answer:If Sean has 32 more whistles than Charles, we can find out how many whistles Charles has by subtracting 32 from the number of whistles Sean has. So, Charles has 45 - 32 = boxed{13} whistles.

answer:If Martin initially had 150 fruits and he ate half of them, he would be left with half of 150, which is: 150 / 2 = 75 fruits Now, out of these 75 fruits, we know that 50 are oranges. To find out how many limes he has, we subtract the number of oranges from the total number of fruits he has left: 75 (total fruits left) - 50 (oranges) = 25 limes Now, to find the ratio of the number of oranges to the number of limes, we divide the number of oranges by the number of limes: 50 (oranges) : 25 (limes) To simplify the ratio, we divide both numbers by the greatest common divisor, which in this case is 25: 50 / 25 : 25 / 25 = 2 : 1 So, the ratio of the number of oranges to the number of limes Martin has now is boxed{2:1} .

answer:To determine the nature of the triangle ( triangle ABC ), we start with the given complex numbers ( a, b, c ) and the relation: [ a + omega b + omega^2 c = 0, ] where ( omega = -frac{1}{2} + frac{sqrt{3}}{2} i ) is a primitive cube root of unity. Let us solve this step-by-step: 1. **Introduction of Relation**: Given: [ a + omega b + omega^2 c = 0 ] 2. **Substituting ( omega ) and ( omega^2 )**: We know: [ omega = -frac{1}{2} + frac{sqrt{3}}{2} i quad text{and} quad omega^2 = -frac{1}{2} - frac{sqrt{3}}{2} i ] Additionally, (omega^3 = 1) and (1 + omega + omega^2 = 0). 3. **Rewriting the Given Equation**: Rewrite the given complex equation: [ a = - (omega b + omega^2 c) ] 4. **Changing Perspective**: Substitute (omega) and (omega^2): [ a - c = -omega (b - c) ] 5. **Using Polar Form and Angle**: Recall: [ omega = e^{2pi i / 3} = cos 120^circ + i sin 120^circ ] Hence, multiplying by (omega) implies a rotation by (120^circ). 6. **Connecting Differences**: Consequently, [ a - c = (cos 120^circ + i sin 120^circ)(c - b) ] 7. **Conceptual Understanding**: Since the difference (a - c) is a (120^circ) rotation of the difference (c - b), and the differences correspond to the sides of the triangle originating from a common vertex: 8. **Properties of (omega)**: Such rotations illustrate that the angles between sides (AB), (BC), and (CA) are equal to (120^circ). 9. **Conclusion**: Given that all internal angles of a triangle sum up to (180^circ), each angle in (triangle ABC) must be: [ angle A = angle B = angle C = 60^circ. ] Hence, (triangle ABC) is an **equilateral triangle**. (boxed{text{C. Equilateral triangle}})

answer:Let x represent the number of gallons the tank holds when it is full. The gas used causes the tank's level to decrease from frac{2}{3} to frac{1}{3}, and this difference corresponds to 18 gallons. Setting up the equation: [ frac{2}{3}x - frac{1}{3}x = 18 ] Simplify on the left: [ frac{1}{3}x = 18 ] To find x, multiply both sides of the equation by 3: [ x = 18 cdot 3 ] [ x = 54 ] Thus, the tank holds boxed{54} gallons when it is full.