John borrowed Rs. $2,10,000$ from a bank at an interest rate of $10 \%$ per annum, compounded annually. The loan was repaid in two equal installments, the first after one year and the second after another year. The first installment was interest of one year plus part of the principal amount, while the second was the rest of the principal amount plus due interest thereon. Then each installment, in Rs., is

Entered answer:

✅ Correct Answer: 121000

John borrowed Rs. 210000 at 10% compound interest annually. He repays in two equal installments - one after 1 year and another after 2 years.

We call each installment Rs. x.

The present value of all future payments must equal the borrowed amount.

If you receive money in the future, it's worth less today because of interest. We need to "discount" future payments to find their value today.

Present Value Formula: If you receive Rs. P after n years at r% interest, its present value = $\dfrac{P}{(1+\tfrac{r}{100})^n}$

For our problem:

Interest rate = 10% = 0.10

So, (1 + 0.10) = 1.1

Present value of 1st installment (received after 1 year):

$\dfrac{x}{1.1}$

Present value of 2nd installment (received after 2 years):

$\dfrac{x}{1.1^2} = \dfrac{x}{1.21}$

Since present value of all installments = borrowed amount:

$\dfrac{x}{1.1} + \dfrac{x}{1.1^2} = 210000$

$\dfrac{x}{1.1} + \dfrac{x}{1.21} = 210000$

Finding common denominator (1.21):

$\dfrac{x \times 1.1}{1.21} + \dfrac{x}{1.21} = 210000$

$\dfrac{1.1x + x}{1.21} = 210000$

$\dfrac{2.1x}{1.21} = 210000$

$2.1x = 210000 \times 1.21$

$2.1x = 254100$

$x = \dfrac{254100}{2.1} = 121000$

Each installment = Rs. 121000

Remember that with compound interest problems involving equal installments, always use the present value approach. It's the fastest and most reliable method!

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