Question

Simple interest on Rs. 500 for 4 years at 6.25 % p.a. is equal to the simple interest on Rs. 400 at 5 % for a certain period of time. The period time is
A. 4 years
B. 5 years
C. \[6\dfrac{1}{4}\] years
D. \[8\dfrac{2}{3}\] years

Answer 1

Hint: To find the simple interest, we use the formula\[S.I = \dfrac{{P \times T \times R}}{{100}}\] where, S.I is the simple interest, P is principal amount, R is the rate percent yearly and T is the number of years. Since, all the values are known and it is given, hence by substituting the values in the above formula we get the required simple interest.

Complete step-by-step solution:
Here in this question, we have to find the value of time where principal amount and the rate is given.
Here we have two simple interests, we equate both the simple interest and then we determine the value of period time.
The interest rate per annum, \[{R_1} = 6.25\]
The principal amount, \[{P_1} = Rs.500\]
The number of years, \[{T_1} = 4\]
The simple interest is calculated by using the formula \[S.I = \dfrac{{{P_1} \times {T_1} \times {R_1}}}{{100}}\]
Substituting the values to the formula we get,
\[ \Rightarrow S.I = \dfrac{{500 \times 4 \times 6.25}}{{100}}\] ----- (1)
We have another set of data, so it is defined as
The interest rate per annum, \[{R_2} = 5\]
The principal amount, \[{P_2} = Rs.400\]
The number of years, \[{T_2} = x\]
The simple interest is calculated by using the formula \[S.I = \dfrac{{{P_2} \times {T_2} \times {R_2}}}{{100}}\]
Substituting the values to the formula we get,
\[ \Rightarrow S.I = \dfrac{{400 \times x \times 5}}{{100}}\] ---- (2)
Here we have to find the value of x . It can be determined by equating the equation (1) and the equation (2)
On equating we have
\[ \Rightarrow \dfrac{{500 \times 4 \times 6.25}}{{100}} = \dfrac{{400 \times x \times 5}}{{100}}\]
Cancel the denominators we have
\[ \Rightarrow 500 \times 4 \times 6.25 = 400 \times x \times 5\]
On simplifying the terms for x we have
\[ \Rightarrow x = \dfrac{{500 \times 4 \times 6.25}}{{400 \times 5}}\]
On further simplifying we get
\[ \Rightarrow x = 6.25\]
This can be written as
\[ \Rightarrow x = 6\dfrac{1}{4}\] years.

Hence the correct answer is option ‘C’.

Note: The compound interest is interest calculated on the amount that includes principal and accumulated interest of the previous period whereas simple interest is interest on the invested amount for entire period. This is the difference between the simple interest and compound interest. To find the value of amount where principal amount, rate of interest and time is known we use the standard formula \[A = P{\left[ {1 + \dfrac{r}{{100}}} \right]^n}\] to determine the value of A. We can also determine the compound interest by subtracting the initial principal amount from the amount.