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# A man A borrows Rs 8000 at 12% per annum simple interest and B borrows Rs 9100 at 10% per annum simple interest. In how many years will their amounts of debts be equal?  Hint: Assume a variable t which represents the time in years after which the amounts of debts of A and B becomes equal. In simple interest, we have a formula to calculate the amount after t years for a principal amount p at a rate of interest r per annum simple interest. This formula is amount = $p+\dfrac{p\times r\times t}{100}$ . Use this formula to get an equation in t which can be solved to get the value of t.

Before proceeding with the question, we must know all the formulas that will be required to solve this question.
For a principal amount Rs p at a rate of interest r per annum simple interest, the amount after t years is given by the formula,
Amount = $p+\dfrac{p\times r\times t}{100}.............\left( 1 \right)$
In the question, A borrows Rs 8000 at 12% per annum simple interest and B borrows Rs 9100 at 10% per annum simple interest. We are required to find the number of years after which their amounts become equal.
Let us assume that after t years, their amounts become equal.
Since A borrows Rs 8000 at 12% per annum simple interest, using formula $\left( 1 \right)$, the amount after t years is equal to,
\begin{align} & 8000+\dfrac{8000\times 12\times t}{100} \\ & \Rightarrow 8000+960t.............\left( 2 \right) \\ \end{align}
Since B borrows Rs 9100 at 10% per annum simple interest, using formula $\left( 2 \right)$, the amount after t years is equal to,
\begin{align} & 9100+\dfrac{9100\times 10\times t}{100} \\ & \Rightarrow 9100+910t.............\left( 3 \right) \\ \end{align}
Since the amounts we got in $\left( 2 \right)$ and $\left( 3 \right)$ are equal, we get,
\begin{align} & 8000+960t=9100+910t \\ & \Rightarrow 50t=1100 \\ & \Rightarrow t=22 \\ \end{align}
Hence, the answer is 22 years.

Note: There is a possibility that one may commit a mistake while applying the formula for the amount. It is a very common mistake that one forgets to divide $p\times r\times t$ term by 100 in the formula we used to calculate the amount and this mistake may lead us to an incorrect answer.
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