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A radio is available at $\text{₹} 27780/-$ cash price, or three equal annual installments at $15\%$ per annum under $CI$ compounding annually. Each installment amount, therefore, is _______.

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Installment is paid annually and $3$ times. i.e., after first year, second year and third year.

After $3$ years, our original amount $27780$ becomes $27780\times (1.15^3)$ compounding at $15\%$ annually.

Let our installment be $x$. After $2$ year this will be

$x + x \times (1.15)$

After $3$ years, paid amount becomes

$x + x \times (1.15) +  x \times (1.15)^2.$

Thus we get

$$27780 \times (1.15)^3 = x + x \times (1.15) + x \times (1.15^2)$$

$\implies x = \frac{27780 \times 1.15^3}{1+1.15+1.15^2} = 12167.$
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After giving equal instalment at $15\%$ rate per annum for $3$ years, the final amount will be equal to $Rs. 27780 \times (1.15)^3 = 42249.90$

Here, we need to find the initial or principal amount. then after $2^{nd}$ year it'll be $principal * (1.15)$ and after $3^{rd}$ year it'll be $Principal * (1.15)^2$. As all the installments should be equal.

$\therefore P + P \times \left (1 + \dfrac{15}{100} \right ) + P \times \left (1 + \dfrac{15}{100} \right )^2 = 27780 \times (1.15)^3\\ \implies P + P \times (1.15) + P \times (1.15)^{2} = 42249.90 \\ \implies 3.4725P = 42249.90 \\ \implies P = \dfrac{42249.90}{3.4725} \\ \implies P = 12166.99$

The instalment amount should be $Rs. 12166.99 \approx 12167$

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Formula :Amount = principal ( 1/ 1 + r/100 )t

if installment for 1 year is 115 which is due to principal 100 and R 15%

now, here Amount = 27780 , each installment amount = principal , so let  installment = x 

x * ( 100/115 ) + x * ( 100/115 )2  +  x * ( 100/115 )3   = 27780

solving we get x = 12,169.26

Answer:

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