Question

In a factory, the production of scooters rose to $48400$ from $40000$ in $2$ years. Find the rate of growth per annum.

Answer 1

We have given that the production of scooters rose from 40000 to 48400 in two years. We have to find the rate of growth. Firstly we consider the rate of growth equal to $R$% we take the production of scooter equal P and production of the scooter after 2 years equal to $A$. Then we apply the formula.
$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}$
Here ‘$n$’ represents the number of years since $A$, $P$ and n is known ... the formula. So we can calculate the value of $R$.

Complete step-by-step answer:
The production of scooters is equal to $P\; = {\text{ }}40000$
The production of scooters after two years $A = {\text{ }}48400$
Time period is equal to $\;n = {\text{ }}2{\text{ }}years$
Let R be the rate of growth of the scooter per annum
Now we have the formula
$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^2}$
Putting the value of $P$, $A$ and n in the formula, we get
$\Rightarrow 48400 = 40000{\left( {1 + \dfrac{R}{{100}}} \right)^2}$
$\Rightarrow \dfrac{{48400}}{{40000}} = {\left( {1 + \dfrac{R}{{100}}} \right)^2}$
$\Rightarrow \dfrac{{484}}{{400}} = {\left( {1 + \dfrac{R}{{100}}} \right)^2}$
Taking square root on both sides
$\Rightarrow \dfrac{{484}}{{400}} = {\left( {1 + \dfrac{R}{{100}}} \right)^2}$
$\Rightarrow \sqrt {{{\left( {\dfrac{{22}}{{20}}} \right)}^2}} = \sqrt {{{\left( {1 + \dfrac{R}{{100}}} \right)}^2}}$
The square cancel square root on both sides then we left with
\Rightarrow $$$$\dfrac{{22}}{{20}} = 1 + \dfrac{R}{{100}}
In this step +1 is transfer to other side, by this there is change of sign that is -1
\Rightarrow $$$$\dfrac{R}{{100}} = \dfrac{{22}}{{20}} - 1
\Rightarrow $$$$\dfrac{{22 - 20}}{{20}} = \dfrac{2}{{20}}
In this next step we change $\dfrac{{22 - 20}}{{20}}$ with $\dfrac{R}{{100}}$
\Rightarrow $$$$\dfrac{R}{{100}} = \dfrac{2}{{20}}
\Rightarrow $$$$R = \dfrac{{2 \times 100}}{{20}}
\Rightarrow $$$$R = 10\%

So, the rate of growth of scooter per annum is equal to $10\%$.

Note: The growth rate refers to the percentage change in a specific variable within a specific time. For inventors growth rate typically represents the compounded annualized rate of growth of a company’s revenues. Growth rates are used to express the annual change in a variable as a percentage.Growth rate can be beneficial in accessing a company’s performance.