To find the value of the expression, we can add the fractions as follows: (\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = \frac{11}{6}).

To find the time when the second train will overtake the first train, we can use the formula: (Time = \frac{Distance}{Speed Difference}). The distance between the two trains at 11:00 AM is 60 miles (since the first train has been traveling for 1 hour). The speed difference is 75 mph - 60 mph = 15 mph. Therefore, the time it takes for the second train to overtake the first train is (\frac{60}{15} = 4) hours. Adding this to 11:00 AM, we get 12:30 PM as the time when the second train will overtake the first train.

To solve the equation, we can use the quadratic formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}). Here, (a = 2, b = -5, c = 2). Plugging these values into the formula, we get: (x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(2)}}{2(2)} = \frac{5 \pm \sqrt{25 - 16}}{4} = \frac{5 \pm \sqrt{9}}{4} = \frac{5 \pm 3}{4}). Therefore, the solutions are (x = 1) and (x = 2).

The area of a circle is given by the formula (A = \pi r^2). Here, the radius (r = 10) centimeters. Plugging this value into the formula, we get: (A = \pi (10)^2 = 100\pi) square centimeters.

The statement 'In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides' is known as the Pythagorean Theorem.

The Fibonacci sequence starts with 0 and 1, and each subsequent number is the sum of the two preceding numbers. Therefore, the 10th number in the sequence can be found by adding the 9th and 8th numbers. The 9th number is 34 and the 8th number is 21, so the 10th number is 34 + 21 = 55.

The derivative of the function (f(x) = x^3 - 2x^2 + 3x - 4) can be found using the power rule of differentiation. The power rule states that if (f(x) = x^n), then (f'(x) = nx^(n-1)). Applying this rule to each term of the function, we get: (f'(x) = 3x^2 - 4x + 3).

To find the probability of selecting a green ball, we need to divide the number of green balls (20) by the total number of balls (10 + 15 + 20 = 45). Therefore, the probability is (\frac{20}{45} = \frac{2}{5}).

The value of (\log_{10} 1000) can be found using the logarithmic property (\log_{10} 10^n = n). Since (1000 = 10^3), we have (\log_{10} 1000 = \log_{10} 10^3 = 3).

The perimeter of a regular hexagon is equal to the sum of the lengths of all six sides. Since each side is 10 centimeters long, the perimeter is (6 \times 10 = 60) centimeters.

To find the equation of the line, we can use the point-slope form: (y - y_1 = m(x - x_1)), where ((x_1, y_1)) is one of the given points and (m) is the slope of the line. Using the point ((2, 3)), we have: (y - 3 = m(x - 2)). To find the slope, we can use the other given point ((5, 7)): (7 - 3 = m(5 - 2)). Simplifying this, we get (m = 2). Substituting this value of (m) back into the point-slope form, we get the equation of the line: (y - 3 = 2(x - 2)), which can be simplified to (y = 2x + 1).

The area of a rectangle is calculated by multiplying its length and width. Therefore, the area of the garden is (12 \times 8 = 96) square meters.

The value of (\sin 45^\circ) can be found using the unit circle or by using the trigonometric ratio (\sin \theta = \frac{opposite}{hypotenuse}). In a 45-45-90 triangle, the opposite side and the hypotenuse are both equal to (\sqrt{2}). Therefore, (\sin 45^\circ = \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}).

The volume of a cube is calculated by cubing its side length. Therefore, the volume of the cube with a side length of 5 centimeters is (5^3 = 125) cubic centimeters.

When flipping a coin, there are two possible outcomes: head or tail. Since both outcomes are equally likely, the probability of getting a head is (\frac{1}{2}).