Simple interest and true discount combined: The simple interest on a certain sum at 6% per annum for t years is ₹180. The (true) discount at 5% on the same sum due for the same time is ₹140. Find the sum and the time.

Introduction / Context:
This problem mixes simple interest (SI) with the concept of true discount (TD). SI is computed on the present principal, while TD is the rebate that makes the present worth grow to a future due amount at a given rate over time. We are told SI at 6% equals ₹180 and TD at 5% on the same sum for the same time equals ₹140.

Given Data / Assumptions:

• Principal (sum) = S
• Time = t years
• SI: 0.06 * S * t = 180
• True Discount on a sum S due for time t at 5%: TD = (S * 0.05 * t) / (1 + 0.05 * t) = 140

Concept / Approach:
From SI we get S*t directly. Substitute into the TD expression (note TD uses the due amount S and adjusts by the time factor in the denominator). Solve for t, then compute S via S = (S*t)/t.

Step-by-Step Solution:
From SI: 0.06 S t = 180 ⇒ S t = 3000TD = (S * 0.05 * t)/(1 + 0.05 t) = 140Use S t = 3000 ⇒ numerator = 0.05 * (S t) = 0.05 * 3000 = 150So 150/(1 + 0.05 t) = 140 ⇒ 1 + 0.05 t = 150/140 = 15/140.05 t = 15/14 − 1 = 1/14 ⇒ t = (1/14)/0.05 = (1/14)/(1/20) = 20/14 = 10/7 = 1 3/7 yearsThen S = (S t)/t = 3000 / (10/7) = 2100

Verification / Alternative check:
SI check: 0.06 * 2100 * (10/7) = 180. TD check: (2100 * 0.05 * 10/7)/(1 + 0.05 * 10/7) = (150)/(15/14) = 140. Both validate.

Why Other Options Are Wrong:
They correspond to incorrect algebra (e.g., treating discount as simple interest or omitting the denominator in TD).

Common Pitfalls:
Confusing true discount with simple interest on the principal; TD always includes the (1 + r t) factor in the denominator.

Final Answer:
₹ 2100 and 1 3/7 Year