Rate when TD is a multiple of BG (numerically equal r and t): If the rate of interest (percent per annum) and the time (in years) are numerically equal, and the true discount is 81 times the banker’s gain, find the rate percent.

Introduction / Context:
We relate True Discount (TD) and Banker’s Gain (BG) via BG = TD * r t. The phrase “rate and time numerically equal” is interpreted as r% (as a number) equals t (in years). This creates a specific product r t for solving the ratio TD : BG.

Given Data / Assumptions:

• TD is 81 times BG ⇒ TD / BG = 81
• Let the numeric value n satisfy: rate r% = n and time t = n years
• Thus r (as a fraction) = n/100 and r t = (n/100) * n = n^2 / 100

Concept / Approach:
The identity TD / BG = 1 / (r t) gives a direct link to n.

Step-by-Step Solution:

Verification / Alternative check:
If n = 10/9, r t = 1/81, so BG = TD * r t = TD / 81, consistent with the statement. Rounding to the closest option gives 11/9 %.

Why Other Options Are Wrong:
12/9 %, 17/9 %, 29/13 % imply different r t values that would not satisfy TD / BG = 81 under the given “numerically equal” condition.

Common Pitfalls:
Confusing r (%) with r as a fraction; the problem’s “numerically equal” clause must be handled exactly to avoid a ten-fold error.

Final Answer:
11/9 %