Question

A sum of money invested at 8% per annum amounts to Rs. 12,122 in 2 years. What will it amount to in 2 years 8 months at 9% per annum?

Answer 1

Hint: We will assume the principal to be x and then we will substitute all the values given in the formula of simple interest \[\text{S}\text{.I=}\dfrac{\text{P }\!\!\times\!\!\text{ R }\!\!\times\!\!\text{ T}}{\text{100}}\] and we will substitute \[\text{S}\text{.I=A-P}\] in this formula and hence we will calculate the principal.

Complete step-by-step answer:
Before proceeding with the question we should understand the concept of principal, amount and simple interest rate.
Simple Interest (S.I) is the method of calculating the interest amount for some principal amount of money. We often have to borrow money from banks in the form of a loan. During payback, apart from the loan amount, we pay some more money that depends on the loan amount as well as the time for which we borrow. This is called simple interest.
The Formula for simple interest helps you to find the interest amount if the principal amount, rate of interest and time periods are given. Simple interest formula is given as \[\text{S}\text{.I=}\dfrac{\text{P }\!\!\times\!\!\text{ R }\!\!\times\!\!\text{ T}}{\text{100}}.......\text{(1)}\] where SI is simple interest, P is principal, R is interest rate, T is time period.
In order to calculate the total amount, the following formula is used \[\text{A=P+S}\text{.I}.....\text{(2)}\] where A is the amount and P is the principal. Amount(A) is the total money paid back at the end of the time period for which it was borrowed.
Rate of interest(R) given is 8% and amount(A) is Rs. 12,122. Also the time period(T) is 2 years. So let the principal be x. Using this information and substituting all the values in equation (1) and equation (2), we get,
\[\text{S}\text{.I=}\dfrac{x\text{ }\!\!\times\!\!\text{ 8 }\!\!\times\!\!\text{ 2}}{\text{100}}.......\text{(3)}\] and \[12122-x\text{=S}\text{.I}.....\text{(4)}\]
Now from equation (4) substituting the value of S.I in equation (3) we get,
\[12122-x\text{=}\dfrac{x\text{ }\!\!\times\!\!\text{ 8 }\!\!\times\!\!\text{ 2}}{\text{100}}.......\text{(5)}\]
Rearranging and solving for x we get,
\[\begin{align}
  & \Rightarrow 100(12122-x)=16x \\
 & \Rightarrow 1212200=100x+16x \\
 & \Rightarrow 116x=1212200 \\
 & \Rightarrow x=\dfrac{1212200}{116}=10450.......(6) \\
\end{align}\]
Now we have calculated the principal as Rs. 10,450. Now the time period(T) is \[2+\dfrac{8}{12}=\dfrac{8}{3}\] years and the rate of interest(R) is 9%. So using this information we calculate simple interest.
\[\text{S}\text{.I=}\dfrac{\text{10450 }\!\!\times\!\!\text{ 9 }\!\!\times\!\!\text{ }\dfrac{8}{3}}{\text{100}}=2508.........(7)\]
Substituting the value of S.I and principal in equation (2) we get,
\[\text{A=10450+2508=12958}\]
Hence the principal amounts to Rs. 12,958 in 2 years 9 months.

Note: We must not get confused in converting the time period 2 years 8 months in years. Remembering the formula of simple interest and amount is the key here. We in a hurry may commit a mistake in solving for x in equation (5) so we need to be careful with the basic calculation steps.