Question

$10000$ was lent for one year at $10\%$ per annum. By how much more will the interest be, if the sum was lent at $10\%$ per annum, interest being compounded half-yearly?

Answer 1

Hint: We have given that an amount is lent for one year at a rate of $10\%$. We have to find how much more will be the interest if it is lent for $1$ year. Since interest being compounded half-yearly and interest $10\%$ per annum, we will divide the rate by 2.
Then, firstly, we calculate compound interest by the formula of compound interest then we calculate the simple interest by the formula of simple interest. After we take the difference and this will give us the required result.

Complete step-by-step answer:
We have given that Amount lent = Principal ($P$) = Rs.$10,000$
Rate of interest = $10\% $
As the amount is being compounded half-yearly, so
Rate = $\dfrac{{10}}{2} = 5\% $
Time period = $1$ year
As the amount is being compounded half-yearly, so time period = $1$ year = $2$ half years
Now, we know that amount is given as
Amount = Principal $\left( {1 + \dfrac{R}{{100}}} \right)n$
Here $R$ represents the rate of interest and $n$ represents the time period.
Amount $ = 10,000 \times {\left( {1 + \dfrac{5}{{100}}} \right)^2}$
$ = 10,000 \times {\left( {1 + \dfrac{1}{{20}}} \right)^2}$
$ = 10,000 \times {\left( {\dfrac{{21}}{{20}}} \right)^2}$
$ = 10,000 \times \dfrac{{21}}{{20}} \times \dfrac{{21}}{{20}}$
$ = 100 \times \dfrac{{21}}{2} \times \dfrac{{21}}{2}$
$ = 25 \times 21 \times 21$
$ = $ Rs.$11,025$
So, the amount is equal to Rs. $11,025$
Compound Interest is given as:
C.I. = Amount – Principal
$ = 11,025 - 10,000$
C.I. $ = $ Rs. $1,025$
Now we find if the amount is lent per annum at a rate of $10\% $ for $1$ year.
Principal $ = $ Rs. $10,000$
Time (T) $ = 1$ year
Rate $ = 10\% $
Amount $ = $ Principal ${\left( {1 + \dfrac{R}{{100}}} \right)^n}$
$ = 10,000{\left( {1 + \dfrac{{10}}{{100}}} \right)^1}$
$ = 10,000\left( {1 + \dfrac{1}{{10}}} \right)$
$ = 10,000\left( {\dfrac{{11}}{{10}}} \right)$
Amount $ = 1000 \times 11$
 $ = Rs11,000$
C.I. = Amount – Principal
$ = 11,000 - 10,000$
C.I.$ = $ RS. $1,000$
The interest will be more by
$ = 1,025 - 1,000$

$ = $Rs. $25$

Note: Compound interest is the addition of interest to the principal sum of a loan or deposit. In other words, it is the interest 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.