The true discount on a bill of ₹ 540 is ₹ 90. The banker's discount is

### Concept & Formula The Banker's Discount (B.D.) is the simple interest on the Face Value (or Bill Value), while the True Discount (T.D.) is the simple interest on the Present Worth (P.W.). The relation between Banker's Discount, True Discount, and Present Worth is given by: $$P.W. = Bill Value - T.D.$$ $$B.D. = \frac{T.D. \times Bill Value}{P.W.}$$ ### Step-by-Step Solution 1. **Given:** Bill Value (Sum Due) = ₹ 540, True Discount (T.D.) = ₹ 90. 2. **Calculate Present Worth:** P.W. = Bill Value - T.D. P.W. = $540 - 90 = 450$ 3. **Calculate Banker's Discount:** B.D. = $\frac{Bill Value \times T.D.}{P.W.}$ B.D. = $\frac{540 \times 90}{450}$ B.D. = $\frac{540}{5} = 108$ ### Exam Strategy & Shortcut A much faster direct formula relating B.D., T.D., and Bill Value is $Bill Value = \frac{B.D. \times T.D.}{B.D. - T.D.}$. However, the ratio method works wonderfully here: Since T.D. is calculated on P.W. and B.D. on Bill Value, $\frac{B.D.}{T.D.} = \frac{Bill Value}{P.W.}$. Plugging in the numbers: $\frac{B.D.}{90} = \frac{540}{450} = \frac{6}{5}$. Thus, $B.D. = 90 \times \frac{6}{5} = 108$. ### Common Pitfall A common mistake is assuming that the Banker's Discount is calculated on the Present Worth. Remember, bankers calculate their discount on the face value of the bill, which always makes B.D. greater than T.D. ### Final Answer Therefore, the correct answer is **₹ 108**.