Question

The difference between the CI and SI on an amount of RS. 18000 in 2 years was RS. 405. What is the rate of interest in % per annum?
(a) 10%
(b) 15%
(c) 20%
(d) 25%

Answer 1

Hint: We solve this problem by using the simple formulas of simple interest (SI) and compound interest (CI).
We have the formula for the Compound interest as
$$CI=P{(1+{\displaystyle \frac{R}{100}})}^T-P$$
Where, $${}^{\prime }{P}^{\prime }$$ is the principal amount, $${}^{\prime }{R}^{\prime }$$ is the rate of interest and $${}^{\prime }{T}^{\prime }$$ is time period.
We have the formula for the simple interest as
$$SI={\displaystyle \frac{P\times T\times R}{100}}$$
Where, $${}^{\prime }{P}^{\prime }$$ is the principal amount, $${}^{\prime }{R}^{\prime }$$ is the rate of interest and $${}^{\prime }{T}^{\prime }$$ is time period.
By using these formulas and the given condition that is the difference between CI and SI is RS. 405 we find the value of rate of interest keeping in mind that CI will always be greater than SI.

Complete step by step answer:
We are given that the principal amount as RS. 18000
Let us assume that the principal amount as
$$\Rightarrow P=18000$$
We are given that the time period as 2 years.
Let us assume that the time period as
$$\Rightarrow T=2$$
We are given that the difference between CI and SI as RS. 405
We know that the compound interest will always be greater than the simple interest
By using this condition, let us convert the given statement into mathematical equation then we get
$$\Rightarrow CI-SI=405....equation(i)$$
Let us assume that the rate of interest per annum as $${}^{\prime }{R}^{\prime }$$
We know that the formula for the Compound interest as
$$CI=P{(1+{\displaystyle \frac{R}{100}})}^T-P$$
Where, $${}^{\prime }{P}^{\prime }$$ is the principal amount, $${}^{\prime }{R}^{\prime }$$ is the rate of interest and $${}^{\prime }{T}^{\prime }$$ is time period.
By using this formula to given parameters we get
$$\Rightarrow CI=18000{(1+{\displaystyle \frac{R}{100}})}^2-18000........equation(ii)$$
We know that the formula for the simple interest as
$$SI={\displaystyle \frac{P\times T\times R}{100}}$$
Where, $${}^{\prime }{P}^{\prime }$$ is the principal amount, $${}^{\prime }{R}^{\prime }$$ is the rate of interest and $${}^{\prime }{T}^{\prime }$$ is time period.
By using this formula to given parameters we get
$$\Rightarrow SI=18000\left({\displaystyle \frac{2R}{100}}\right).....equation(iii)$$
Now by substituting both equation (ii) and equation (iii) in equation (i) we get
$$\begin{array}{rl}& \Rightarrow 18000{(1+{\displaystyle \frac{R}{100}})}^2-18000-18000\left({\displaystyle \frac{2R}{100}}\right)=405\\ & \Rightarrow 18000({(1+{\displaystyle \frac{R}{100}})}^2-{\displaystyle \frac{2R}{100}})=18000+405\\ & \Rightarrow {(1+{\displaystyle \frac{R}{100}})}^2-{\displaystyle \frac{2R}{100}}={\displaystyle \frac{18405}{18000}}\end{array}$$
We know that the formula for square of sum of two numbers that is
$${(a+b)}^2=a^2+2ab+b^2$$

Now, by using this formula to above equation we get
$$\begin{array}{rl}& \Rightarrow 1+{\displaystyle \frac{2R}{100}}+{\displaystyle \frac{R^2}{10000}}-{\displaystyle \frac{2R}{100}}={\displaystyle \frac{18405}{18000}}\\ & \Rightarrow {\displaystyle \frac{R^2}{10000}}={\displaystyle \frac{18405}{18000}}-1\\ & \Rightarrow {\displaystyle \frac{R^2}{10000}}={\displaystyle \frac{405}{18000}}\end{array}$$
Now, by cross multiplying the terms from LHS to RHS in above equation we get’
$$\begin{array}{rl}& \Rightarrow R^2={\displaystyle \frac{405}{18000}}\times 10000\\ & \Rightarrow R=\sqrt{225}\\ & \Rightarrow R=15\mathrm{\%}\end{array}$$
Therefore, the rate of interest is 15%

So, the correct answer is “Option b”.

Note: Students may make mistakes in taking the formulas for the compound interest.
The compound interest is applied based on the principal amount of the starting of the year which changes for every year. So, the formula for the compound interest will be
$$CI=P{(1+{\displaystyle \frac{R}{100}})}^T-P$$
Where, $${}^{\prime }{P}^{\prime }$$ is the principal amount, $${}^{\prime }{R}^{\prime }$$ is the rate of interest and $${}^{\prime }{T}^{\prime }$$ is time period.
Here we subtracted the principal amount because the expression $$P{(1+{\displaystyle \frac{R}{100}})}^T$$ gives the final amount after the time period of $${}^{\prime }{T}^{\prime }$$ but, we need only compound interest. So, we need to subtract the principal amount from the total amount to get the compound interest only.
Students may miss point and takes the formula as
$$CI=P{(1+{\displaystyle \frac{R}{100}})}^T$$
This gives the wrong answer because the above formula gives the total amount after $${}^{\prime }{T}^{\prime }$$ years but not compound interest.