Q.For the same bill of ₹10,000 due 6 months hence at 8% per annum, find

The true discount is charged on the present value, not the face value, so it is a little smaller than the banker's discount found in Example 2.

The true discount works out to ₹384.62 and the present value to ₹9,615.38.

Here $F = 10000$, $r = 8\%$, $t = 0.5$ year, so $rt = 8 \times 0.5 = 4$.

1. True discount:

   $TD = \dfrac{F\,rt}{100 + rt} = \dfrac{10000 \times 4}{100 + 4} = \dfrac{40000}{104} = 384.615\ldots \approx 384.62.$

2. Present value:

   $PV = F - TD = 10000 - 384.62 = 9615.38,$

   or directly $PV = \dfrac{100F}{100+rt} = \dfrac{100 \times 10000}{104} = \dfrac{1000000}{104} = 9615.38.$ Cross-check (independent). The true discount must equal the simple interest on the present value: $9615.38 \times 0.08 \times 0.5 = 9615.38 \times 0.04 = 384.62$, which matches. Note it is indeed a little less than the banker's discount of ₹400 from Example 2 — as it must be.
   ✓Final answer

   (i) True discount = ₹384.62; (ii) present value = ₹9,615.38.

The two formulas for PV agree: $F - TD$ and $100F/(100+rt)$ are algebraically identical, so either can be used as the check on the other.

Computing the true discount on the face value (that gives the banker's discount, not the true discount), or writing $100 - rt$ in the denominator instead of $100 + rt$.