Question

Romesh borrowed a sum of Rs245760 at 12.5% per annum, compounded annually. On the same day, he lent out his money to Ramu at the same rate of interest but compounded semi-annually. Find his gain after 2 years.

Answer 1

Hint: First, find the amount for the compound interest compounded yearly by applying the formula $A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}$ which Romesh borrowed. Then, find the amount for the compound interest compounded half-yearly by applying the formula $A = P{\left( {1 + \dfrac{r}{{2\times 100}}} \right)^{t \times 2}}$ which Romesh lent to Ramu. Then subtract the amount compounded annually from the amount compounded semi-annually to find the gain.

Formula used: The formula for compound interest compounded yearly is,
$A = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}$
The formula for compound interest compounded half-yearly is,
$A = P{\left( {1 + \dfrac{r}{{2 \times 100}}} \right)^{t \times 2}}$
Where P = Principal
r = rate
t = time

Complete step-by-step answer:
Principal, P = Rs. 245760
Time, t = 2 years
Rate, r = 12.5 % p.a.
For the compound interest compounded yearly,
$P = 245760$
$t = 2$
$r = 12.5\% $
Substitute the values in the formula for compounded interest compounded yearly,
$ \Rightarrow A = 245760{\left( {1 + \dfrac{{12.5}}{{100}}} \right)^2}$
Cancel out common factors inside the bracket,
$ \Rightarrow A = 245760{\left( {1 + \dfrac{1}{8}} \right)^2}$
Take LCM inside the bracket,
$ \Rightarrow A = 245760{\left( {\dfrac{{8 + 1}}{8}} \right)^2}$
Add the terms,
$ \Rightarrow A = 245760{\left( {\dfrac{9}{8}} \right)^2}$
Square the term,
$ \Rightarrow A = 245760 \times \dfrac{{81}}{{64}}$
Cancel out the common factor,
$ \Rightarrow A = 3840 \times 81$
Multiply the terms,
$\therefore A = 311040$ ………..….. (1)
So, the amount to be paid by Romesh after 2 years is Rs. 311040.
For the compound interest compounded half-yearly,
$P = 245760$
$t = 2$
$r = 12.5\% $
Substitute the values in the formula for compounded interest compounded semi-yearly,
$ \Rightarrow A = 245760{\left( {1 + \dfrac{{12.5}}{{2 \times 100}}} \right)^{2 \times 2}}$
Cancel out common factors inside the bracket and multiply the power,
$ \Rightarrow A = 245760{\left( {1 + \dfrac{1}{{16}}} \right)^4}$
Take LCM inside the bracket,
$ \Rightarrow A = 245760{\left( {\dfrac{{16 + 1}}{{16}}} \right)^4}$
Add the terms,
$ \Rightarrow A = 245760{\left( {\dfrac{{17}}{{16}}} \right)^4}$
Multiply the term,
$ \Rightarrow A = 245760 \times \dfrac{{83521}}{{65536}}$
Cancel out the common factor,
$ \Rightarrow A = 3.75 \times 83521$
Multiply the terms,
$\therefore A = 313203.75$ ………..... (2)
So, the amount to be received by Romesh after 2 years is Rs. 313203.75.
For the gain find the difference between two amounts is by subtracting equation (1) from equation (2),
$\therefore 313203.75 - 311040 = {\text{Rs}}{\text{. 21}}63.75$

Hence, the gain of Romesh after 2 years is Rs. 2162.75.

Note: Compound interest is the interest calculated on the principal and the interest accumulated over the previous period. It is different from the simple interest where interest is not added to the principal while calculating the interest during the next period.