Mathematics

The investments are Rs. 2666.67, Rs. 3666.67, and Rs. 4666.67 respectively (since the second invests Rs.1000 more than the first, and the third Rs.1000 more than the second, totaling Rs.9000). The profit is shared in the ratio of their investments times time (2 years each). The third person's share is (4666.67 * 2) / ( (2666.67+3666.67+4666.67)*2 ) = 4666.67 / 11000 = 0.4242. Multiplying by total profit Rs.5400 gives approximately Rs. 2290.91. However, since the options are fixed, recalculating precisely: Let first = x, second = x+1000, third = x+2000; sum = 3x + 3000 = 9000 => 3x=6000 => x=2000. So investments are 2000, 3000, 4000. Profit share ratio is 2000:3000:4000 = 2:3:4. Total parts = 9. Third person's share = (4/9)*5400 = Rs.2400. Option A is Rs.2400, so correct answer is A, not B. Therefore, correct answer is A.

The profit sharing ratio is given as 3:1. A invested Rs. 85,000 for the full 12 months, and B invested Rs. 42,500 for x months. The profit ratio is proportional to the product of capital and time invested. So, (85000 * 12) : (42500 * x) = 3 : 1. Simplifying, 1020000 : 42500x = 3 : 1, which gives 1020000 / (42500 * x) = 3, so x = 1020000 / (42500 * 3) = 8 months. Therefore, B joined 8 months into the year, which corresponds to option D (8 months).

The financier claims to lend at simple interest of 10%, but by adding interest every six months to the principal, he effectively compounds the interest semi-annually. The nominal rate is 10% per year, so the semi-annual periodic rate is 5%. The effective annual interest rate is calculated using the formula for compound interest: (1 + 0.05)^2 - 1 = 0.1025 or 10.25%. Therefore, the effective rate of interest charged is 10.25%, which corresponds to option B.

A's capital contribution changes after 8 months: initially Rs.3000 for 8 months, then Rs.2000 for 4 months (after withdrawing Rs.1000). B's capital is Rs.4000 for 8 months, then Rs.5000 for 4 months (after advancing Rs.1000). Calculating weighted capitals: A = 3000*8 + 2000*4 = 24000 + 8000 = 32000; B = 4000*8 + 5000*4 = 32000 + 20000 = 52000. Total = 84000. A's share = 32000/84000 = 8/21. Profit share = (8/21)*630 = Rs.240.

B invested Rs. 15,000 for the entire 2 years, so his capital-time product is 15,000 × 24 = 360,000. A invested Rs. 20,000 for 2 years (20,000 × 24 = 480,000). C joined after 6 months, investing Rs. 20,000 for 1.5 years (20,000 × 18 = 360,000). Total capital-time = 480,000 + 360,000 + 360,000 = 1,200,000. B's share = 360,000 / 1,200,000 = 3/10 of profit. So, B's profit = 3/10 × 25,000 = Rs. 9,000.

The odd one out is option B (2Y4) because in the other options, the two outer digits correspond to the alphabetical position of the middle letter: - A: 1P6 → 1 and 6 form 16, and P is the 16th letter of the alphabet. - C: 2T0 → 2 and 0 form 20, and T is the 20th letter. - D: 1R8 → 1 and 8 form 18, and R is the 18th letter. However, in option B (2Y4), the two outer digits 2 and 4 form 24, but Y is the 25th letter, so it does not fit the pattern. Therefore, B is the odd one out.

The profit ratio between two partners is proportional to the product of their investment and the time for which the investment is made. Given the investment ratio of P and Q as 7:5 and the profit ratio as 7:10, and knowing P invested for 5 months, we set up the equation: (7 × 5) : (5 × t) = 7 : 10. Simplifying, 35 : 5t = 7 : 10, which leads to 35 × 10 = 7 × 5t, so 350 = 35t, and t = 10 months. Therefore, Q invested the money for 10 months, making option B the correct answer.

The question asks to find the number which does not belong to the group among the options 30, 27, 36, and 45. - 30 is an even number (2 × 3 × 5), divisible by 3 and 6, but not a perfect cube. - 27 is an odd number equal to 3³, making it a perfect cube and divisible by 3 but not by 6. - 36 is an even number (2² × 3²), divisible by 3 and 6, but not a perfect cube. - 45 is an odd number (3² × 5), divisible by 3 but not by 6, and not a perfect cube. The key distinction is that 27 is the only perfect cube among the options, while the others are not. Also, 27 is not divisible by 6, unlike 30 and 36, but 45 is also not divisible by 6. However, the perfect cube property is the clearest distinguishing feature. Therefore, option B (27) is the odd one out, making it the correct answer.

The profit share among partners is proportional to the product of their investment and the time period of investment. Calculating each partner's investment*time: A = 6000*5 = 30000, B = 3600*6 = 21600, C = 7500*3 = 22500. Total = 30000 + 21600 + 22500 = 74100. A's share of the profit = (30000 / 74100) * 7410 = Rs.3000. Therefore, the correct answer is Rs.3000, which corresponds to option B.

Profit sharing is based on the product of capital and time invested. Ameen invested for 36 months (70,000 × 36 = 2,520,000), Rahim for 30 months (105,000 × 30 = 3,150,000), and Sameer for 24 months (140,000 × 24 = 3,360,000). The ratio of their profit shares is 2,520,000 : 3,150,000 : 3,360,000, which mathematically simplifies to 12:15:16.