Question

What is the difference between the compound interests on \[Rs.5000\] for \[1\dfrac{1}{2}\]years at \[4\% \] per annum compounded yearly and half yearly?
1. \[Rs.2.04\]
2. \[Rs.3.06\]
3. \[Rs.4.80\]
4. \[Rs.8.30\]

Answer 1

Hint: Here, in the given question, we need to calculate the difference in the compound interests. At first, we need to calculate the amount in both cases and then we will calculate the compound interests in both of the cases. After this, we will subtract the values of the interests to find the difference of the interest.
 In the $1^{st}$ case, we have to find the compound interest compounded annually. We will find the amount by applying the formula $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}$ for the first year and then we will use $A = P{\left( {1 + \dfrac{R}{{2 \times 100}}} \right)^{t \times 2}}$ formula for the next half year. Here, we divided the rate by $2$ and multiplied the time by $2$ in the general formula. This is because, half-yearly is two times in a year, so the number of years is doubled and the rate is halved. Then we will subtract the principal from the amount to get the interest.
In $2^{nd}$ case, to find the amount for the compound interest compounded half-yearly, we will find this by applying the formula $A = P{\left( {1 + \dfrac{R}{{2 \times 100}}} \right)^{t \times 2}}$. Then we will subtract the principal from the amount to get the interest.
Formula used:
$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^{n \times t}}$ where,
$A$ = Amount
$P$ = Principal
$R$ = Rate
$t$ = Time
$n$ = Number of times interest applied per time period. ($n = 1$ for interest compounded annually and $n = 2$ for interest compounded half-yearly)

Complete step-by-step solution:
Case 1: When interest is compounded annually,
Principal $\left( P \right)$ = $Rs.5,000$
Time $\left( t \right)$ = $1\dfrac{1}{2}$ years.
Rate $\left( R \right)$ = $4\% $
We will calculate the amount for $1$ year. Then, we will calculate the amount for $\dfrac{1}{2}$ year.
For $1$ year,
$P = 5000$, $t = 1$ and $r = 4\% $.
On substituting the values in $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}$ formula, we get
$ \Rightarrow A = 5000{\left( {1 + \dfrac{4}{{100}}} \right)^1}$
On cancelling out common factors, we get
$ \Rightarrow A = 5000\left( {1 + \dfrac{1}{{25}}} \right)$
Take LCM
$ \Rightarrow A = 5000\left( {\dfrac{{25 + 1}}{{25}}} \right)$
$ \Rightarrow A = 5000 \times \dfrac{{26}}{{25}}$
On multiplication and division of terms, we get
$ \Rightarrow A = Rs.5,200$
Now this would act as a principal value for the next $\dfrac{1}{2}$ year.
Now, for $\dfrac{1}{2}$ year,
$P = 5200$, $t = \dfrac{1}{2}$ and $r = 4\% $
On substituting the values in $A = P{\left( {1 + \dfrac{R}{{2 \times 100}}} \right)^{t \times 2}}$ formula, we get
$ \Rightarrow A = 5200{\left( {1 + \dfrac{4}{{2 \times 100}}} \right)^{\dfrac{1}{2} \times 2}}$
On cancelling the common factors, we get
$ \Rightarrow A = 5200\left( {1 + \dfrac{1}{{50}}} \right)$
Take LCM
$ \Rightarrow A = 5200\left( {\dfrac{{50 + 1}}{{50}}} \right)$
On cancelling the common factors, we get
$ \Rightarrow A = 104 \times 51$
On multiplication of terms, we get
$ \Rightarrow A = Rs.5304$
So, interest is,
$I = A - P$
$ \Rightarrow I = 5,304 - 5000$
On subtraction of terms, we get
$ \Rightarrow I = RS.304$
Case 2: When interest is compounded half-yearly,
$P = 5000$
$t = 1\dfrac{1}{2}$ years = $\dfrac{3}{2}$ years
$r = 4\% $
On substituting the values in $A = P{\left( {1 + \dfrac{R}{{2 \times 100}}} \right)^{t \times 2}}$ formula, we get
$ \Rightarrow A = 5000{\left( {1 + \dfrac{4}{{2 \times 100}}} \right)^{\dfrac{3}{2} \times 2}}$
On cancelling the common factors, we get
\[ \Rightarrow A = 5000{\left( {1 + \dfrac{1}{{50}}} \right)^3}\]
Take LCM
\[ \Rightarrow A = 5000{\left( {\dfrac{{50 + 1}}{{50}}} \right)^3} = 5000{\left( {\dfrac{{51}}{{50}}} \right)^3}\]
On expansion, we get
\[ \Rightarrow A = 5000 \times \dfrac{{51}}{{50}} \times \dfrac{{51}}{{50}} \times \dfrac{{51}}{{50}}\]
On cancelling the common factors, we get
\[ \Rightarrow A = 51 \times \dfrac{{51}}{5} \times \dfrac{{51}}{5}\]
On multiplication and division of terms, we get
$ \Rightarrow A = Rs.5,306.04$
So, compound interest is,
$I = A - P$
$ \Rightarrow I = 5,306.04 - 5000$
$ \Rightarrow I = Rs.306.04$
Difference between two compound interests = Compound interest compounded half-yearly – Compound interest compounded annually
$ \Rightarrow Rs.\left( {306.04 - 304} \right)$
$ \Rightarrow Rs.{\text{ }}2.04$
Thus, the difference between the compound interest compounded yearly and half-yearly is \[Rs.{\text{ 2}}{\text{.04}}\]
Therefore, the correct option is 1.

Note: Here, in the given question, we calculated compound interest when interest is compounded yearly and half-yearly. But in half-yearly interest, we divided the rate by $2$ and multiplied the time period by $2$. Similarly, when we calculate the interest quarterly we divide the rate by $4$ and multiply the time period by $4$. Take care of the calculations so as to be sure of our final answer.