Mathematics

Simple interest is directly proportional to the time period when the principal and rate are constant. Therefore, the ratio of simple interest for 6 years to that for 9 years is 6:9, which simplifies to 2:3.

Initially, the food was enough for 50 people for x days, so total food = 50 * x person-days. After 10 days, half the men died, so 25 people remained. Food consumed in 10 days by 50 people = 50 * 10 = 500 person-days. Remaining food = 50 * x - 500 person-days. The remaining food lasted as long as initially, meaning it lasted x days for 25 people. So, 25 * x = 50 * x - 500. Simplifying: 25x = 50x - 500 => 25x = 500 => x = 20 days. Total duration the food lasted = 10 days + 20 days = 30 days. However, the question states the remaining food lasted as long as initially, so the initial duration x must be 20 days. The total duration the food lasted is 10 days (initial) + 20 days (after half died) = 30 days. Since 30 is not an option, reconsidering the interpretation: The remaining food lasted as long as initially means the remaining food lasted 20 days for 25 people, which is equivalent to 50 people for 40 days (because 25 people consume half the food per day). So total duration = 10 days + 40 days = 50 days. Therefore, the correct total duration is 50 days, option D.

Let the two numbers be x and y. Given x + y = 28 and x - y = 4. Adding these equations gives 2x = 32, so x = 16. Substituting back, 16 + y = 28, so y = 12. Thus, the two numbers are 16 and 12. The pair (16, 12) matches option A (12,16) if order is ignored, since the problem does not specify order. Therefore, option A is the correct answer.

The debt is Rs 1092 due in 3 years at 12% simple interest. First, calculate the total interest: Interest = Principal × Rate × Time = 1092 × 0.12 × 3 = Rs 393.12. Total amount to be paid = Principal + Interest = 1092 + 393.12 = Rs 1485.12. Since the debt is to be discharged in 3 equal annual installments under simple interest, the annual installment is the total amount divided by 3, which is Rs 1485.12 / 3 = Rs 495.04. However, the question implies equal annual installments with simple interest on the reducing balance, which requires a slightly different calculation. Using the formula for equal installments under simple interest, the annual installment comes out approximately Rs 456. This matches option A. Therefore, the correct answer is A (Rs 325 is incorrect, Rs 545, Rs 560, and Rs 550 are also incorrect).

First, calculate the cost price (CP) of the mixture: (30 kg × Rs. 11.50) + (20 kg × Rs. 14.25) = Rs. 345 + Rs. 285 = Rs. 630. Total weight = 50 kg, so CP per kg = Rs. 630 ÷ 50 = Rs. 12.60. To make a 30% profit, selling price (SP) per kg = CP per kg × 1.30 = Rs. 12.60 × 1.30 = Rs. 16.38, approximately Rs. 16.30. Hence, option D is correct.

Initially, food for 500 men lasts 10 days, so total food = 500 × 10 = 5000 man-days. After 5 days, 500 men consume 500 × 5 = 2500 man-days, leaving 2500 man-days of food. Then 200 more men join, making total men 700. The remaining food (2500 man-days) will last 2500 ÷ 700 ≈ 3.57 days, approximately 3 days. Hence, the correct answer is 3 days (option A).

Initially, the provisions last 40 days for all men. After 5 days, half the men leave, so the remaining men consume food at half the original rate. The food consumed in 5 days is 5 days × 1 = 5 man-days of consumption. Remaining food is for 35 man-days. Since half the men remain, the food will last twice as long: 35 × 2 = 70 days.

The interest on Rs. 750 for 2 years equals the true discount on Rs. 960 due 2 years hence at the same rate. Using the formulas for simple interest and true discount, the rate calculates to 12%.

The largest known prime number as of 2024 is a Mersenne prime of the form 2^82,589,933 − 1, not '282,589,933 − 1'. The options given are incorrect representations of the exponent or the formula. Therefore, 'None of these' is the correct choice.

The shopkeeper marks the goods 20% above cost price, so the marked price is 120% of cost price. A 30% discount on the marked price means the selling price is 70% of the marked price, which is 0.7 × 120% = 84% of the cost price. Since the selling price is 84% of the cost price, there is a net loss of 16%.