Question

A certain sum of money $Q$ was deposited for ${\rm{5}}$ years and $4$ months at $4.5\% $ simple interest and mounted to $Rs.200$. Then the value of $Q$ is

Answer 1

Hint: In the solution, first, we have to convert the time in years. For that we have to divide the number of months by 12. The formula for the amount received for a simple interest is $A = P + PRT$, where $P$ is the principle amount, $R$ is the interest rate and $T$ is the time. In the question, the rate of interest, total amount received and the number of years are given. We have to substitute the given values in the formula. From that we have to calculate the principal amount.

Complete step by step solution:
Given that, the total amount received after ${\rm{5}}$ years and $4$ months $ = {\rm{Rs}}.\;248$.
Principle amount $ = Q$
Rate of interest $ = 4.5\% = 0.045$
Total deposited period = ${\rm{5}}$ years and $4$ months
Converting ${\rm{5}}$ years and $4$ months into years,
Time $\begin{array}{c} = 5 + \dfrac{4}{{12}}\\{\rm{ = }}\dfrac{{{\rm{16}}}}{{\rm{3}}}\;{\rm{years}}\end{array}$

Now, we have to calculate the principal amount $Q$.
It is known that the total amount received for a simple interest rate is $A = P + PRT$, where $P$ is the principal amount, $R$ is the interest rate, $T$ is the time and $A$ is the amount received.
Substituting the value of $Q$ for $P$, 0.045 for $R$, $\dfrac{{{\rm{16}}}}{{\rm{3}}}$ for $T$ and 248 for $A$.
Therefore, we get,
$ \Rightarrow 248 = Q + Q \times 0.045 \times \dfrac{{16}}{3}$
Now, solving the equation for $Q$, we get
$\begin{array}{l} \Rightarrow 248 = Q\left[ {1 + 0.045 \times \dfrac{{16}}{3}} \right]\\ \Rightarrow 248 = Q\left[ {1 + \dfrac{{24}}{{100}}} \right]\end{array}$

Simplify the above equation.
$\begin{array}{c}
248 = Q\left[ {\dfrac{{100 + 24}}{{100}}} \right]\\
\Rightarrow 248 = Q\left[ {\dfrac{{124}}{{100}}} \right]\\
\Rightarrow 248 = 1.24Q\\
\Rightarrow Q = \dfrac{{248}}{{1.24}}\\
\Rightarrow Q = 200
\end{array}$

Hence, the required value of $Q$ is ${\rm{Rs}}{\rm{.}}\;200$.


Note: Simple interest is calculated on the basis of a basic amount borrowed for the entire period at a particular rate of interest. Here we have to determine the value of $Q$ for the given data. Since the formula for the total amount received for the simple interest is known, so by substituting the given values of the rate of interest, the amount received and the time, we can easily calculate the required value of $Q$.