Mathematics

The profit is shared in the ratio of their investments: 22500:35000 = 9:14. Total parts = 9 + 14 = 23. Raees's share = (14/23) × 13800 = Rs. 8400.

The profit sharing ratio is based on the product of capital and the time for which it was invested. A invests Rs.8000 for 4 months fully, then Rs.4000 (half) for 8 months. B invests Rs.9000 for 6 months fully, then Rs.6000 (two-thirds) for 6 months. Calculating: A's capital-time = (8000*4) + (4000*8) = 32000 + 32000 = 64000; B's capital-time = (9000*6) + (6000*6) = 54000 + 36000 = 90000. The ratio is 64000:90000 = 32:45, which corresponds to option C.

The series appears to be factorial numbers: 1! = 1, 1! = 1, 2! = 2, 3! = 6, 4! = 24, but the next number should be 5! = 120, not 96. Therefore, 96 is the wrong number in the series.

The question asks to find the one which does not belong to the group among the options PMS, ROU, GDJ, and KIM. The key is to identify the pattern in the letter sequences. PMS, ROU, and GDJ each have letters that follow a pattern of descending alphabetical order with specific steps: PMS (P->M->S) has a pattern of -3, +6 in alphabetical positions; ROU (R->O->U) has -3, +6; GDJ (G->D->J) has -3, +6. However, KIM (K->I->M) has a pattern of -2, +4, which does not match the others. Therefore, KIM does not belong to the group. This reasoning is supported by the detailed analysis of letter positions and patterns in the search results and the literature on odd one out pattern recognition questions. The explanation in the original question was unclear, but the corrected explanation based on the verified pattern is that options A, B, and C have a consistent pattern of letter shifts (-3, +6), while option D does not, making it the odd one out.

The profit sharing ratio is based on the product of capital and time. Let B join after x months. Then, A's capital × 12 months = 45000 × 12 = 540000, B's capital × time = 5400 × x. Given profit ratio A:B = 2:1, so 540000 : 5400x = 2 : 1. Solving for x gives x = 5 months. Since 5 months is not an option, the closest correct option is 4 months (option A). The original explanation incorrectly chose 6 months as the closest option, but 4 months is closer to 5 months than 6 months.

The options consist of a letter followed by a number. The letters correspond to their position in the alphabet: B=2, E=5, D=4, I=9. The numbers are 4, 25, 16, and 91. The numbers 4 (2²), 25 (5²), and 16 (4²) are perfect squares, while 91 is not a perfect square. Therefore, option D (I91) is the odd one out because 91 is not a perfect square, unlike the others.

The profit shares in a partnership where investments are made for different durations are proportional to the product of the amount invested and the time period of investment. Here, A invests Rs. 10,000 for 8 months, so A's capital-time product is 10,000 × 8 = 80,000. B invests Rs. 10,000 for 12 months, so B's capital-time product is 10,000 × 12 = 120,000. The total capital-time product is 80,000 + 120,000 = 200,000. A's share of the profit is (80,000 / 200,000) × 25,000 = Rs. 10,000, and B's share is (120,000 / 200,000) × 25,000 = Rs. 15,000. This matches option D, making it the correct answer.

Z's share of the profit is Rs. 14,000 out of Rs. 50,000, which is 14,000/50,000 = 0.28 or 28%. The capitals and times invested determine the profit share. Let Z join after x months. X and Y invested for 12 months, Z for (12 - x) months. The ratio of investments is 20000*12 : 25000*12 : 30000*(12 - x) = 240000 : 300000 : 30000(12 - x). The share of Z is 30000(12 - x) / (240000 + 300000 + 30000(12 - x)) = 0.28. Solving gives x = 4 months. So, Z joined after 4 months from the beginning.

Sameena invested Rs. 50,000 for 3 years (36 months), while Nida joined after 6 months, so she invested Rs. 80,000 for 2.5 years (30 months). The profit sharing ratio is calculated based on the product of capital and time: (50,000 × 36) : (80,000 × 30) = 1,800,000 : 2,400,000 = 3 : 4. Total profit is Rs. 24,500, so Sameena's share = (3/7) × 24,500 = Rs. 10,500. This matches option B exactly, confirming the correct answer.

Options A (41), B (43), and C (53) are all prime numbers, whereas option D (57) is not prime because it is divisible by 3 and 19. Therefore, 57 does not belong to the group of prime numbers.