Amit borrowed ₹ 20,000 at 12% per annum compound interest. If he pays 40% of the sum borrowed at the end of the first year and 40% of the sum borrowed at the end of the second year, find the amount of loan outstanding at the beginning of the third year.

For the first year:

P = ₹ 20000, R = 12 %, T = 1 year

$\text{Interest} = \dfrac{P \times R \times T}{100}\\[1em] = \dfrac{20,000 \times 12 \times 1}{100}\\[1em] = \dfrac{240,000}{100}\\[1em] = ₹ 2,400$

Amount at the end of first year = P + I

= ₹ 20,000 + 2,400

= ₹ 22,400

Amit paid 40% of 20,000 at the end of the first year = $\dfrac{40}{100}$ x 20,000 = ₹ 8,000

Amount outstanding at the beginning of the second year = ₹ 22,400 - ₹ 8,000 = ₹ 14,400

For the second year:

P = ₹ 14,400, R = 12 %, T = 1 year

$\text{Interest} = \dfrac{P \times R \times T}{100}\\[1em] = \dfrac{14,400 \times 12 \times 1}{100}\\[1em] = \dfrac{172,800}{100}\\[1em] = ₹ 1728$

Amount at the end of second year = P + I

= ₹ 14,400 + 1,728

= ₹ 16,128

Amit again paid 40% of ₹ 20,000 at the end of the second year, which is 8000.

Amount outstanding at the beginning of the third year = ₹ 16,128 - ₹ 8,000 = ₹ 8,128

Hence, the loan amount outstanding at the beginning of the third year is ₹ 8,128.