The simple interest and the true discount on a certain sum of money for a given time at a given rate are Rs. 85 and Rs. 80 respectively. What is the sum of money?

Introduction / Context:
Here both simple interest and true discount are given for the same sum, same time, and same rate, but the time and rate themselves are not specified. The task is to find the principal sum. Using the known relationship between simple interest and true discount allows us to compute the effective rate time product and then the principal without explicitly knowing the rate or time separately.

Given Data / Assumptions:

• Simple interest SI = Rs. 85.
• True discount TD = Rs. 80.
• Sum S is due after some time t at rate r percent per annum.
• The same S, t, and r are used for both SI and TD.

Concept / Approach:
We use the standard formulas: SI = S * r * t / 100 TD = S * r * t / (100 + r * t) Let R = r * t. Then: SI = S * R / 100, and TD = S * R / (100 + R). We also know that: SI / TD = (100 + R) / 100. From the numerical SI and TD, we can compute SI / TD, then find R, then use one of the expressions to solve for S.

Step-by-Step Solution:
Step 1: Compute the ratio SI / TD. SI / TD = 85 / 80 = 17 / 16 = 1.0625. Step 2: Set this equal to (100 + R) / 100. 17 / 16 = (100 + R) / 100. Step 3: Cross multiply: 17 * 100 = 16 * (100 + R). 1700 = 1600 + 16R. Step 4: 16R = 1700 − 1600 = 100. Step 5: R = 100 / 16 = 6.25. So R = r * t = 6.25 percent for the entire period.
Step 6: Use SI formula to find S. SI = S * R / 100 = S * 6.25 / 100. 85 = S * 6.25 / 100. Step 7: S = 85 * 100 / 6.25. Note that 6.25 = 25 / 4, so 100 / 6.25 = 100 * 4 / 25 = 16. Thus S = 85 * 16 = 1360.

Verification / Alternative check:
Compute TD using S = 1360 and R = 6.25. Then: TD = S * R / (100 + R) = 1360 * 6.25 / 106.25. First, 1360 * 6.25 = 8500. 106.25 is 425 / 4, so 8500 divided by 106.25 is 80. Thus TD = 80 and SI = 85, exactly as given, confirming that S = 1360 is correct.

Why Other Options Are Wrong:
Other choices such as Rs. 1200, Rs. 1316, Rs. 1350, or Rs. 1500 do not satisfy the simultaneous conditions SI = 85 and TD = 80 when used with the same rate and time. They either produce different interest or different discount, not matching the given values.

Common Pitfalls:
Common errors include treating SI and TD as equal or attempting to guess rate and time separately without using the ratio method. Miscalculating the ratio 85 / 80 or mishandling the fractional rate 6.25 can also lead to wrong answers. Using the simplified parameter R = r * t and carefully solving the resulting algebraic equations avoids these issues.

Final Answer:
The sum of money is Rs. 1360.