Rs. 10 is the true discount on a bill of Rs. 110 due after time t. What is the true discount on the same sum due after double the time (rate unchanged)?

Given data

• A = Rs. 110
• True discount TD₁ = Rs. 10 at time t
• Find TD₂ at time 2t with same rate r

Concept / Approach

For simple interest, true discount is: TD = A × (r t) / (1 + r t) Let x = r t.

Step-by-step calculation
10 = 110 × x / (1 + x)10 + 10x = 110x ⇒ 10 = 100x ⇒ x = 0.1At double time: x₂ = 2x = 0.2TD₂ = 110 × 0.2 / 1.2 = 110 / 6 = 18.33

Verification
PW at t: 110/1.1 = 100 ⇒ TD = 10. At 2t: 110/1.2 = 91.67 ⇒ TD ≈ 18.33.

Common pitfalls

• Assuming TD doubles with time (it increases sublinearly because of the denominator 1 + r t).

Final Answer
True discount after double time = Rs. 18.33.