Question

Shikha invested Rs.6400 for 3 years at the rate of $10\% $ per annum compounded annually. Sneha invested the same amount at the same rate for the same time but on simple interest. Who gets more interest and by how much?
A) Sneha, \[{\text{Rs}}.198.40\]
B) Sneha, \[{\text{Rs}}.146.50\]
C) Shikha, \[{\text{Rs}}.146.50\]
D) Shikha, \[{\text{Rs}}.198.40\]

Answer 1

Hint: Here, we are required to find the compound interest and the simple interest earned by Shikha and Sneha respectively, when they invest the same amount for the same time period and for the same rate of interest. We will substitute the given values in the formula of Simple and Compound interests and find which interest is greater. Then, we will subtract the other interest from the same and get the required answer.

Formula Used:
$C.I = P{\left( {1 + \dfrac{R}{{100}}} \right)^n} - P$
$S.I = \dfrac{{P.R.T}}{{100}}$

Complete step by step solution:
According to the question,
Shikha invested ${\text{Rs}}.6,400$ for 3 years at the rate of $10\% $ per annum compounded annually.
Hence, the amount invested by Shikha, Principal,$P = {\text{Rs}}.6,400$
The amount is invested for a total duration of 3 years, $n = 3$
The rate of interest per annum, $R = 10\% $
Since, the principal invested by Shikha is compounded annually.
Hence, we will use the formula of Compound Interest.
Its formula is:
$C.I = P{\left( {1 + \dfrac{R}{{100}}} \right)^n} - P$
Where, $C.I$ is the Compound Interest, $P$ is the Principal, $R$ is the rate of interest per annum and $n$ is the time period.
Hence, substituting the given values in this formula, we get,
$C.I = 6400{\left( {1 + \dfrac{{10}}{{100}}} \right)^3} - 6400$
$ \Rightarrow C.I = 6400{\left( {1 + 0.1} \right)^3} - 6400$
$ \Rightarrow C.I = 6400{\left( {1.1} \right)^3} - 6400$
Taking 6400 common,
$ \Rightarrow C.I = 6400\left( {1.331 - 1} \right) = 6400 \times 0.331$
$ \Rightarrow C.I = 6400 \times \dfrac{{331}}{{1000}} = \dfrac{{331 \times 64}}{{10}} = \dfrac{{21184}}{{10}}$
$ \Rightarrow C.I. = 2118.40$
Hence, total interest received by Shikha is ${\text{Rs}}.2118.40$
Now, according to the question,
Sneha invested the same amount at the same rate for the same time but on simple interest.
Hence, the amount invested by Sneha, Principal,$P = {\text{Rs}}.6,400$
The amount is invested for a total duration of 3 years, $T = 3$
The rate of interest per annum, $R = 10\% $
Since, the principal invested by Sneha is on simple interest.
Hence, we will use the formula:
$S.I = \dfrac{{P.R.T}}{{100}}$
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.
Hence, substituting the given values in this formula, we get,
$S.I = \dfrac{{\left( {6400} \right)\left( {10} \right)\left( 3 \right)}}{{100}}$
$ \Rightarrow S.I = 64 \times 30 = 1920$
Hence, total interest received by Sneha is ${\text{Rs}}.1920.00$
Therefore, Shikha gets more interest than Sneha.
The difference between their interests is:
$2118.40 - 1920.00 = {\text{Rs}}.198.40$
Hence, Shikha gets more interest than Sneha by ${\text{Rs}}.198.40$.

Therefore, option D is the correct answer.

Note:
In this question we have used the formula of Simple Interest. Simple Interest is the interest earned on the Principal or the amount of loan. Its formula is, as we have discussed,
$S.I = \dfrac{{P.R.T}}{{100}}$
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.
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’.
Its formula is:
$C.I = P{\left( {1 + \dfrac{R}{{100}}} \right)^n} - P$
Where, $C.I.$ is the Compound Interest, $P$ is the Principal, $R$ is the rate of interest per annum and $n$ is the time period.