True discount over double time: If the true discount on a bill of $110 due at the end of some time is $10, what is the true discount on the same amount due at the end of double that time (same rate, simple interest)?

Introduction / Context:
True discount (TD) on a sum A for time fraction R = r * t (simple) is TD = A * R / (1 + R). Knowing TD for one time lets you solve for R, then compute TD at double time 2R. This avoids needing the actual rate r or time t separately.

Given Data / Assumptions:

• A = $110 (face value).
• TD at time t is $10.
• Simple interest setting; same rate for double time.

Concept / Approach:
Let R = r * t. From TD = A * R / (1 + R) = 10, solve for R. Then for double time, use 2R in the same formula: TD(2t) = A * (2R) / (1 + 2R).

Step-by-Step Solution:

Verification / Alternative check:
Present worth at time t: PW = A − TD = 110 − 10 = 100; consistent with R = 0.1. At double time, TD drops below $22 due to the denominator (1 + 2R), giving ≈ $18.33.

Why Other Options Are Wrong:
$20 or $22 assume linear scaling; true discount is not linear with time. $21.81, $24.20 do not match the exact formula.

Common Pitfalls:
Treating TD like simple interest (proportional to time) instead of applying the TD formula with denominator 1 + R.

Final Answer:
$ 18.33