Question

ICICI lent $Rs.1$ lakh to captain Ram Singh at $6\% $ per annum of simple interest for $10$ years period. Meanwhile, ICICI offered a discount in rate of interest for armed forces. Thus the rate of interest ICICI decreased to $4\% $ . In this way, Ram Singh had to pay a total amount of $1.48$ lakh. After how many years Ram Singh got the discount in rate of interest?
 (A) $3{\text{ years}}$
(B) ${\text{4 years}}$
(C) ${\text{6 years}}$
(D) ${\text{5 years}}$

Answer 1

Hint:
Remember that in the simple interest the principal amount never changes throughout the time period of interest. Use the amount lent as the principal amount and the interest amount can be calculated by subtracting the total amount paid by principle. Use the formula for simple interest, i.e. $S.I. = \dfrac{{{\text{Principle}} \times {\text{Rate}} \times {\text{Time}}}}{{100}}$ to calculate the interest of two terms. Assume some variable ‘m’ for the time for interest without the discount. Now apply formula and form an equation for total interest. Solve the equation with only one unknown to get the answer.

Complete step by step solution:
Here in this problem, Ram Singh is taking a loan of $Rs.100000$ from the bank at a simple interest of $6\% $ for $10$ years. But before completion of ten years, the bank decides to give a discount on his rate of interest. Now the new rate of interest is $4\% $ for the rest of the time till completion of ten years. At the end of ten years, Ram Singh had to pay $Rs.148000$ . Now with this information, we need to calculate the time at which the discount in interest was applied.
Since the principal amount that was lent was $Rs.100000$ and the amount after ten years with interest was $Rs.148000$ , therefore total interest paid $ = 148000 - 100000 = Rs.48000$
And this interest is given at two different rates but in a total of ten years.
So, let us assume that the discount on interest was given after completion of $'m'$ years.
Therefore, we can write that:
$ \Rightarrow $ Total interest amount$ = $ Interest at the rate of $6\% $ in $'m'$ years $ + $ Interest at the rate of $4\% $ in $'10 - m'$ years
As we know that the simple interest on can calculated by multiplying principle amount with rate and time in years.
$ \Rightarrow $ Simple Interest $ = \dfrac{{{\text{Principle}} \times {\text{Rate}} \times {\text{Time}}}}{{100}}$
So, using the above formula for simple interest and substituting values of interest paid we can write the equation as:
$ \Rightarrow 48000 = \dfrac{{100000 \times 6 \times m}}{{100}} + \dfrac{{100000 \times 4 \times \left( {10 - m} \right)}}{{100}}$
Now we can solve the above equation to find the value for the only unknown. Let’s take out the common from the parenthesis, this will give:
$ \Rightarrow 48000 = \dfrac{{100000 \times 6 \times m}}{{100}} + \dfrac{{100000 \times 4 \times \left( {10 - m} \right)}}{{100}} = \dfrac{{100000}}{{100}}\left( {6m + 4\left( {10 - m} \right)} \right)$
This can be further simplified by dividing both sides by $1000$ as:
$ \Rightarrow 48000 = \dfrac{{100000}}{{100}}\left( {6m + 4\left( {10 - m} \right)} \right) \Rightarrow 48 = \left( {6m + 4\left( {10 - m} \right)} \right)$
Now let’s open the inside brackets and expand the equation:
$ \Rightarrow 48 = \left( {6m + 4\left( {10 - m} \right)} \right) \Rightarrow 48 = 6m + 40 - 4m$
Transposing unknown terms on one side of the equation, we get:
\[ \Rightarrow 48 = 6m + 40 - 4m \Rightarrow 6m - 4m = 48 - 40 \Rightarrow 2m = 8\]
Therefore, we get: $m = 4$
Thus, the discount interest is applied after four years of the whole ten years

Hence, the option (B) is the correct answer.

Note:
In a question like this, we should always try to substitute all the given values in the formula and then figure out a way to find the rest of the unknown. Simple interest is a quick and easy method of calculating the interest charge on a loan. Simple interest is determined by multiplying the daily interest rate by the principal by the number of days that elapse between payments.