Question

In a factory, the production of scooters rose to 48400 from 40000 in 2 years. Find the rate of growth per annum.
A. 20%
B. 10%
C. 30%
D. 8%

Answer 1

Hint: We will use the formula which states that:-
Present Production = Initial Production $$\times {\left(1+{\displaystyle \frac{R}{100}}\right)}^n$$, where R = Rate of Increase, n = number of years
We will put in the given values and thus eventual finding R.

Complete step-by-step answer:
We have Initial Production = 40000 and Production after two years = 48400
So, now we will use the formula:- Present Production = Initial Production $$\times {\left(1+{\displaystyle \frac{R}{100}}\right)}^n$$,
where R = Rate of Increase and n = number of years.
$\therefore$ we have 48400 = 40000 $$\times {\left(1+{\displaystyle \frac{R}{100}}\right)}^2$$
Now, we will bring 40000 from R.H.S. to L.H.S.
$\therefore {\displaystyle \frac{48400}{40000}}={\left(1+{\displaystyle \frac{R}{100}}\right)}^2$
We will take square roots on both sides.
$\therefore \sqrt{{\displaystyle \frac{48400}{40000}}}=1+{\displaystyle \frac{R}{100}}$ ……(1)
We will rewrite 48400 and 40000 as squares of some numbers like:-
48400 = $220\times 220={220}^2$
40000 = $200\times 200={200}^2$
We will put these values in (1). We will get:-
$\therefore \sqrt{{\left({\displaystyle \frac{220}{200}}\right)}^2}=1+{\displaystyle \frac{R}{100}}$
We know that the square root will cut off the square. So, we have:-
$\therefore {\displaystyle \frac{220}{200}}=1+{\displaystyle \frac{R}{100}}$
We can rewrite it as:-
${\displaystyle \frac{R}{100}}+1={\displaystyle \frac{220}{200}}$
We will now subtract 1 from both sides.
So, we get:- ${\displaystyle \frac{R}{100}}+1-1={\displaystyle \frac{220}{200}}-1$
Simplifying it by taking L.C.M. on R.H.S.
${\displaystyle \frac{R}{100}}={\displaystyle \frac{220-200}{200}}$
Now simplifying it more, we have:-
${\displaystyle \frac{R}{100}}={\displaystyle \frac{20}{200}}$
Now, we will take the 100 from the denominator of L.H.S. to R.H.S.
$\therefore R={\displaystyle \frac{20\times 100}{200}}={\displaystyle \frac{2000}{200}}=10$
Hence, the rate is 10%.

So, the correct answer is “Option B”.

Note: We need to learn the formula Present Production = Initial Production $$\times {\left(1+{\displaystyle \frac{R}{100}}\right)}^n$$,
where R = Rate of Increase and n = number of years.
We need to remember the units of n in general formulas are years but we may be sometimes provided with data in months or weeks, then we will have to convert that data in years to use the formula correctly. Always remember to take the $n^{th}$root because we have n in the formula. Here, we have taken n to be 2. So, it is a possibility that we may learn the formula with 2 instead of n.