Question

A borrows Rs. 800 at the rate of 12% p.a. simple interest and B borrows Rs. 910 at the rate of 10% p.a. simple interest. In how many years will their amount of debts be equal?
$\left( a \right)$ 18 years
$\left( b \right)$ 20 years
$\left( c \right)$ 22 years
$\left( d \right)$ 24 years

Answer 1

Hint: In this particular type of question use the concept that total amount on any principal amount is the sum of the principal amount and the simple interest, where simple interest is given as, S.I = $\dfrac{{Prt}}{{100}}$, where, P = principal amount, r = rate of interest and t = time period, so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given data,
A borrows Rs. 800 at the rate of 12% p.a. simple interest.
So principal (${P_1}$) amount for A = Rs. 800
Rate (${r_1}$) of interest = 12% p.a.
As we know that the simple interest on any principal is calculated as
S.I = $\dfrac{{{P_1}{r_1}{t_1}}}{{100}}$, where, P = principal amount, r = rate of interest and t = time period.
So the total amount he has to pay after ${t_1}$ years, ${A_1}$ = ${P_1}$ + S.I
$ \Rightarrow {A_1} = {P_1} + \dfrac{{{P_1}{r_1}{t_1}}}{{100}}$........... (1)
Now B borrows Rs. 910 at the rate of 10% p.a. simple interest.
So principal (${P_2}$) amount for B = Rs. 910
Rate (${r_2}$) of interest = 10% p.a.
As we know that the simple interest on any principal is calculated as
S.I = $\dfrac{{{P_2}{r_2}{t_2}}}{{100}}$, where, P = principal amount, r = rate of interest and t = time period.
So the total amount he has to pay after ${t_2}$ years, ${A_2}$ = ${P_2}$ + S.I
$ \Rightarrow {A_2} = {P_2} + \dfrac{{{P_2}{r_2}{t_2}}}{{100}}$........... (2)
Now let after n years their amount of debts are equal.
Therefore, ${t_1} = {t_2} = n$ and ${A_1} = {A_2}$
So from equation (1) and (2) we have,
\[ \Rightarrow {P_1} + \dfrac{{{P_1}{r_1}n}}{{100}} = {P_2} + \dfrac{{{P_2}{r_2}n}}{{100}}\]
Now substitute the values we have,
\[ \Rightarrow 800 + \dfrac{{800\left( {12} \right)n}}{{100}} = 910 + \dfrac{{910\left( {10} \right)n}}{{100}}\]
Now simplify e have,
\[ \Rightarrow \dfrac{{800\left( {12} \right)n}}{{100}} - \dfrac{{910\left( {10} \right)n}}{{100}} = 910 - 800\]
\[ \Rightarrow \left( {\dfrac{{9600}}{{100}} - \dfrac{{9100}}{{100}}} \right)n = 110\]
\[ \Rightarrow \left( {\dfrac{{500}}{{100}}} \right)n = 110\]
\[ \Rightarrow 5n = 110\]
\[ \Rightarrow n = \dfrac{{110}}{5} = 22\] Years.
So after 22 years their amount of debts are equal.
So this is the required answer.
Hence option (C) is the correct answer.

Note – Whenever we face such types of questions the key concept we have to remember is that always recall the formula of simple interest which is stated above then first calculate the amount paid by A in ${t_1}$ time and then calculate the amount paid by B in ${t_2}$ time, then according the equation equate these amounts such that ${t_1} = {t_2} = n$ and calculate the number of years as above in which the debts of A and B are equal.