Mathematics

Given the LCM (495) and HCF (5) of two numbers and their sum (100), we can find the numbers using the relationships: product = HCF × LCM = 5 × 495 = 2475, and sum = 100. Let the numbers be a and b. Then a + b = 100 and ab = 2475. Using the formula for difference: (a - b)^2 = (a + b)^2 - 4ab = 100^2 - 4 × 2475 = 10000 - 9900 = 100, so a - b = 10. Therefore, the difference between the two numbers is 10, which corresponds to option A.

Given the floor area (length × width) = 20 m², longer wall area (length × height) = 15 m², and shorter wall area (width × height) = 12 m², we can find the dimensions of the room. Let length = L, width = W, and height = H. From the equations: L × W = 20, L × H = 15, and W × H = 12. Multiplying all three gives (L × W × H)² = 20 × 15 × 12 = 3600, so L × W × H = 60 m³, which is the volume of the room. The question asks for the volume of the cube with the same volume as the room, so the cube's volume is also 60 m³. Therefore, the correct answer is option C (60 m³).

The total surface area of a sphere is given by the formula 4πr². A hemisphere is half of a sphere, but its total surface area includes both the curved surface area and the base (a circle). The curved surface area of a hemisphere is half that of the sphere, i.e., 2πr², and the base area is πr². Therefore, the total surface area of a hemisphere is 3πr². The ratio of the total surface area of a sphere to that of a hemisphere is 4πr² : 3πr², which simplifies to 4:3. Hence, the correct ratio is 4:3, corresponding to option B.

The number of plants leaves a remainder of 5 when divided by 6, 8, 10, and 12, but is divisible by 13. The least common multiple (LCM) of 6, 8, 10, and 12 is 120. The number of plants is therefore 120k + 5. Checking multiples of 120 plus 5 for divisibility by 13, 845 (120*7 + 5) is divisible by 13, so the number of plants is 845.

First, convert all dimensions to the same unit (cm). The wall volume is 2500 cm * 200 cm * 75 cm = 37,500,000 cm³. The brick volume is 20 cm * 10 cm * 7.5 cm = 1500 cm³. Dividing the wall volume by the brick volume gives 37,500,000 / 1500 = 25,000 bricks. However, this calculation assumes no mortar gaps. The options given suggest the closest correct answer is 25,000, which corresponds to option D. But since the calculation matches 25,000 exactly, the correct answer should be D, not A. Therefore, the original correct_answer 'D' is correct. The explanation clarifies the calculation steps.

First, convert the room dimensions to centimeters: length = 624 cm, width = 432 cm. The largest square tile size that can exactly cover the floor is the greatest common divisor (GCD) of 624 and 432, which is 48 cm. The number of tiles needed is (624/48) × (432/48) = 13 × 9 = 117 tiles. However, since 117 is option D, and the calculation confirms 117 tiles, the correct answer is D, not C. Therefore, the original correct_answer 'D' is correct. The explanation clarifies the method to find the least number of square tiles.

The volume of each cube is the cube of its edge length: 6³ = 216 cm³, 8³ = 512 cm³, and 10³ = 1000 cm³. The total volume after melting is 216 + 512 + 1000 = 1728 cm³. The edge of the new cube is the cube root of 1728, which is 12 cm.

First, convert the flow speed from km/h to m/min: 3 km/h = 3000 m/60 min = 50 m/min. The cross-sectional area of the river is depth × width = 2 m × 45 m = 90 m². The volume flow per minute is area × speed = 90 m² × 50 m/min = 4500 m³/min. However, option A is 4500 m³, which matches this calculation. Therefore, correct answer is A, not D.

The volume of a cone is given by (1/3)πr²h. If the height is increased by 100%, it doubles, so the new volume is twice the original volume, meaning a 100% increase.

The surface area of a sphere is proportional to the square of its diameter. Given the diameters are in ratio 1:2, the surface areas are in ratio 1²:2² = 1:4.