Question

A sum of 16,400 is borrowed to be paid in 2 years by two equal annual payments, allowing 5% compound interest. The annual payment is
A. Rs 8000
B. Rs 8050
C. Rs 8820
D. Rs 9000

Answer 1

Hint: In this question it is given that a sum of 16,400 is borrowed to be paid in 2 years by two equal annual payments, allowing 5% compound interest. We have to find the annual payment, i.e, the amount that he pays each year. So to find the N equal instalment we have use one formula, which is,
$$P=\dfrac{x}{\left( 1+\dfrac{r}{100} \right)^{1} } +\dfrac{x}{\left( 1+\dfrac{r}{100} \right)^{2} } +\cdots +\dfrac{x}{\left( 1+\dfrac{r}{100} \right)^{N} }$$......(1)
Where,
P=Principle amount.
r=Rate of interest.
N=Number of equal installments.
x=amount of equal instalment.

Complete step-by-step answer:
Given that the principal amount(P) =16400, rate of interest(r) =5 and number of instalment(N) =2,
Since we have to pay two times and the paying amount is equal, so we can write from the equation (1),
$$P=\dfrac{x}{\left( 1+\dfrac{r}{100} \right)^{1} } +\dfrac{x}{\left( 1+\dfrac{r}{100} \right)^{2} }$$
$$\Rightarrow 16400=\dfrac{x}{\left( 1+\dfrac{5}{100} \right)^{1} } +\dfrac{x}{\left( 1+\dfrac{5}{100} \right)^{2} }$$ [putting the values]
$$\Rightarrow 16400=\dfrac{x}{\left( 1+\dfrac{1}{20} \right)^{1} } +\dfrac{x}{\left( 1+\dfrac{1}{20} \right)^{2} }$$
$$\Rightarrow 16400=\dfrac{x}{\left( \dfrac{21}{20} \right) } +\dfrac{x}{\left( \dfrac{21}{20} \right)^{2} }$$
Now as we know that $$\dfrac{1}{\left( \dfrac{x}{y} \right) } =\dfrac{y}{x}$$, so by using this the above equation can be written as,
$$ 16400=\dfrac{20x}{21} +\dfrac{x}{\left( \dfrac{441}{400} \right) }$$
$$\Rightarrow 16400=\dfrac{20x}{21} +\dfrac{400x}{441}$$
$$\Rightarrow 16400=\dfrac{20x\times 21+400x}{441}$$ [since, LCM(21,441) = 441]
$$\Rightarrow 16400=\dfrac{820x}{441}$$
$$\Rightarrow x=\dfrac{16400\times 441}{820}$$
$$\Rightarrow x=8820$$
Therefore the annual payment is Rs. 8820.
Hence the correct option is option C.

Note: While solving this type of problems, always remember that the value of N is the total number of times where each time the amount is compounded and the term might be a whole year, half-year or a quarter-year.