Ratio, Proportion and Partnership

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 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 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 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 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.

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.

A invests Rs.3000 for 6 months and then Rs.6000 (double) for the next 6 months, so his total investment time product is (3000×6) + (6000×6) = 18000 + 36000 = 54000. B invests Rs.4000 for 12 months, so total is 4000×12 = 48000. The profit ratio is 54000:48000 = 39:49.

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.

The given capitals are in the ratio 6 : 8 : 10. To simplify this ratio, divide each term by their greatest common factor, which is 2. This gives the simplified ratio 3 : 4 : 5. Standard practice in mathematics is to express ratios in their simplest form, so the correct answer is option A (3:4:5).

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.

The shopkeeper sells the mixture at Rs. 177 per kg with an 18% profit, so the cost price of the mixture is Rs. 177 / 1.18 = Rs. 150 per kg. Let the cost of the other brand be x. The mixture cost per kg = (2*200 + 3*x) / 5 = 150. Solving gives x = Rs. 120 per kg.

The ratio of men to women is 3:5, so the total parts are 3 + 5 = 8. Each part represents 24 ÷ 8 = 3 people. Therefore, the number of women is 5 parts × 3 = 15.