The banker's gain of a certain sum due 2 years hence at 10% per annum is ₹ 24. The present worth is

### Concept & Formula Banker's Gain (B.G.) is the simple interest on the True Discount (T.D.). Present Worth (P.W.) relates to T.D. similarly, where T.D. is the simple interest on the P.W. $$B.G. = \frac{T.D. \times R \times T}{100}$$ $$T.D. = \frac{P.W. \times R \times T}{100}$$ ### Step-by-Step Solution 1. **Given:** B.G. = ₹ 24, Time ($T$) = 2 years, Rate ($R$) = $10\%$. 2. **Find True Discount (T.D.):** B.G. = $\frac{T.D. \times R \times T}{100}$ $24 = \frac{T.D. \times 10 \times 2}{100}$ $24 = \frac{T.D. \times 20}{100} = \frac{T.D.}{5}$ T.D. = $24 \times 5 = 120$ 3. **Find Present Worth (P.W.):** T.D. = $\frac{P.W. \times R \times T}{100}$ $120 = \frac{P.W. \times 10 \times 2}{100}$ $120 = \frac{P.W.}{5}$ P.W. = $120 \times 5 = 600$ ### Exam Strategy & Shortcut Combine the formulas to relate B.G. directly to P.W.: $B.G. = P.W. \times (\frac{R \times T}{100})^2$. Here, $\frac{R \times T}{100} = \frac{20}{100} = \frac{1}{5}$. So, $24 = P.W. \times (\frac{1}{5})^2 \Rightarrow P.W. = 24 \times 25 = 600$. This calculates the answer in one single step! ### Common Pitfall A common error is confusing Banker's Gain with True Discount. Always remember that B.G. is interest *on* the interest (T.D.), not the principal (P.W.). ### Final Answer Therefore, the correct answer is **₹ 600**.