A certain sum is due after 2 years. Its true discount is ₹ 1800, and the simple interest on the same sum (for the same time and rate) is ₹ 2232. Find the sum due (the face value).

Introduction / Context:
This question connects True Discount (TD) and Simple Interest (SI) on the same face value over the same time. Using TD and SI together lets us solve for the face value efficiently.

Given Data / Assumptions:

• TD = 1800
• SI on the same face value and time = 2232
• Time and rate are the same for both measures (simple interest basis).

Concept / Approach:
Let face value be F and x = r*t. Then: TD = F * x / (1 + x) SI on F = F * x Using both equations eliminates x and gives F directly.

Step-by-Step Solution:
From SI: F * x = 2232 → x = 2232 / F. TD = F * x / (1 + x) = 1800. So, 2232 / (1 + 2232/F) = 1800. 2232 = 1800 + (1800 * 2232)/F. 432 = (1800 * 2232)/F → F = (1800 * 2232)/432 = 9300.

Verification / Alternative check:
Compute x = SI/F = 2232/9300 = 0.24. Then TD = F * x/(1 + x) = 9300 * 0.24/1.24 = 2232/1.24 = 1800, confirming correctness.

Why Other Options Are Wrong:
Values like 9450 or 9600 give inconsistent x, resulting in TD not equal to 1800. 9350 and 9000 likewise fail the joint TD–SI condition.

Common Pitfalls:
Treating TD as SI on face value (it is not); ignoring the (1 + x) denominator; or attempting to use present worth directly without eliminating variables.

Final Answer:
₹ 9300