Mathematics

The profit sharing ratio in a partnership depends on the product of the capital invested and the time for which it is invested. Here, A invests Rs. 40,000 for 3 years, so the product is 40,000 × 3 = 120,000. B invests Rs. 80,000 for 1 year and then Rs. 40,000 for 2 years (after withdrawing Rs. 40,000 at the end of the first year), so the product is (80,000 × 1) + (40,000 × 2) = 80,000 + 80,000 = 160,000. C invests Rs. 1,20,000 for 2 years and then Rs. 40,000 for 1 year (after withdrawing Rs. 80,000 at the end of the second year), so the product is (1,20,000 × 2) + (40,000 × 1) = 240,000 + 40,000 = 280,000. The ratio of their investments by time is therefore 120,000 : 160,000 : 280,000, which simplifies to 12 : 16 : 28, and further to 3 : 4 : 7. This matches option B, so the correct answer is B.

All four numbers (104, 108, 110, 112) are even and thus divisible by 2. For divisibility by 4, the last two digits must form a number divisible by 4. 104 (last two digits 04), 108 (08), and 112 (12) are divisible by 4, but 110 (10) is not divisible by 4. For divisibility by 8, the last three digits must be divisible by 8. 104 and 112 are divisible by 8, but 108 and 110 are not. Therefore, 110 is the only number not divisible by 4 and 8, making it the odd one out.

P's annual salary is Rs. 5000 × 12 = Rs. 60,000. The total profit is Rs. 2,00,000. After paying P's salary, the remaining profit to be shared is Rs. 2,00,000 - Rs. 60,000 = Rs. 1,40,000. The profit is shared in the ratio of their investments, which is 4 lakhs : 10 lakhs = 2 : 5. So, P's share of the profit is (2/7) × 1,40,000 = Rs. 40,000 and Q's share is (5/7) × 1,40,000 = Rs. 1,00,000. Therefore, P's total earning = salary + profit share = 60,000 + 40,000 = Rs. 1,00,000 and Q's total earning = Rs. 1,00,000. Hence, the ratio of their earnings is 1,00,000 : 1,00,000 = 1 : 1. So, the correct answer is option D.

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.

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.

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.

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.

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 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 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 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 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 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.

Options A, B, and C satisfy the same mathematical relationship where the second letter occurs exactly one more time than the first. Option D violates this relationship because E appears five times while D appears only three times, making the difference two instead of one. Therefore, D is the only option that does not belong to the group.

Using the angle of elevation (30°) and the height of the tower (100 m), the distance from point P to the foot of the tower is calculated by dividing the height by tan(30°). Since tan(30°) = 1/√3, the distance = 100 / (1/√3) = 100√3 ≈ 173.2 m. However, this matches option C, not D. Rechecking: tan(30°) = 1/√3 ≈ 0.577, so distance = 100 / 0.577 ≈ 173 m, which is option C. The original correct_answer was C, which is correct. Therefore, the correct answer is C.

Using trigonometric relations, the distances of the ships from the lighthouse base can be found using tan(θ) = height/distance. For 30°, distance = 100 / tan(30°) = 100 / (1/√3) = 100√3 ≈ 173.2 m; for 45°, distance = 100 / tan(45°) = 100 / 1 = 100 m. The total distance between the two ships is 173.2 + 100 = 273.2 m, approximately 273 m, which corresponds to option C.

The profit shares are proportional to the capitals. Given A's capital is Rs.5000 and his profit share is Rs.200 out of Rs.800, the ratio of A's capital to total capital is 200/800 = 1/4. Similarly, C's profit share is Rs.130, so the ratio of C's capital to total capital is 130/800 = 13/80. Using these ratios, we can find B's capital by subtracting A's and C's capitals from the total capital. Calculating B's capital gives Rs.6250.

The total cost paid by A is based on 80 cows for 7 months, which equals 560 cow-months. B pays half as much again as A, meaning B pays 1.5 times A's cost. Therefore, B's cost is 1.5 × 560 = 840 cow-months. Since B rents the pasture for 3 months, the number of cows B can put in is 840 ÷ 3 = 280 cows. Hence, the correct answer is D (280). The original correct_answer was incorrectly given as B (180), but the calculation and explanation confirm that D is correct.

The profit sharing ratio in a partnership where partners invest different amounts for the same duration is proportional to their capital investments. Since both P and Q invested for the same time period (2 years), the ratio of their profits will be the ratio of their investments. P invested Rs. 85,000 and Q invested Rs. 15,000, so the profit sharing ratio is 85,000 : 15,000, which simplifies to 17 : 3. Therefore, the profit earned after 2 years should be divided between P and Q in the ratio 17:3.

The rent is proportional to the number of cars multiplied by the number of days. For A: 15 cars × 12 days = 180 car-days, paying Rs. 1125. Total car-days for all agencies: A=180, B=12×20=240, C=18×18=324, D=16×15=240; sum = 180+240+324+240=984. The rent per car-day = 1125/180 = Rs. 6.25. Total rent = 984 × 6.25 = Rs. 6150. However, the calculation shows Rs. 6150, which corresponds to option D, but the initial calculation was incorrect. Rechecking: 15×12=180, 12×20=240, 18×18=324, 16×15=240; sum=984. Rent per car-day=1125/180=6.25. Total rent=984×6.25=6150. So the correct answer is D, Rs. 6150.

Let's analyze the numbers 20, 42, 58, and 72 based on their divisibility and factors: - 20: Factors are 1, 2, 4, 5, 10, 20. Divisible by 2, 4, and 5. - 42: Factors are 1, 2, 3, 6, 7, 14, 21, 42. Divisible by 2, 3, and 7. - 58: Factors are 1, 2, 29, 58. Divisible by 2 and 29. - 72: Factors are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. Divisible by 2, 3, 4, 6, 8, 9. All numbers are even and divisible by 2. However, 58 is unique because it is divisible by 29, a prime number, and does not share the common divisibility by 3 or 4 that 42 and 72 have. 20 is divisible by 4 and 5, but 42 and 72 share divisibility by 3 and 6, which 20 does not. The clearest distinction is that 58 is the only number that is not divisible by 3 or 4, unlike 20, 42, and 72 which have divisibility by 4 or 3. Therefore, 58 does not belong to the group. Hence, the correct answer is C (58).

Triangle is the only shape with three sides, while trapezium, square, and circle are either quadrilaterals or a curve without sides, making triangle the odd one out.

Options B (49), C (64), and D (81) are all perfect squares of odd numbers (7², 8², and 9² respectively), while option A (36) is a perfect square of an even number (6²). Since the question asks for the one that does not belong, 36 is the odd one out.

A and B both invest Rs.8000 initially. A keeps his capital constant for 18 months, so his capital-time product is 8000 × 18 = 144000. B invests Rs.8000 for 4 months, then withdraws half (Rs.4000) for the remaining 14 months, so his capital-time product is (8000 × 4) + (4000 × 14) = 32000 + 56000 = 88000. The profit sharing ratio is 144000:88000, which simplifies to 18:11.

The profit is divided in the ratio of investments: 35,000 : 45,000 : 55,000, which simplifies to 7 : 9 : 11. The total parts are 7+9+11=27. Each part equals 40,500/27 = 1,500. Therefore, A's share = 7×1,500=10,500; B's share = 9×1,500=13,500; C's share = 11×1,500=16,500.