Question

If the simple interest on a sum of money for \[2\]years at $5\% $per annum is $Rs.50$, what will be the compound interest on the same sum at the same rate for the same time

Answer 1

Hint:First, we will need to know about simple interest and compound interest.
Simple interest calculates the interest charged on a given amount of money, we can calculate the total amount by adding the principal amount and the interest.
Compound interest calculates the principal and interest accumulated over the previous period of time.

Formula used:
Simple interest $SI = \dfrac{{P \times R \times T}}{{100}}$
Compound interest $A = P{(1 + \dfrac{R}{{100}})^T}$where A is the amount,
$CI = A - P$

Complete step-by-step solution:
Given that the simple interest is $Rs.50$where the rate of the interest is $5\% $and the time taken is \[2\]years.
Apply all this information into the given simple interest formula, thus we get $SI = \dfrac{{P \times R \times T}}{{100}} \Rightarrow 50 = \dfrac{{P \times 5 \times 2}}{{100}}$ (where $R = 5,T = 2,SI = 50$from the given)
Thus, solving the equation we get $50 = \dfrac{{P \times 5 \times 2}}{{100}} \Rightarrow P \times 5 \times 2 = 100 \times 50$
$P \times 5 \times 2 = 100 \times 50 \Rightarrow P = \dfrac{{5000}}{{10}}$(By the multiplication operation)
Hence, we get the principal amount is $P = \dfrac{{5000}}{{10}} \Rightarrow P = 500$
Thus, we get the principal amount and we already know the time taken, Rate of interest.
To find the amount in the compound interest we use the formula $A = P{(1 + \dfrac{R}{{100}})^T}$where A is the amount,
Applying the know values we get, $A = P{(1 + \dfrac{R}{{100}})^T} \Rightarrow A = 500{(1 + \dfrac{5}{{100}})^2}$ (where $P = 500,R = 5,T = 2$)
Further solving we get, $A = 500{(1 + \dfrac{5}{{100}})^2} \Rightarrow A = 500{(1 + \dfrac{1}{{20}})^2} \Rightarrow 500 \times \dfrac{{21}}{{20}} \times \dfrac{{21}}{{20}}$ (by cross multiplication)
Hence, we get, the amount as $A = 500 \times \dfrac{{21}}{{20}} \times \dfrac{{21}}{{20}} \Rightarrow 551.25$(by division operation)
Since we got the amount of the compound interest and we already know the principal amount
Thus apply the two things into the Compound interest formula we get $CI = A - P \Rightarrow 551.25 - 500$
Solving this we get, $CI = 51.25Rs$
Hence the compound interest is $Rs.51.25$

Note:Since P is the principal amount, R is the rate of interest, T is the time taken, A is the amount, SI is the simple interest and CI is the compound interest.
Frist from the given that we can easily to the solution using the formula of simple interest and then we can able to find the principal amount which is unknown from the given and after that we used the formula of the amount and finally, we used compound interest formula which is amount subtracts the principal amount $CI = A - P$ and thus we get the requirement.
The comparison of simple and compound interest is $CI > SI$
We can also able to prove that from the given question we hence $SI = 50,CI = 51.25$and hence we get $CI > SI \Rightarrow 51.25 > 50$