Question

Geeta borrowed ${\text{Rs}}15,000$ for 18 months at a certain rate of interest compounded semi-annually. If at the end of six months it amounted to ${\text{Rs}}15,600$; calculate:
1) The rate of interest per annum
2) The total amount of money that Geeta must pay at the end of 18 months in order to clear the account.

Answer 1

Hint:
Here, we will use the formula of simple interest and find the rate of interest per annum. Then, we will substitute the values in the formula of simple interest for all the three semi-annual years and using the logic of compound interest, we will be able to find the required amount of money to be paid by Geeta at the end of 18 months.

Formula Used:
$S.I = \dfrac{{P.R.T}}{{100}}$, where $S.I$ is the Simple Interest, $P$ is the Principal, $R$ is the rate of interest per annum and $T$ is the time period.

Complete step by step solution:
Sum of money borrowed by Geeta, i.e. Principal, $P = {\text{Rs}}15,000$
Time period for which the money has been borrowed, $T = 18$ months
Let the rate of interest be $R$, where the interest has been compounded semi-annually.
Now, we know that,
Total Amount is equal to the sum of principal and interest
Hence, Amount, $A = P + I$
Here, substituting $P = {\text{Rs}}15,000$ and according to the question, amount $A = {\text{Rs}}15,600$ at the end of six months.
$ \Rightarrow 15600 = 15000 + I$
$ \Rightarrow I = 600$
Therefore, the interest , $I = 600$
Now, we know that for the first year or the first semi-annual year, the compound interest is the same as the simple interest.
Thus, using the formula of simple interest,
$S.I = \dfrac{{P \cdot R \cdot T}}{{100}}$
Or $R = \dfrac{{S.I \times 100}}{{P \times T}}$
Hence, substituting the known values, we get,
$R = \dfrac{{600 \times 100}}{{15000 \times \dfrac{1}{2}}}$
This is because, semi-annually can be written as $\dfrac{1}{2}$ years.
$ \Rightarrow R = \dfrac{{60}}{{15}} \times 2 = 8\% $
Therefore, the rate of interest per annum $ = 8\% $
Now, in compound interest, the same amount is carried forwarded to the next year as a principal.
Hence, for the ${2^{nd}}$ half year,
Principal, $P = 15,600$
Rate, $R = 8\% $
And, the time, $T = \dfrac{1}{2}$year
Hence, we will again use simple interest for the second half year,
Thus, we get,
$S.I = \dfrac{{P \cdot R \cdot T}}{{100}}$
$ \Rightarrow S.I = \dfrac{{15600 \times 8 \times 1}}{{100 \times 2}} = 156 \times 4 = 624$
Therefore, total amount after the second half year, will be:
$A = P + I$
$ \Rightarrow A = 15600 + 624 = {\text{Rs}}16,224$
Hence, this amount will be the principal for the next half year.
Therefore, for the ${3^{rd}}$ half year,
Principal, $P = {\text{Rs}}16,224$
Rate, $R = 8\% $
And, the time, $T = \dfrac{1}{2}$ year
Hence, we will again use simple interest for the third half year,
Thus, we get,
$S.I = \dfrac{{P \cdot R \cdot T}}{{100}}$
$ \Rightarrow S.I = \dfrac{{16224 \times 8 \times 1}}{{100 \times 2}} = \dfrac{{16224 \times 4}}{{100}} = \dfrac{{64896}}{{100}} = 648.96$
Therefore, total amount after the third half year, will be:
$A = P + I$
$ \Rightarrow A = 16224 + 648.96 = {\text{Rs}}16,872.96$

Therefore, the total amount of money that Geeta must pay at the end of 18 months in order to clear the account is ${\text{Rs}}16,872.96$
Hence, this is the required answer.

Note:
In this question, we have used the formula of Simple Interest as well as Compound Interest. Simple Interest is the interest earned on the Principal or the amount of loan. There is another type of interest, which is the Compound Interest. Compound Interest is calculated both on the Principal as well as on the accumulated interest of the previous year. Hence, this is also known as ‘interest on interest’.