A money lender gives a loan of Rs. 8,000 for 6 months and deducts Rs. 640 as interest at the time of lending. The effective rate of interest per annum charged by the lender is (Given \(\left(\displaystyle\frac{25}{23}\right)^2 = 1.1815\))
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 18.15%
Since the interest is deducted in advance, the borrower actually receives only
\(8000 - 640 = 7360\)
So Rs. 640 is the interest on Rs. 7,360 for 6 months, and the rate per half-year is
\( \displaystyle i = \frac{640}{7360} = \frac{2}{23} \)
There are two half-years in a year, so the effective rate is
\( \displaystyle r_{\mathrm{eff}} = (1 + i)^2 - 1 = \left(\frac{25}{23}\right)^2 - 1 \)
\(= 1.1815 - 1 = 0.1815 = 18.15\%\)
Why the other options are incorrect
Option (1) 16% — Incorrect. This is 2 × 8%, which takes the interest on the full Rs. 8,000 and also ignores compounding.
Option (2) 16.64% — Incorrect. This is \((1.08)^2 - 1\), which treats Rs. 640 as interest on Rs. 8,000; the borrower only gets Rs. 7,360.
Option (3) 17.39% — Incorrect. This is 2 × (2/23), the nominal rate per annum; it leaves out the interest earned on the interest of the first half-year.