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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.

Complete step-by-step answer:

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.

Complete step-by-step answer:

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.

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