{20}} \\

& \Rightarrow {{P}_{2008}}=5.4\times {{10}^{7}}{{\left( 1.024 \right)}^{20}} \\

& \Rightarrow {{P}_{2008}}=5.4\times {{10}^{7}}\left( 1.61 \right) \\

& \Rightarrow {{P}_{2008}}=8.694\times {{10}^{7}} \\

\end{align} $

Rounding off the above number we get,

$ {{P}_{2008}}=8.7\times {{10}^{7}} $

From the above calculation, we have the population of Andhra Pradesh in 2008 is $ 8.7\times {{10}^{7}} $ .

**Note**: It has observed that generally, in writing the formula students forgot to divide the rate per annum by 100 so the formula if you forgot to divide rate by 100 is:

$ {{P}_{2008}}={{P}_{1988}}{{\left( 1+r \right)}^{n}} $

The above formula is a wrong formula. To save yourself from making this mistake you can remember as the rate is always written in $ \% $ and percent is out of 100 so from that you can remember to divide the rate by 100.

$ {{P}_{2008}}={{P}_{1988}}{{\left( 1+\dfrac{r}{100} \right)}^{n}} $

One more important thing to be noted here is that “n” in the above formula is in years so sometimes question gives the time duration in months so first of all convert into years by dividing months by 12 then substitute the value of n like time duration given as 8 months so converting 8 months to year by dividing 8 by 12.

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