Question

The population of the city is $2,50,000$. The death rate is $4.5\% $ and the birth rate is $6.5\% $. What will be the city’s population after $3$ years?

Answer 1

Hint: First, the population of the city is calculated in overall years that means the city is now with $2,50,000$people.
If the death rate is $4.5\% $ and the birth rate is $6.5\% $
From this given information we have to find the population of the given city after three years.
Formula used: $P{[1 + \dfrac{R}{{100}}]^t}$ where R is the rate of growth in population, P is the initial amount and t is the time is given (three years)

Complete step by step answer:
Given that the population of the city is $2,50,000$.
To find the annual growth of the given population we need to calculate using the death and birth rate because in the birth the population will be increased and in the death, the population will be decreased.
Hence to get the annual growth in the given population we use the subtraction operation, thus we get $R = B.R - D.R$ where R is the required annual rate, birth rate minus death rate will give the requirement.
Hence applied the value of birth and death we get, $R = B.R - D.R \Rightarrow 6.5\% - 4.5\% $
Thus, further solving this we get, $R = B.R - D.R \Rightarrow 2\% $ which is the annual rate of the growth in the given population.
Hence, we have the annual rate R, we know the time t is three years (a requirement in the given question), and also, we know that initial population P is $2,50,000$
Therefore, substituting every value in the given formula, we get, $P{[1 + \dfrac{R}{{100}}]^t} = 2,50,000{[1 + \dfrac{2}{{100}}]^3}$
Further solving using the division, multiplication, and addition operation we get, \[250000 \times {[1 + \dfrac{2}{{100}}]^3} = 250000 \times {[\dfrac{{102}}{{100}}]^3}\](cross multiplied)
\[250000 \times {[\dfrac{{102}}{{100}}]^3} \Rightarrow 250000 \times {[1.02]^3}\](By division)
\[250000 \times {[1.02]^3} \Rightarrow 250000 \times 1.0612\](By the power cube using the multiplication)
Hence finally we get, $P{[1 + \dfrac{R}{{100}}]^t} = 265300$is the population after three years.

Note: We are also able to calculate the population for after one and two years like in the given same formula just apply it as one or two, we get the resultant.
Since the rate of growth is annual because the given death rate and also the birth rate are given in the calculation annually.
Hence subtracting these we get the original growth rate.
If the initial population is unknown where the after three years the population is given, then we need to apply the same formula like $P{[1 + \dfrac{2}{{100}}]^3} = 265300$ to find the initial population, we further solve this we get the same $2,50,000$.