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Amal invests Rs $12000$ at $8\%$ interest, compounded annually, and Rs $10000$ at $6\%$ interest, compounded semi-annually, both investments being for one year. Bimal invests his money at $7.5\%$ simple interest for one year. If Amal and Bimal get the same amount of interest, then the amount, in Rupees, invested by Bimal is ______

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Given that,

Amal $ \Rightarrow P=12000, R=8\%$, at $1$ year compound interest.

$\qquad \Rightarrow P=10000, R=6\%$, at semi-annually compound interest.

When interest is calculated semi-annually then, the rate will be half and time will be doubled.

- $R=3\%, T=2$ years.

Amal first $CI:$

$ \boxed {\text{Amount } = P \left ( 1+\frac{R}{100} \right)^T}$

$ \Rightarrow A=12000 \left (1+ \frac{8}{100} \right)^{1}$

$ \Rightarrow A=12000 \times \frac{108}{100}$

$ \Rightarrow A=12960$

$\because CI = A-P $

$\quad = 12960-12000$

$ \therefore \boxed{CI=960}$

Amal second $CI:$

$ \Rightarrow A=10000 \left (1+ \frac{3}{100} \right)^2$

$\Rightarrow A= 10000 \times \frac{103}{100} \times \frac{103}{100}$

$ \Rightarrow A=10609$

$\because CP=A-P$

$ \qquad=10609-10000$

$\therefore \boxed{CI=609}$

Total interest of Amal $=960+609=1569$

$\because$ Amal and Bimal get the same amount of interest.

Amal interest $=$ Bimal interest $=1569$

Bimal $\Rightarrow R=7.5\%$, at $1$ year simple interest.

Let the amount invested by Bimal $ = P.$

$ \boxed {SI= \frac{P\times R\times T}{100}}$

$ \Rightarrow 1569= \frac {P \times 7.5 \times 1}{100}$

$ \Rightarrow1569= \frac {P \times 75 \times 1}{100 \times 10}$

$ \Rightarrow P=20920$

$ \therefore$ The amount invested by Bimal $=20920.$

Correct Answer $:20920$