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# A sum of money at simple interest amounts to Rs. 9440 in 3 years, if the rate of interest is increased by 25% the same sum amounts to Rs. 9800 at the same time. The original rate of interest is:${\text{A}}{\text{. 10% p}}{\text{.a}}{\text{.}} \\ {\text{B}}{\text{. 8% p}}{\text{.a}}{\text{.}} \\ {\text{C}}{\text{. 7}}{\text{.5% p}}{\text{.a}}{\text{.}} \\ {\text{D}}{\text{. 6% p}}{\text{.a}}{\text{.}} \\$

Hint: Simple interest is the accumulated interest on the principal amount while the total amount accumulated in a defined time period is the sum of the principal amount and the interest accumulated on the principal amount. The formula used for the calculation of the simple interest is $SI = \dfrac{{prt}}{{100}}$ where, $p$ is the principal amount, $r$ is the rate of interest annually (in percentage), and $t$ is the time for which the interest to be determined (in years). In this question, the rate of interest needed to be determined for two different total amounts. So, we need to get the ratio of the amounts by establishing a relation between the new interest rate and the original interest rate.

Complete step by step solution: Let the original rate of interest be $r\%$.
Then, the amount with the simple interest for the principal amount$(P)$ in the time period of 3 years will be given as:
$\dfrac{{{\text{Prt}}}}{{{\text{100}}}}{\text{ + P = 9440 - - - - - - (i)}}$
Now the rate of interest has been increased by 25% of the initial amount, we get:
$r' = r + 25\% {\text{of }}r \\ = 1.25r \\$
The amount with the simple interest for the principal amount$(P)$ in the time period of 3 years will be given as:
$\dfrac{{{\text{Pr't}}}}{{{\text{100}}}}{\text{ + P = 9800 - - - - - (ii)}}$
Dividing equation (i) and equation (ii) to determine the value of the original rate of interest as:
$\dfrac{{\left( {\dfrac{{{\text{Prt}}}}{{{\text{100}}}}{\text{ + P}}} \right)}}{{\left( {\dfrac{{{\text{Pr't}}}}{{{\text{100}}}}{\text{ + P}}} \right)}}{\text{ = }}\dfrac{{{\text{9440}}}}{{{\text{9800}}}} \\ \dfrac{{3r + 100}}{{3(1.25r) + 100}} = \dfrac{{944}}{{980}} \\ 980(3r + 100) = 944(3.75r + 100) \\ 2940r - 3540r = 94400 - 98000 \\ - 600r = - 3600 \\ r = \dfrac{{ - 3600}}{{ - 600}} \\ = 6\% {\text{ p}}{\text{.a}}{\text{.}} \\$
Hence, the value of the original rate of interest is 6% per annum.
Option D is correct.

Note: The candidates should not get confused with the term 25%, it is not the new interest rate but the increase in the rate of the interest from the original one. Moreover, the total amount should not be considered as the principal amount.