answer:To determine the coordinates of the point P(-1,2) with respect to the origin, we simply need to look at the given coordinates. The coordinates are already provided in the format (x, y), where x represents the horizontal distance from the origin, and y represents the vertical distance from the origin. Given that the point is P(-1,2), it means: - The x-coordinate is -1, indicating the point is 1 unit to the left of the origin. - The y-coordinate is 2, indicating the point is 2 units above the origin. However, it seems there was a mistake in the standard solution provided. The coordinates of the point P(-1,2) with respect to the origin are correctly given as (-1,2), not (1,-2) as stated. Therefore, the correct answer should be directly read from the question, without any changes needed, as the coordinates given are already with respect to the origin. Thus, the correct encapsulation of the final answer, following the original options provided, should be: - There is no option that matches (-1,2) exactly as it is the correct representation of the point P with respect to the origin. The standard solution's answer choice **B** (1,-2) is incorrect based on the coordinates given in the problem statement. Given the discrepancy, it's important to note that the correct representation of the point P(-1,2) with respect to the origin is indeed (-1,2), and none of the provided options match this. If we strictly follow the instructions and the format, we should highlight that the provided solution has an error, and based on the correct interpretation of the coordinates: boxed{text{None of the provided options}}

answer:1. **Understanding the Ratio**: The ratio of boys to girls in Ms. Lee's science class is 3:4, implying that for every 3 boys, there are 4 girls. 2. **Setting Up Variables**: Let the number of boys be 3x and the number of girls be 4x. Here, x is a common multiplier. 3. **Total Students Equation**: The total number of students is given as 42. The equation becomes: [ 3x + 4x = 7x ] [ 7x = 42 ] 4. **Solving for x**: [ x = frac{42}{7} = 6 ] 5. **Calculating Number of Boys and Girls**: - Number of boys = 3x = 3 times 6 = 18 - Number of girls = 4x = 4 times 6 = 24 6. **Finding the Difference**: The difference in number between girls and boys is: [ 4x - 3x = x = 6 ] 7. **Conclusion**: There are 6 more girls than boys in Ms. Lee's science class. The final answer is 6. The final answer is boxed{6} (Choice C).

answer:Let's call the amount John spent at the arcade "A". After spending at the arcade, he had 3.45 - A left. The next day, he spent one third of the remaining allowance at the toy store. So, he spent (1/3) * (3.45 - A) at the toy store. After spending at the toy store, he had the following amount left: 3.45 - A - (1/3) * (3.45 - A) Then, he spent his last 0.92 at the candy store. So, we can set up the equation: 3.45 - A - (1/3) * (3.45 - A) = 0.92 Now, let's solve for A: 3.45 - A - (1/3) * 3.45 + (1/3) * A = 0.92 3.45 - A - 1.15 + (1/3) * A = 0.92 2.30 - A + (1/3) * A = 0.92 2.30 - (2/3) * A = 0.92 Now, let's isolate A: (2/3) * A = 2.30 - 0.92 (2/3) * A = 1.38 To find A, we multiply both sides by the reciprocal of (2/3), which is (3/2): A = 1.38 * (3/2) A = 2.07 So, John spent 2.07 at the arcade. To find the fraction of his allowance that he spent at the arcade, we divide the amount spent at the arcade by his total allowance: Fraction spent at the arcade = 2.07 / 3.45 To simplify this fraction, we can divide both the numerator and the denominator by the greatest common divisor of the two numbers. However, since these numbers don't share a common divisor other than 1, the fraction is already in its simplest form. Therefore, the fraction of his allowance that John spent at the arcade is boxed{2.07} / 3.45.

answer:By the Cauchy-Schwarz inequality in its multiplicative form, we consider: [((x + 3) + 3(2y + 3)) left( frac{1}{x + 3} + frac{1}{2y + 3} right) ge (1 + sqrt{3})^2.] This simplifies to: [x + 3 + 6y + 9 ge 4(1 + sqrt{3})^2 = 16 + 8 sqrt{3},] so, arranging the terms, [x + 6y + 12 ge 16 + 8 sqrt{3}.] And thus: [x + 3y ge frac{16 + 8sqrt{3} - 12}{2} = 2 + 4sqrt{3}.] Equality occurs when (x + 3)^2 = 3(2y + 3)^2, or x + 3 = sqrt{3}(2y + 3). Substituting this into frac{1}{x + 3} + frac{1}{2y + 3} = frac{1}{4}, we get: [frac{1}{sqrt{3}(2y + 3)} + frac{1}{2y + 3} = frac{1}{4}.] Solving, [ left(frac{1}{sqrt{3}} + frac{1}{2}right)frac{1}{2y + 3} = frac{1}{4}, ] which reduces to [ left(frac{2 + sqrt{3}}{2sqrt{3}}right)frac{1}{2y + 3} = frac{1}{4}.] This gives 2y + 3 = 4sqrt{3}(2 + sqrt{3})/3. Finding 2y and then y, we get y = frac{4sqrt{3}(2 + sqrt{3})/3 -3}{2}, and using x + 3 = sqrt{3}(2y + 3), we find x accordingly. Upon solving, the minimum value is boxed{2 + 4 sqrt{3}}.