True & Banker's Discount

A bill of ₹F is due after T years. Paying it today should cost less. How much less has two different answers, and the whole topic is knowing which is which.

• Present worth (PW) — the sum that, left at R% simple interest for T years, would grow to exactly ₹F. PW = F × 100/(100 + RT).
• True discount (TD) = F − PW = interest on the present worth. TD = F × RT/(100 + RT).
• Banker's discount (BD) — the bank is simpler and charges interest on the face value: BD = F × RT/100.
• Banker's gain (BG) = BD − TD = TD²/PW, i.e. simple interest on the true discount itself.

The single most common mistake is computing the true discount on the face value. Remember the difference in one line: BD is interest on F, TD is interest on PW, and since PW is smaller, TD is always smaller than BD. If your TD comes out larger, you have swapped them.

Note the giveaway in the denominator: 100 + RT means true discount, plain 100 means banker's discount.

Working backwards, F = TD × BD / (BD − TD).

PW = F×100/(100+RT) · TD = F − PW = F×RT/(100+RT) · BD = F×RT/100 · BG = BD − TD = TD²/PW · F = TD×BD/(BD−TD)

A bill of ₹2,200 is due 2 years hence at 5% p.a. Find PW, TD, BD and BG. → RT = 10. PW = 2200 × 100/110 = ₹2,000; TD = 2200 − 2000 = ₹200; BD = 2200 × 10/100 = ₹220; BG = 220 − 200 = ₹20. Sanity check twice: ₹2,000 at 5% simple interest for 2 years earns ₹200 and grows to exactly ₹2,200; and TD²/PW = 200² ÷ 2000 = 40000 ÷ 2000 = ₹20, matching BG.