answer:Given the ratio 3:5:6:8, we can determine each person's share based on Amanda's share of 45: - Multiply each ratio part by the factor that makes Amanda's share equal to 45. Since Amanda corresponds to the part '3', we find the multiplying factor by dividing 45 (Amanda's share) by 3 (her part in the ratio): [ text{Multiplier} = frac{45}{3} = 15 ] - Apply this multiplier to all other parts in the ratio: - Ben's share: 5 times 15 = 75 dollars - Carlos's share: 6 times 15 = 90 dollars - Diana's share: 8 times 15 = 120 dollars - Calculate the total shared sum: [ text{Total amount} = 45 + 75 + 90 + 120 = boxed{330} text{ dollars} ]

answer:Let's denote the number of days it takes for David to do the work alone as ( D ). Kim can do the work in 3 days, so her work rate is ( frac{1}{3} ) of the work per day. David's work rate is ( frac{1}{D} ) of the work per day. When they work together, their combined work rate is ( frac{1}{3} + frac{1}{D} ) of the work per day. Since they finish the work together, their combined work rate results in 1 complete work, so: ( frac{1}{3} + frac{1}{D} = 1 ) Now, let's find out the share of the payment they should get based on the amount of work they do. The total payment is Rs. 150, and Kim's share is Rs. 60. Therefore, David's share must be Rs. 90 (since 150 - 60 = 90). The ratio of their shares should be equal to the ratio of the work they do, so: ( frac{Kim's , share}{David's , share} = frac{Kim's , work , rate}{David's , work , rate} ) Given that Kim's share is Rs. 60 and David's share is Rs. 90, we have: ( frac{60}{90} = frac{frac{1}{3}}{frac{1}{D}} ) Simplifying the left side of the equation gives us ( frac{2}{3} ), so: ( frac{2}{3} = frac{frac{1}{3}}{frac{1}{D}} ) Cross-multiplying gives us: ( 2 cdot frac{1}{D} = frac{1}{3} cdot 3 ) Simplifying the right side of the equation gives us: ( 2 cdot frac{1}{D} = 1 ) Dividing both sides by 2 gives us: ( frac{1}{D} = frac{1}{2} ) Therefore, ( D = 2 ) days. So, David can do the work alone in boxed{2} days.

answer:We start by rewriting each term with base 4: [ 4^{frac{1}{2}} cdot 16^{frac{1}{4}} cdot 64^{frac{1}{8}} cdot 256^{frac{1}{16}} dotsm = 4^{frac{1}{2}} cdot (4^2)^{frac{1}{4}} cdot (4^3)^{frac{1}{8}} cdot (4^4)^{frac{1}{16}} dotsm. ] This simplifies to: [ 4^{frac{1}{2}} cdot 4^{frac{1}{2}} cdot 4^{frac{3}{8}} cdot 4^{frac{1}{4}} dotsm = 4^{left(frac{1}{2} + frac{1}{2} + frac{3}{8} + frac{1}{4} + dotsright)}. ] Let ( S ) be the sum of these exponents: [ S = frac{1}{2} + frac{1}{2} + frac{3}{8} + frac{1}{4} + frac{5}{16} + frac{3}{8} + dots. ] Similar to how (2S) was handled in the original problem, we attempt to solve (S) through a series relationship. This forms a non-standard decreasing pattern: [ 2S = 1 + 1 + frac{3}{4} + frac{1}{2} + frac{5}{8} + frac{3}{4} + dots. ] Subtracting these, we get: [ S = 1 + frac{1}{4} + frac{1}{8} + frac{1}{16} + dots. ] This is a geometric series where: [ S = 1 + sum_{n=2}^infty left(frac{1}{4}right)^{n-1} = 1 + frac{frac{1}{4}}{1 - frac{1}{4}} = 1 + frac{1}{3} = frac{4}{3}. ] So, [ 4^{frac{1}{2}} cdot 16^{frac{1}{4}} cdot 64^{frac{1}{8}} cdot 256^{frac{1}{16}} dotsm = 4^S = 4^{frac{4}{3}} = boxed{4^{frac{4}{3}}}. ]

answer:Let's assume the investment of C is Rs. x. According to the given information: B invests two-thirds of what C invests, so B's investment is (2/3)x. A invests 3 times as much as B, so A's investment is 3 * (2/3)x = 2x. Now, the total investment is the sum of investments by A, B, and C, which is 2x + (2/3)x + x = (7/3)x. The profit is distributed in the ratio of their investments. So, the share of each partner in the profit will be proportional to their investment. The share of B in the profit = (Investment of B / Total Investment) * Total Profit = ((2/3)x / (7/3)x) * 6600 = (2/7) * 6600 = 2 * 942.8571 (approx) = 1885.7142 (approx) So, the share of B in the profit is approximately Rs. boxed{1885.71} .