The banker's discount on ₹1600 at 15% p.a. equals the true discount on ₹1680 for the same time and rate. What is the time?

Given data

• BD on ₹1600 at r = 15% for time t equals TD on ₹1680 at same r and t.

Concept/ApproachUse BD = S r t and TD = S r t ÷ (1 + r t) (simple interest/discount).

Step-by-Step calculation BD = 1600 × 0.15 × t = 240 t TD = 1680 × 0.15 × t ÷ (1 + 0.15 t) = 252 t ÷ (1 + 0.15 t) Set 240 t = 252 t ÷ (1 + 0.15 t) ⇒ 240 = 252 ÷ (1 + 0.15 t) 1 + 0.15 t = 252 ÷ 240 = 21 ÷ 20 = 1.05 0.15 t = 0.05 ⇒ t = 1÷3 year = 4 months

Verification/AlternativeAt t = 1÷3, BD = 1600 × 0.15 × 1÷3 = ₹80; TD on 1680 = 1680 × 0.15 × 1÷3 ÷ (1 + 0.05) = 84 ÷ 1.05 = ₹80.

Final Answer4 months