Question

Calculate the compound interest for ₹. $15000$for $4$ years at $16\% $compounded semi-annually.
$
  A)\,\,12763 \\
  B)\,\,11159 \\
  C)\,\,10159 \\
  D)\,\,9159 \\
 $

Answer 1

Hint:For compound interest when interest is compounded semi-annually, we first change rate and time according to semi-annually. And then substituting values in formula to calculate amount and then finding difference of amount and principal to get required compound interest.
Formulas used: $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$, C.I. = A – P where is P is principal , R is rate per annum, T is time and A is amount due on principal and I is compound interest

Complete step-by-step Solution:
Given,
Principal = $15000$
Rate = $16\% $
Time = $4$ years.
Since, it is given that interest is compounded half yearly.
Therefore, we change the rate and time according to half yearly.
For this we divide the given rate by two and multiply time by two.
Rate = $\dfrac{1}{2} \times 16 = 8\% (for\,\,half\,\,yearly)$
Time=$4 \times 2(for\,\,half\,\,yearly)$
Also, we know that amount in compound interest is given as: $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$
Substituting values in above formula. We have,
$A = 15000{\left( {1 + \dfrac{8}{{100}}} \right)^8}$
Or
$A = 15000{\left( {1 + 0.08} \right)^8}$
To evaluate above we use a binomial expansion formula and take only the first four terms and neglecting rest as they are very small.
$={\left( {1 + x} \right)^n} = 1 + nx + \dfrac{{n(n - 1)}}{{2!}}{x^2} + \dfrac{{n(n - 1)(n - 2}}{{3!}}{x^3} + ............$
$
 = {\left( {1 + 0.08} \right)^8} = 1 + 8 \times 0.08 + \dfrac{{8(8 - 1)}}{{2!}}{x^2} + \dfrac{{8(8 - 1)(8 - 2)}}{{3!}}{x^3} + .............neglecting \\
   = 1 + 1 + 0.64 + 0.1792 + 0.028672 + 0.0028672 \\
 $
\[ = 1.8507392\]
Usding, in above we have:
$
  A = 15000(1.850739) \\
   \Rightarrow A = 27761.085 \\
 $
Also, we know that compound interest is given as: A – P
$
  \therefore C.I. = A - P \\
   = 27761.085 - 15000 \\
   = 12761.085 \\
 $
As, we have taken only four terms of binomial expansion.
So, we are not getting the exact value of compound interest but an approximate value.
Therefore, from above we see that compound interest on a given amount is ₹. $12761.085$ Or we can say ₹.$12763$

Hence, from given options we can say correct option is option (A)

Note:While rounding off it is the dividend that is being rounded off because the divisor already has only $1$ significant digit so it cannot be further rounded off to $2$, since $2$ being smaller than $5$, will otherwise make the denominator $0$ and the division will become invalid, since anything divided by $0$ is not defined.