Mathematics

The original provisions last 30 days for the full number of men. If 2/3 of the men do not attend the mess, only 1/3 of the men consume the provisions. Since the same amount of food is consumed by fewer men, the food will last longer. Specifically, the food will last 3 times longer because the consumption rate is reduced to 1/3. Therefore, the new duration is 30 days × 3 = 90 days. Hence, the correct answer is 90 days (Option B).

The original principal is Rs. 800, and it amounts to Rs. 920 in 3 years, so the interest earned is Rs. 120. The simple interest formula is I = P * R * T / 100. Using this, the rate R = (I * 100) / (P * T) = (120 * 100) / (800 * 3) = 5%. If the rate is increased by 3%, the new rate is 8%. The new interest = (800 * 8 * 3) / 100 = Rs. 192. The new amount = principal + interest = 800 + 192 = Rs. 992. However, Rs. 992 is option A, which was originally marked correct, but the explanation confirms it is correct. So the original correct_answer 'A' is correct. Therefore, the original correct_answer is correct, but the explanation was missing and has been added. Difficulty is easy based on the calculation complexity.

Using the simple interest formula I = P × R × T / 100, where I = 5400, R = 12%, and T = 3 years, we solve for P: P = (I × 100) / (R × T) = (5400 × 100) / (12 × 3) = 150000 / 36 = Rs. 10,000. Hence, the principal amount is Rs. 10,000.

To compare the two offers, we calculate the present value of the credit offer using simple interest at 18% per annum for 8 months. Present value = 12,880 / (1 + (18% × 8/12)) = 12,880 / 1.12 = Rs. 11,500, which is less than Rs. 12,000 cash. Therefore, the cash offer is better.

The present worth (P) is calculated using the formula P = A / (1 + rt), where A = Rs. 132, r = 5% = 0.05, and t = 2 years. So, P = 132 / (1 + 0.05*2) = 132 / 1.1 = Rs. 120. However, this matches option C, but the question asks for simple interest, and the calculation is correct. Rechecking: P = 132 / (1 + 0.05*2) = 132 / 1.1 = 120. So option C is correct. The original correct_answer was C, which matches the calculation. The explanation was missing and is now added.

The expenditure is proportional to the number of people and the number of days. First, calculate the daily expenditure per person for the 12-member family: Rs. 850 in 10 days means Rs. 850 / (12 * 10) = Rs. 7.08 per person per day. For the 8-member family spending Rs. 340, the number of days = Rs. 340 / (8 * 7.08) ≈ 6 days. Therefore, the correct answer is 6 days (option C).

The sum of the first n even numbers can be calculated using the formula n(n+1). For n=70, the sum is 70 × 71 = 4970. Alternatively, using the arithmetic series sum formula S = n/2 × (first term + last term), where the first term is 2 and the last term is 140, the sum is 70/2 × (2 + 140) = 35 × 142 = 4970. Therefore, the correct answer is 4970, which corresponds to option A.

The speed of the train is 54 km/hr, which converts to 15 m/s. The train passes a man in 20 seconds, so the length of the train is speed × time = 15 m/s × 20 s = 300 m. The train passes the platform in 36 seconds, so the total distance covered is 15 m/s × 36 s = 540 m. The length of the platform is the total distance minus the length of the train, which is 540 m - 300 m = 240 m. Therefore, the correct answer is 240 meters (option C).

The simple interest earned is Rs. 3875 - Rs. 2500 = Rs. 1375. Using the formula SI = (P × R × T) / 100, we have 1375 = (2500 × R × 4) / 100. Solving for R gives R = (1375 × 100) / (2500 × 4) = 13.75%. However, rechecking the calculation: (1375 × 100) / (2500 × 4) = (137500) / (10000) = 13.75%. So the rate is 13.75%, which corresponds to option D. Therefore, the original correct answer D is correct. The initial explanation was missing, so it is added now.

Initially, the provisions were stored to last 40 days for a certain number of soldiers. After 10 days, the remaining provisions would last 30 days for the original number of soldiers. However, the number of soldiers doubled after 10 days, so the remaining provisions must now be shared among twice as many soldiers, effectively halving the remaining duration. Therefore, the remaining provisions will last 30 / 2 = 15 days after the number of soldiers doubled.