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# A borrows Rs. 800 at the rate of 12% per annum simple interest and B borrows Rs. 910 at the rate of 10% per annum simple interest. In how many years will their amounts of debt be equal?

Last updated date: 17th Mar 2023
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Hint: Assume that the amounts of both A and B will be equal after t years. Then find out the amount on the given sum of money at respective simple interest rates after t years of time. Then compare both amounts.

Let the amounts of debt of A and B will be equal after $t$ years.
Now, according to the question, A is borrowing Rs. 800 at the rate of 12% per annum.
Principal, $P = 800$ and rate, $r = 12\%$
We know that the amount of a sum on simple interest for $t$ years can be calculated as:
$\Rightarrow {\text{ Amt}}{\text{. }} = P\left( {1 + \dfrac{{r \times t}}{{100}}} \right)$
Using this formula, A’s amount of debt after $t$ years will be:
$\Rightarrow {\text{ }}{{\text{A}}_{{\text{amt}}{\text{.}}}} = 800 \times \left( {1 + \dfrac{{12 \times t}}{{100}}} \right) .....(i)$
Similarly, B is borrowing Rs. 910 at the rate of 10% per annum. So in this case, we have:
Principal, $P = 910$ and rate, $r = 10\%$.
Using the same formula, B’s amount of debt after $t$ years will be:
$\Rightarrow {\text{ }}{{\text{B}}_{{\text{amt}}}} = 910 \times \left( {1 + \dfrac{{10 \times t}}{{100}}} \right) .....(ii)$
As we have discussed earlier, amounts of debt of both A and B will be equal after $t$ years. Therefore we have:
$\Rightarrow {{\text{A}}_{{\text{amt}}}} = {\text{ }}{{\text{B}}_{{\text{amt}}}}$
Putting values from equation $(i)$ and $(ii)$, we’ll get:
$\Rightarrow 800 \times \left( {1 + \dfrac{{12 \times t}}{{100}}} \right) = 910 \times \left( {1 + \dfrac{{10 \times t}}{{100}}} \right), \\ \Rightarrow 80 \times \left( {1 + \dfrac{{12t}}{{100}}} \right) = 91 \times \left( {1 + \dfrac{{10t}}{{100}}} \right), \\ \Rightarrow 80 + \dfrac{{96t}}{{10}} = 91 + \dfrac{{91t}}{{10}}, \\ \Rightarrow \dfrac{{96t}}{{10}} - \dfrac{{91t}}{{10}} = 91 - 80, \\ \Rightarrow \dfrac{{5t}}{{10}} = 11, \\ \Rightarrow t = 22 \\$
Thus the amounts of debt of A and B will be equal after 22 years.

Note: If the sum is kept on compound interest instead of simple interest, then the amount is calculated as:
$\Rightarrow {\text{ Amt}}{\text{.}} = P{\left( {1 + \dfrac{r}{{100}}} \right)^t}$, where P is the principal sum kept initially, r is the rate of compound interest and t is the time period.