Present worth from banker’s gain: The banker’s gain on a certain sum due in 2 years at 5% per annum is $8. Find the present worth (true present value).

Introduction / Context:
Banker’s gain BG equals BD − TD, where BD is banker’s discount on the face value S for time t, and TD is the true discount. For a given rate and time, BG has a simple proportional relationship to S, which we can exploit to find the face value and then compute the true present worth PW = S / (1 + r t).

Given Data / Assumptions:

• t = 2 years, r (discount rate) = 5% per annum.
• BG = $8.

Concept / Approach:
With d = 5% and t = 2, BD = S * d * t / 100 = 0.10 S. TD = S − S/(1 + 0.10) = S * (0.10/1.10) = 0.090909 S. Hence BG = 0.0090909 S = S/110. Therefore S = 110 * BG. PW is then S divided by (1 + 0.10) since true present worth discounts the amount using simple interest on the present-worth base.

Step-by-Step Solution:

Verification / Alternative check:
BD = 0.10 * 880 = 88; TD = 880 − 800 = 80; BG = 88 − 80 = 8, consistent.

Why Other Options Are Wrong:

• $1600, $1200, $880: These correspond to other quantities (face value or proceeds) and not the true present worth.

Common Pitfalls:
Mistaking BD or face value for present worth. PW uses the true-discount (present worth) relationship, not the banker’s discount proceeds.

Final Answer:
$ 800