Question

A sum of money compounded annually amounts to $1375$ in $5$ years and$1980$ in$7$ years. Find the annual rate of interest.
A) $12\% $
B) $20\% $
C) $15\% $
D) $10\% $

Answer 1

Hint - To solve this question, first we have to understand the concept behind simple interest, compound interest and then by using these parameters and putting the given values, we can calculate the sum of money compounded annually and hence, solve this question.

Complete step by step solution:
Compound interest: Compound interest is the addition of the interest to the principal sum of a loan or deposit, or in other words interest on interest .It can also be defined as the result of investing interest again, rather than paying out, so that interest in the next period is the earned on the principal sum plus previously accumulated interest.
Compound interest is interest calculated on the initial principal, which also includes all the accumulated interest from the previous periods on a deposit or loan. compound interest is the product of the initial principal amount by one plus the annual interest rate raised to the number of compound periods minus one.
Compound interest -
$
  A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}} \\
    \\
$
$
  A = {\text{final amount}} \\
  P = {\text{initial principal balance}} \\
  r\,{\text{ = }}\,{\text{interest}}\,{\text{rate}} \\
  n\, = {\text{number of times interest applied per time period }} \\
  t\, = {\text{number of time period elapsed}} \\
    \\
$
Now let’s solve our query –
In this query, Interest is compounded annually (i.e., we will be using compound interest formula),
 so, we have -
Amount $A\, = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}$
So, for the first situation,
$1375 = P \times {\left( {1 + \dfrac{R}{{100}}} \right)^5} \to \left( {eq\,1} \right)$
And for the second situation,
$1980 = P \times {\left( {1 + \dfrac{R}{{100}}} \right)^7} \to \left( {eq\,2} \right)$
 By comparing equation 2 by equation 1, we get
$
  \dfrac{{1980}}{{1375}} = {\left( {1 + \dfrac{R}{{100}}} \right)^2} \\
   \Rightarrow \dfrac{{396}}{{275}} = {\left( {1 + \dfrac{R}{{100}}} \right)^2} \\
   \Rightarrow \dfrac{{36}}{{25}} = {\left( {1 + \dfrac{R}{{100}}} \right)^2} \\
   \Rightarrow \left( {1 + \dfrac{R}{{100}}} \right) = \dfrac{6}{5} \\
   \Rightarrow \dfrac{R}{{100}} = \dfrac{1}{5} \\
   \Rightarrow R = 20\% \\
$
So, the correct answer is option B – 20%

Note - Principal is the money which is originally agreed to pay back. Interest is the cost applied on borrowing the principal. Simple interest is imposed on the original (principal) amount. However, Compound interest is calculated on principal, as well as on the interest accumulated on previous periods. For example, in this question, the annual rate of interest applied on the money is $20\% $.