Question

A sum of money lent out at simple interest amounts \[{\text{Rs}}.720\] after \[2{\text{yrs}}\] and to \[{\text{Rs}}.1020\] after a further period of \[{\text{5yrs}}\]. What is the sum?
A) \[{\text{Rs}}.200\]
B) \[{\text{Rs}}.350\]
C) \[{\text{Rs}}.475\]
D) \[{\text{Rs}}.600\]

Answer 1

Hint: We will solve this question by calculating the simple interest first of all because the given value is the amount. And the amount equals the addition of principal value and the interest on it for a particular period.
\[{\text{Amount}} = {\text{Principal}} + {\text{S}}{\text{.I}}\]

Complete step-by-step solution:
Step 1: It is given that the Amount for \[2{\text{yrs}}\] at simple interest is \[{\text{Rs}}.720\] and after further, \[{\text{5yrs}}\] the Amount at simple interest is \[{\text{Rs}}.1020\] which means that total time will be equals to \[\left( {{\text{2 + 5}}} \right){\text{yrs = 7yrs}}\].
By writing the above information in the form of an equation, we get:
\[ \Rightarrow {\text{Principal}} + {\text{S}}{\text{.I for 2yrs = Rs}}{\text{.720}}\] …………….. (1)
\[ \Rightarrow {\text{Principal}} + {\text{S}}{\text{.I for 7yrs = Rs}}{\text{.1020}}\] …………… (2)
Step 2: By subtracting equation (2) from equation (1), we get:
\[
  {\text{Principal}} + {\text{S}}{\text{.I for 7yrs = Rs}}{\text{.1020}} \\
  {\text{Principal}} + {\text{S}}{\text{.I for 2yrs = Rs}}{\text{.720}} \\
  \overline {{\text{S}}{\text{.I for 5yrs}} = {\text{Rs}}.300} \\
 \]
Step 3: Now, for calculating the simple interest for \[{\text{2yrs}}\] we will solve the particular equation as below:
\[
  {\text{S}}{\text{.I for 5yrs = Rs}}{\text{.300}} \\
  {\text{S}}{\text{.I for 1yrs = }}\dfrac{{{\text{Rs}}{\text{.300}}}}{5} \\
  {\text{S}}{\text{.I for 2yrs = }}\dfrac{{{\text{Rs}}{\text{.300}}}}{5} \times 2 \\
 \]
By solving the RHS side of the equation \[{\text{S}}{\text{.I for 2yrs = }}\dfrac{{{\text{Rs}}{\text{.300}}}}{5} \times 2\], we get:
\[ \Rightarrow {\text{S}}{\text{.I for 2yrs = Rs}}{\text{.120}}\]
Step 4: Now for calculating the principal value \[{\text{2yrs}}\], we will use the formula \[{\text{Amount}} = {\text{Principal}} + {\text{S}}{\text{.I}}\], and by substituting the values of the amount and simple interest, we get:
\[ \Rightarrow {\text{Rs}}.720 = {\text{Principal}} + {\text{Rs}}.120\]
By bringing \[{\text{Rs}}.120\] into the LHS side of the equation, we get:
\[ \Rightarrow {\text{Rs}}.720 - {\text{Rs}}.120 = {\text{Principal}}\]
By doing subtraction in the LHS side of the above equation, we get:
\[ \Rightarrow {\text{Rs}}.600 = {\text{Principal}}\]

\[\because \] Option D is correct.

Note: Students generally get confused in the terms amount and principal value. You should remember that the amount of value is the addition of the principal or base value and the interest applying on it for a particular year. Also, the formula for calculating the amount is shown as below:
\[{\text{Amount = P}}\left( {1 + {\text{rt}}} \right)\], where \[{\text{P}} = {\text{Principal}}\], \[{\text{r}} = {\text{Rate}}\] and \[{\text{t}} = {\text{time}}\].