💡Okay, so I have this problem about a singing competition where 18 students made it to the finals. Their scores are listed in a table with scores ranging from 9.40 to 9.90, and the number of students who got each score. I need to find the median and the mode of these scores. Hmm, let's break this down step by step.First, let's recall what median and mode mean. The median is the middle value when all the numbers are arranged in order. Since there are 18 students, which is an even number, the median will be the average of the 9th and 10th scores. The mode is the score that appears most frequently.Looking at the table, the scores are already in ascending order, which is helpful. The number of students for each score is given, so I can list out all the scores in order to find the median and mode.Let me write down the scores with their frequencies:- 9.40: 2 students- 9.50: 3 students- 9.60: 5 students- 9.70: 4 students- 9.80: 3 students- 9.90: 1 studentSo, if I list all the scores in order, it would look like this:9.40, 9.40, 9.50, 9.50, 9.50, 9.60, 9.60, 9.60, 9.60, 9.60, 9.70, 9.70, 9.70, 9.70, 9.80, 9.80, 9.80, 9.90Let me count to make sure there are 18 scores:1. 9.402. 9.403. 9.504. 9.505. 9.506. 9.607. 9.608. 9.609. 9.6010. 9.6011. 9.7012. 9.7013. 9.7014. 9.7015. 9.8016. 9.8017. 9.8018. 9.90Yes, that's 18 scores. Now, to find the median, which is the average of the 9th and 10th scores. Let's see:9th score: 9.6010th score: 9.60So, the median is (9.60 + 9.60)/2 = 9.60Okay, that seems straightforward.Now, for the mode, which is the score that appears most frequently. Looking at the frequencies:- 9.40: 2- 9.50: 3- 9.60: 5- 9.70: 4- 9.80: 3- 9.90: 1The highest frequency is 5, which corresponds to the score 9.60. So, the mode is 9.60.Wait a minute, but let me double-check. Sometimes, people get confused between the median and the mode. The median is the middle value, and the mode is the most frequent. In this case, both the median and the mode are 9.60. That seems a bit unusual, but it's possible.Let me verify the median calculation again. Since there are 18 scores, the median is the average of the 9th and 10th scores. Counting to the 9th and 10th positions:1. 9.402. 9.403. 9.504. 9.505. 9.506. 9.607. 9.608. 9.609. 9.6010. 9.6011. 9.7012. 9.7013. 9.7014. 9.7015. 9.8016. 9.8017. 9.8018. 9.90Yes, both the 9th and 10th scores are 9.60, so the median is definitely 9.60.And the mode, as I calculated earlier, is also 9.60 because it appears more times than any other score.So, both the median and the mode are 9.60. Looking back at the options provided:A: 9.70, 9.60B: 9.60, 9.60C: 9.60, 9.70D: 9.65, 9.60The correct answer should be option B: 9.60, 9.60.I think that makes sense. I don't see any mistakes in my calculations. Both the median and the mode are 9.60.

💡Okay, so I have this problem about handshakes at a grand opening of an art museum. Let me try to understand it step by step. There are 15 married couples, so that means there are 15 men and 15 women, making a total of 30 people at the event. The problem says that each woman shook hands with every other woman, but no handshakes took place between men. Also, nobody shook hands with their spouse. So, I need to figure out how many handshake interactions there were among these 30 people.First, I think I should focus on the women since they are the ones doing the handshakes. There are 15 women, and each one shakes hands with every other woman. But wait, they don't shake hands with their own spouses, right? So, does that mean each woman doesn't shake hands with one specific man? Or does it mean something else?Hmm, the problem says "nobody shook hands with their spouse." So, if a woman doesn't shake hands with her spouse, but the spouses are men, and men didn't shake hands with anyone else. So, actually, the women are only shaking hands with other women, and not with their own husbands. But since men aren't shaking hands with anyone, maybe the only restriction is that women don't shake hands with their own husbands, but since men aren't participating in handshakes, does that affect the count?Wait, maybe I'm overcomplicating it. Let's break it down. If there are 15 women, and each woman shakes hands with every other woman except her spouse. But since the men aren't shaking hands with anyone, the only handshakes that happen are between women, and each woman doesn't shake hands with her own husband, but since her husband isn't shaking hands with anyone, maybe the only thing we need to consider is that each woman doesn't shake hands with her husband, but since the husband isn't part of the handshaking group, does that mean each woman just shakes hands with all the other women?Wait, no, because the problem says "nobody shook hands with their spouse." So, if a woman is at the event, she doesn't shake hands with her spouse, who is a man. But since men aren't shaking hands with anyone, maybe the only thing we need to do is subtract the handshakes that would have happened between each woman and her spouse. But since those handshakes didn't happen anyway, maybe it doesn't affect the count.Alternatively, maybe I need to think about it differently. If all the women were shaking hands with every other woman, the total number of handshakes would be the combination of 15 women taken 2 at a time, which is C(15,2). That would be 15*14/2 = 105 handshakes. But then, since each woman doesn't shake hands with her spouse, and there are 15 women, does that mean we need to subtract 15 handshakes? Because each woman would have shaken hands with her spouse if it weren't for the rule.Wait, but the problem says that no handshakes took place between men, and only women shook hands with each other. So, the handshakes between women and men didn't happen anyway. So, maybe the initial count of 105 handshakes is correct, and we don't need to subtract anything because the handshakes between women and their spouses weren't part of the women's handshakes in the first place.Let me think again. If there are 15 women, each shaking hands with 14 other women, that would be 15*14 = 210, but since each handshake is counted twice in that method, we divide by 2, giving us 105 handshakes. That seems right.But wait, the problem says "nobody shook hands with their spouse." So, does that mean that each woman didn't shake hands with her spouse, but since the spouse is a man and men aren't shaking hands with anyone, maybe the count remains 105 because the potential handshake between a woman and her spouse wasn't part of the women's handshakes anyway.Alternatively, if we consider that each woman would have shaken hands with 14 other women, but not with her spouse, which is a man, then the count remains 105 because the spouse isn't part of the women's group.Wait, but if we think about it, the total number of possible handshakes among the women is 105, and since the spouses are men and not part of the handshaking group, there's no overlap. So, the total number of handshakes is indeed 105.But let me double-check. If there are 15 women, each shaking hands with 14 others, that's 15*14 = 210, but since each handshake is between two women, we divide by 2, getting 105. That makes sense.So, the answer should be 105, which is option B.

💡First, I need to calculate the value of (87^2 - 78^2). I can use the difference of squares formula, which states that (a^2 - b^2 = (a + b)(a - b)).Applying this formula:[87^2 - 78^2 = (87 + 78)(87 - 78) = 165 times 9]Next, I'll compute the product:[165 times 9 = 1485]Now, I need to find all divisors of 1485 that fall within the range (45 < N < 100). To do this, I'll factorize 1485:[1485 = 5 times 297 = 5 times 3 times 99 = 5 times 3 times 3 times 33 = 5 times 3 times 3 times 3 times 11]So, the prime factorization of 1485 is (3^3 times 5 times 11).Using the prime factors, I'll list all possible divisors of 1485:[1, 3, 5, 9, 11, 15, 27, 33, 45, 55, 99, 135, 165, 297, 495, 1485]From this list, the divisors that satisfy (45 < N < 100) are:[55 text{ and } 99]Therefore, the possible values of (N) are 55 and 99.

💡Okay, so I have this problem here: I need to find the scaling transformation that changes the line x - 2y = 2 into the line 2x' - y' = 4. Hmm, scaling transformations usually involve scaling the coordinates by some factor, right? So, I think this means I need to figure out how much to scale the x and y coordinates so that when I apply the transformation, the original line becomes the new line.Let me start by writing down both equations again to make sure I have them right.Original line: x - 2y = 2Transformed line: 2x' - y' = 4I need to find a scaling transformation that takes any point (x, y) on the original line and maps it to a point (x', y') on the transformed line. Scaling transformations are usually represented as:x' = a * xy' = b * yWhere a and b are the scaling factors for the x and y coordinates, respectively. So, my goal is to find the values of a and b such that when I substitute x' and y' into the transformed line equation, it's equivalent to the original line equation.Let me substitute x' and y' into the transformed line equation:2x' - y' = 4Substituting x' = a * x and y' = b * y:2(a * x) - (b * y) = 4Simplify:2a * x - b * y = 4Now, I want this equation to be equivalent to the original line equation, which is x - 2y = 2. So, I can set up the following equations by comparing coefficients:For the x term: 2a = 1For the y term: -b = -2And for the constant term: 4 = 2Wait, hold on. The constant term on the left side after substitution is 4, but the original equation has a constant term of 2. That seems like a problem. Maybe I need to scale the entire equation or adjust the constants somehow.Let me think. If I have 2a * x - b * y = 4, and I want this to be equivalent to x - 2y = 2, I can divide the entire transformed equation by 2 to make the constant term match:(2a * x - b * y) / 2 = 4 / 2Which simplifies to:a * x - (b/2) * y = 2Now, this should be equal to the original equation x - 2y = 2. So, comparing coefficients again:For x: a = 1For y: - (b/2) = -2So, solving for a and b:From the x term: a = 1From the y term: - (b/2) = -2 => b/2 = 2 => b = 4So, the scaling factors are a = 1 and b = 4. That means the scaling transformation is:x' = 1 * x = xy' = 4 * ySo, the transformation scales the y-coordinate by a factor of 4 and leaves the x-coordinate unchanged.Let me double-check this. If I take a point (x, y) on the original line x - 2y = 2, then after scaling, the point becomes (x, 4y). Plugging into the transformed line equation:2x' - y' = 2x - 4yBut wait, the transformed line is supposed to be 2x' - y' = 4. So, substituting x' = x and y' = 4y:2x - 4y = 4Divide both sides by 2:x - 2y = 2Which is exactly the original equation. So, that checks out.Alternatively, if I take the original equation x - 2y = 2 and apply the inverse transformation, I should get the transformed equation. Let's see:If x' = x and y' = 4y, then x = x' and y = y'/4.Substituting into the original equation:x - 2*(y'/4) = 2 => x - (y'/2) = 2Multiply both sides by 2:2x - y' = 4Which is the transformed equation. So, that also checks out.Therefore, the scaling transformation is x' = x and y' = 4y.I think that makes sense. The x-coordinate isn't scaled at all, so a = 1, and the y-coordinate is scaled by 4, so b = 4. This transformation stretches the original line vertically by a factor of 4, which changes the slope and the intercept accordingly to match the transformed line.Just to visualize, the original line x - 2y = 2 can be rewritten as y = (x - 2)/2, which has a slope of 1/2 and a y-intercept at -1. The transformed line 2x' - y' = 4 can be rewritten as y' = 2x' - 4, which has a slope of 2 and a y-intercept at -4. So, the slope has changed from 1/2 to 2, which is a factor of 4 increase, and the y-intercept has also increased by a factor of 4 from -1 to -4. This aligns with scaling the y-coordinate by 4, as the transformation affects both the slope and the intercept in the y-direction.So, everything seems consistent. I don't think I made any mistakes here. The key was realizing that the constant term needed to be adjusted by scaling, which led me to divide the transformed equation by 2 to match the original equation's constant term. From there, comparing coefficients gave me the scaling factors directly.Another way to think about it is to consider how scaling affects the equation of a line. Scaling the x-coordinate by a factor a and the y-coordinate by a factor b transforms the line equation Ax + By = C into (A/a)x' + (B/b)y' = C. So, in this case, starting from x - 2y = 2, the transformed equation would be (1/a)x' + (-2/b)y' = 2. We want this to equal 2x' - y' = 4, so:(1/a)x' + (-2/b)y' = 2We can set this equal to 2x' - y' = 4 by scaling both sides appropriately. Let's see:If we multiply both sides of the transformed equation by 2:2*(1/a)x' + 2*(-2/b)y' = 4Which simplifies to:(2/a)x' + (-4/b)y' = 4We want this to be equal to 2x' - y' = 4, so:2/a = 2 => a = 1-4/b = -1 => 4/b = 1 => b = 4So, again, we get a = 1 and b = 4. This method also confirms the scaling factors.Therefore, I'm confident that the scaling transformation is x' = x and y' = 4y.**Final Answer**The scaling transformation is boxed{begin{cases} x' = x y' = 4y end{cases}}.

💡Okay, so I have this problem here about a quadratic equation, y equals ax squared plus bx plus c. It says that the graph intercepts the y-axis at y equals 2, and it has a vertex at (1, -1). I need to find a minus b plus c, where a, b, and c are integers. Hmm, let me think about how to approach this.First, I remember that the y-intercept of a quadratic equation is the value of y when x is 0. So, if I plug in x equals 0 into the equation y = ax² + bx + c, I get y = c. The problem says the y-intercept is at y = 2, so that means c must be 2. Okay, so c is 2. That's one value down.Next, the vertex of a quadratic equation is given by the point (h, k), where h is -b/(2a) and k is the value of the function at that point. The vertex here is (1, -1), so h is 1 and k is -1. That means that when x is 1, y is -1. So, I can plug x = 1 and y = -1 into the equation to get another equation involving a and b.So, substituting x = 1 and y = -1 into y = ax² + bx + c, we get:-1 = a(1)² + b(1) + cWhich simplifies to:-1 = a + b + cBut I already know that c is 2, so plugging that in:-1 = a + b + 2Subtracting 2 from both sides:-3 = a + bSo, a + b = -3. That's one equation.Now, I also know that the x-coordinate of the vertex is given by -b/(2a). Since the vertex is at x = 1, that means:1 = -b/(2a)Multiplying both sides by 2a:2a = -bSo, b = -2aNow, I have two equations:1. a + b = -32. b = -2aI can substitute the second equation into the first one. So, replacing b with -2a in the first equation:a + (-2a) = -3Simplifying:-a = -3Multiplying both sides by -1:a = 3Now that I have a = 3, I can find b using the second equation:b = -2a = -2*3 = -6So, a is 3, b is -6, and c is 2. The problem asks for a - b + c. Let me compute that:a - b + c = 3 - (-6) + 2 = 3 + 6 + 2 = 11Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from the beginning, c is 2 because the y-intercept is 2. Then, using the vertex (1, -1), I set up the equation -1 = a + b + 2, which simplifies to a + b = -3. Then, using the vertex formula, I found that b = -2a. Substituting that into a + b = -3 gives a - 2a = -3, so -a = -3, meaning a = 3. Then, b = -6. Plugging back into the equation, y = 3x² -6x + 2. Let me check the vertex: the x-coordinate is -b/(2a) = 6/(6) = 1, which matches. The y-coordinate is y = 3(1)^2 -6(1) + 2 = 3 -6 + 2 = -1, which also matches. So, everything checks out.Therefore, a - b + c is 3 - (-6) + 2, which is 3 + 6 + 2 = 11. I think that's the correct answer.

💡First, determine the total number of marbles in the bag by adding the number of red, blue, and yellow marbles.Next, calculate the probability of drawing a red or blue marble in a single draw by dividing the number of red and blue marbles by the total number of marbles.Since the drawing is done with replacement, the probability of drawing a red or blue marble remains the same for the second draw.Finally, multiply the probability of drawing a red or blue marble in the first draw by the probability of drawing a red or blue marble in the second draw to find the overall probability of both events occurring consecutively.