Question

Find the amount and C.I. on Rs 24000 compounded semi-annually for $1\dfrac{1}{2}$ years at the rate of $10\% $ p.a.

Answer 1

Hint: So to solve this question we first need to find the compound interest for the first two years and after that, we have to find the simple interest for the last $\dfrac{1}{2}$ year where the principal amount will be the amount after 1 year and the summation of the all amount will be the final amount after compounded.

Formula used: The formula for compound interest is,
$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$
The formula for simple interest is,
$I = \dfrac{{P \times R \times T}}{{100}}$
Where P is the principal
R is the rate of interest
T is the time

Complete step-by-step solution:
Here first we will find the compounded amount for the first year,
So, for this,
$P = 24000,R = 10,T = 1$
Therefore, the compound amount will be,
$ \Rightarrow A = 24000{\left( {1 + \dfrac{{10}}{{100}}} \right)^1}$
Cancel out the common factors,
$ \Rightarrow A = 24000\left( {1 + \dfrac{1}{{10}}} \right)$
Take LCM inside the bracket,
$ \Rightarrow A = 24000\left( {\dfrac{{10 + 1}}{{10}}} \right)$
Add the terms and cancel out the common factors,
$ \Rightarrow A = 2400 \times 11$
Multiply the terms,
$ \Rightarrow A = 26400$
Now, this amount will be used as the principal amount for the next half year.
So, for this,
$P = 26400,R = 10,T = \dfrac{1}{2}$
Therefore, the simple interest will be,
$ \Rightarrow I = \dfrac{{26400 \times 10 \times \dfrac{1}{2}}}{{100}}$
Simplify the terms,
$ \Rightarrow I = 1320$
Therefore after 2.5 years, the compounded amount will be,
$ \Rightarrow 26400 + 1320 = 27720$
As we know,
Compound Interest = (Compounded Amount - Principal Amount)
Substitute the values,
$ \Rightarrow $ Compound Interest $ = 27720 - 24000$
Subtract the terms on the right side,
$ \Rightarrow $ Compound Interest $ = 3720$

Hence the amount is Rs 27720 and the compound interest is Rs 3720.

Note: To solve this we need to know that Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. It is the result of reinvesting interest, rather than paying it out, so that interest in the next period is then earned on the principal sum plus previously accumulated interest.
The simple annual interest rate is the interest amount per period, multiplied by the number of periods per year. The simple annual interest rate is also known as the nominal interest rate.