answer:1. **Calculate the new product of integers**: [ 3 times 4 times 5 = 60 ] 2. **Compute the sum of the new fractions**: [ frac{1}{3} + frac{1}{4} + frac{1}{5} ] Find a common denominator, which is the least common multiple of 3, 4, and 5, equal to 60: [ frac{1}{3} = frac{20}{60}, quad frac{1}{4} = frac{15}{60}, quad frac{1}{5} = frac{12}{60} ] Adding these, [ frac{20}{60} + frac{15}{60} + frac{12}{60} = frac{47}{60} ] 3. **Multiply the product of integers by the sum of fractions**: [ 60 times frac{47}{60} = 47 ] 4. **Conclusion**: The expression evaluates to 47. Thus, the final answer is 47. The correct answer is boxed{text{(B)} 47}.

answer:Define [ g(n) = leftlceil frac{119n}{120} rightrceil - leftlfloor frac{120n}{121} rightrfloor. ] We need to find when g(n) = 1. Consider shifting n by the least common multiple of 120 and 121, which is 14520: [ g(n + 14520) = leftlceil frac{119 (n + 14520)}{120} rightrceil - leftlfloor frac{120 (n + 14520)}{121} rightrfloor = leftlceil frac{119n}{120} + 119 rightrceil - leftlfloor frac{120n}{121} + 120 rightrfloor. ] Simplify this as [ leftlceil frac{119n}{120} rightrceil + 119 - leftlfloor frac{120n}{121} rightrfloor - 120 = leftlceil frac{119n}{120} rightrceil - leftlfloor frac{120n}{121} rightrfloor - 1 = g(n) - 1. ] This indicates that g(n + 14520) = g(n) - 1. For each residue class modulo 14520, there is a unique integer n such that g(n) = 1. Thus, the answer is boxed{14520}.

answer:1. **Identify the side lengths of the cubes**: The side lengths of these cubes are 1, 3, 4, 5, 6, 7, 8, and 9 units, with the largest at the bottom. 2. **Determine the visible faces for each cube**: - The bottom cube (side length 9) has all its faces visible except the top face, contributing 5 faces. - The second cube from bottom (side length 8) has all faces visible except the bottom face, contributing 5 faces because it is larger than the cube on top of it. - The cube above (side length 7) has its top and side faces visible, contributing 5 faces. - Continue this pattern, with each cube revealing all but the bottom face until the top cube. 3. **Calculate the surface area of each cube**: - Bottom cube (side length 9): 5 times 9^2 = 5 times 81 = 405 - Second cube (side length 8): 5 times 8^2 = 5 times 64 = 320 - Third cube (side length 7): 5 times 7^2 = 5 times 49 = 245 - Continue the pattern, summing the areas. 4. **Compute the total surface area**: - Total surface area = 405 + 320 + 245 + 200 (5 times 5^2) + 144 (4 times 6^2, only top and four sides visible) + 90 (5 times 3^2) + 4 (4 times 1^2) 5. **Sum the surface areas**: - Total surface area = 405 + 320 + 245 + 200 + 144 + 90 + 4 = 1408 6. **Conclusion**: - The total surface area of the tower is 1408 square units. The final answer is boxed{1408}

answer:1. Calculate the area of the plot of land in cm² since the land is a rectangle: [ text{Area} = text{length} times text{width} = 12 text{ cm} times 8 text{ cm} = 96 text{ cm}^2. ] 2. Convert the area from cm² to miles² knowing that 1 cm² on the map corresponds to 1 mile² in reality (since 1 cm = 1 mile and squaring both sides gives 1 cm² = 1 mile²): [ text{Converted Area} = 96 text{ cm}^2 times 1 text{ mile}^2/text{cm}^2 = 96 text{ miles}^2. ] 3. Convert the area from miles² to acres using the conversion factor 1 mile² = 640 acres: [ text{Area in Acres} = 96 text{ miles}^2 times 640 text{ acres/mile}^2 = 61440 text{ acres}. ] Therefore, the actual size of the plot of land is (boxed{61440 text{ acres}}).