Anil borrows Rs 2 lakhs at an interest rate of 8% per annum, compounded half-yearly. He repays Rs 10320 at the end of the first year and closes the loan by paying the outstanding amount at the end of the third year. Then, the total interest, in rupees, paid over the three years is nearest to

Anil takes a loan of Rs 2 lakhs and pays interest that compounds half-yearly. This means interest is calculated every 6 months (twice a year) and the interest earned in the first 6 months also earns interest in the next 6 months.

When interest compounds half-yearly at 8% per annum, we use 4% per half-year period.

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Formula: $A = P(1 + r/100)^n$

Where $P = Rs 200000$ (principal), $r = 4\%$ (per half-year), and $n = 2$ (two half-year periods in one year).

Amount after 1 year $= 200000 \times (1 + 4/100)^2$

$= 200000 \times (1.04)^2$

$= 200000 \times 1.0816$

$= Rs 216320$

Interest accumulated in first year $= 216320 - 200000 = Rs 16320$

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Anil pays Rs 10320 at the end of first year.

Outstanding amount $= 216320 - 10320 = Rs 206000$

The remaining Rs 206000 will now grow for 2 more years (4 half-year periods).

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The outstanding Rs 206000 compounds for 2 more years which equals 4 half-year periods.

Final amount $= 206000 \times (1.04)^4$

$= 206000 \times 1.16986$

$= Rs 240991$ (approximately)

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Method 1 - Direct Calculation:

Total interest $= Final amount - Original principal$

$= 240991 - 200000$

$= Rs 40991$

Anil also paid Rs 10320 after the first year.

Total interest paid $= 40991 + 10320 = Rs 51311$

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Method 2 - Break it down:

Interest for first year: Rs 16320

Amount paid in first year: Rs 10320

Net interest paid in first year: Rs $16320 - Rs 10320 = Rs 6000$ (this adds to principal)

Interest for remaining 2 years: Rs $240991 - Rs 206000 = Rs 34991$

Total interest over 3 years: Rs $16320 + Rs 34991 = Rs 51311$

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The total interest paid over three years is Rs 51311.

In compound interest problems with partial repayments, we always track how much interest accumulates, how much is actually paid, and what amount carries forward to the next period.