answer：First, determine how many laps Yvonne swims in 5 minutes, which is 10 laps.Next, calculate the number of laps Yvonne's younger sister swims by taking half of Yvonne's laps:[ frac{10}{2} = 5 text{ laps} ]Finally, find out how many laps Joel swims by multiplying his sister's laps by 3:[ 3 times 5 = 15 text{ laps} ]

answer：Okay, so I have this problem where I need to find the value of ( k ) that makes the sum of the distances ( overline{AC} + overline{BC} ) as small as possible. The coordinates given are ( A(4,6) ), ( B(3,0) ), and ( C(k,0) ). Hmm, let me try to visualize this.First, I know that ( A ) is somewhere above the x-axis, and both ( B ) and ( C ) are on the x-axis. So, ( C ) is a point that can move along the x-axis, and I need to find its position such that the total distance from ( A ) to ( C ) and from ( B ) to ( C ) is minimized.I remember something about reflecting points to find the shortest path. Maybe I can use that idea here. If I reflect point ( A ) across the x-axis, its reflection ( A' ) would be at ( (4, -6) ). Then, the distance from ( A ) to ( C ) is the same as the distance from ( A' ) to ( C ). So, ( overline{AC} = overline{A'C} ).Now, the problem becomes finding a point ( C ) on the x-axis such that the sum ( overline{A'C} + overline{BC} ) is minimized. I think this is similar to finding a point where the path from ( A' ) to ( C ) to ( B ) is the shortest possible. In geometry, the shortest path between two points is a straight line, so maybe if I draw a straight line from ( A' ) to ( B ), the point where this line intersects the x-axis will be the optimal ( C ).Let me try to find the equation of the line connecting ( A'(4, -6) ) and ( B(3, 0) ). The slope ( m ) of this line is:[m = frac{0 - (-6)}{3 - 4} = frac{6}{-1} = -6]So, the slope is ( -6 ). Now, using the point-slope form of the equation with point ( B(3, 0) ):[y - 0 = -6(x - 3)][y = -6x + 18]This line intersects the x-axis where ( y = 0 ). Plugging ( y = 0 ) into the equation:[0 = -6x + 18][6x = 18][x = 3]Wait, so the intersection point is at ( (3, 0) ), which is exactly point ( B ). That means the optimal point ( C ) is at ( (3, 0) ), so ( k = 3 ).But let me double-check. If ( C ) is at ( (3, 0) ), then ( overline{AC} ) is the distance from ( (4,6) ) to ( (3,0) ), and ( overline{BC} ) is the distance from ( (3,0) ) to ( (3,0) ), which is zero. So, the total distance is just ( overline{AC} ).Is there a way to get a smaller total distance? If I move ( C ) slightly to the left or right of ( 3 ), would the sum ( overline{AC} + overline{BC} ) decrease?Let's try ( k = 2 ). Then, ( C ) is at ( (2, 0) ). Calculating ( overline{AC} ):[sqrt{(4-2)^2 + (6-0)^2} = sqrt{4 + 36} = sqrt{40} approx 6.32]And ( overline{BC} ):[sqrt{(3-2)^2 + (0-0)^2} = sqrt{1} = 1]Total distance: ( 6.32 + 1 = 7.32 ).Now, if ( k = 3 ), ( overline{AC} ):[sqrt{(4-3)^2 + (6-0)^2} = sqrt{1 + 36} = sqrt{37} approx 6.08]And ( overline{BC} = 0 ), so total distance is ( 6.08 ), which is less than 7.32.What if ( k = 4 )? Then, ( C ) is at ( (4, 0) ). Calculating ( overline{AC} ):[sqrt{(4-4)^2 + (6-0)^2} = sqrt{0 + 36} = 6]And ( overline{BC} ):[sqrt{(3-4)^2 + (0-0)^2} = sqrt{1} = 1]Total distance: ( 6 + 1 = 7 ), which is more than 6.08.So, it seems that ( k = 3 ) gives the smallest total distance. But why does reflecting ( A ) across the x-axis and drawing a straight line to ( B ) give the optimal point? I think it's because the reflection trick helps in finding the shortest path that involves a reflection over a line, which in this case is the x-axis.Another way to think about it is using calculus. Let's express the total distance ( D ) as a function of ( k ):[D(k) = sqrt{(4 - k)^2 + 6^2} + sqrt{(3 - k)^2 + 0^2}][D(k) = sqrt{(4 - k)^2 + 36} + |3 - k|]To minimize ( D(k) ), we can take its derivative with respect to ( k ) and set it to zero.First, let's consider ( k neq 3 ) to avoid the absolute value complication.The derivative ( D'(k) ) is:[D'(k) = frac{-(4 - k)}{sqrt{(4 - k)^2 + 36}} - frac{(3 - k)}{|3 - k|}]Set ( D'(k) = 0 ):[frac{-(4 - k)}{sqrt{(4 - k)^2 + 36}} - frac{(3 - k)}{|3 - k|} = 0]This equation looks a bit complicated, but let's analyze it.Case 1: ( k < 3 ). Then, ( |3 - k| = 3 - k ), so:[frac{-(4 - k)}{sqrt{(4 - k)^2 + 36}} - frac{(3 - k)}{3 - k} = 0][frac{-(4 - k)}{sqrt{(4 - k)^2 + 36}} - 1 = 0][frac{-(4 - k)}{sqrt{(4 - k)^2 + 36}} = 1]But the left side is negative (since ( -(4 - k) ) is negative when ( k < 4 )), and the right side is positive. So, no solution in this case.Case 2: ( k > 3 ). Then, ( |3 - k| = k - 3 ), so:[frac{-(4 - k)}{sqrt{(4 - k)^2 + 36}} - frac{(3 - k)}{k - 3} = 0][frac{-(4 - k)}{sqrt{(4 - k)^2 + 36}} + 1 = 0][frac{-(4 - k)}{sqrt{(4 - k)^2 + 36}} = -1][frac{4 - k}{sqrt{(4 - k)^2 + 36}} = 1]Let me denote ( t = 4 - k ). Then, the equation becomes:[frac{t}{sqrt{t^2 + 36}} = 1]Square both sides:[frac{t^2}{t^2 + 36} = 1][t^2 = t^2 + 36][0 = 36]Which is a contradiction. So, no solution in this case either.Wait, that's strange. It seems like the derivative doesn't equal zero for any ( k neq 3 ). Maybe the minimum occurs at ( k = 3 ), where the derivative isn't defined because of the absolute value.Let me check the behavior of ( D(k) ) around ( k = 3 ).For ( k ) slightly less than 3, say ( k = 2.9 ):[D(k) = sqrt{(4 - 2.9)^2 + 36} + |3 - 2.9| = sqrt{1.21 + 36} + 0.1 approx sqrt{37.21} + 0.1 approx 6.1 + 0.1 = 6.2]For ( k = 3 ):[D(k) = sqrt{(4 - 3)^2 + 36} + 0 = sqrt{1 + 36} = sqrt{37} approx 6.08]For ( k = 3.1 ):[D(k) = sqrt{(4 - 3.1)^2 + 36} + |3 - 3.1| = sqrt{0.81 + 36} + 0.1 approx sqrt{36.81} + 0.1 approx 6.07 + 0.1 = 6.17]So, the function ( D(k) ) reaches its minimum at ( k = 3 ), even though the derivative isn't zero there. This makes sense because ( k = 3 ) is a point where the function isn't differentiable (due to the absolute value), but it's still the point where the function changes direction from decreasing to increasing.Therefore, the value of ( k ) that minimizes ( overline{AC} + overline{BC} ) is ( 3 ).

answer：First, I recognize that the LCM and HCF of two numbers are related to the numbers themselves through the formula:[ text{LCM} times text{HCF} = text{Product of the two numbers} ]Given that the LCM is 2310 and one of the numbers is 462, I can use this formula to find the HCF. However, the other number provided is 150, which doesn't seem to fit because 150 is not a factor of 2310.To find the correct second number, I'll rearrange the formula to solve for the unknown number ( x ):[ text{HCF} = frac{462 times x}{2310} ]Since the HCF must be an integer, ( x ) must be a factor of 2310. Dividing 2310 by 462 gives:[ x = frac{2310}{462} = 5 ]Now that I have the correct second number, which is 5, I can calculate the HCF:[ text{HCF} = frac{462 times 5}{2310} = frac{2310}{2310} = 1 ]Therefore, the HCF of 462 and 5 is 1.

answer：First, I will identify and combine all the constant terms in the expression. This involves adding or subtracting the numbers that do not have any variables attached to them.Next, I will move on to the linear terms, which are the terms that contain the variable ( x ) raised to the first power. I will combine these by adding or subtracting their coefficients.Then, I will handle the squared terms, which are the terms with ( x ) raised to the second power. Similar to the previous steps, I will combine these by performing the necessary addition or subtraction on their coefficients.Finally, I will address the cubic terms, which are the terms with ( x ) raised to the third power. I will combine these by adding or subtracting their coefficients as well.After combining all like terms, I will present the simplified polynomial in standard form, arranging the terms from the highest power of ( x ) to the lowest.

answer：First, I need to determine the probability that the target is hit when all three individuals shoot simultaneously. To do this, I'll first find the probability that none of them hit the target and then subtract that from 1.The probability of person A missing the target is 1 - 1/2 = 1/2.The probability of person B missing the target is 1 - 1/3 = 2/3.The probability of person C missing the target is 1 - 1/4 = 3/4.Next, I'll calculate the probability that all three miss the target by multiplying their individual probabilities of missing:1/2 * 2/3 * 3/4 = 1/4.Finally, the probability that the target is hit is the complement of all three missing:1 - 1/4 = 3/4.

answer：Okay, so I have this problem here with points O, P, and M on a line segment AB. The problem gives me some ratios and tells me that OM is 2 units long. I need to find the length of PM. Hmm, let me try to visualize this first.Alright, AB is a straight line segment. Points O, P, and M are somewhere on this line. AO is 4/5 of AB, which means O is pretty close to A. BP is 2/3 of AB, so P is closer to B. M is the midpoint, so it's right in the middle of AB. Let me write down what I know:1. AO = (4/5)AB2. BP = (2/3)AB3. M is the midpoint, so AM = MB = (1/2)AB4. OM = 2I need to find PM. Hmm, okay. Maybe I can express all these points in terms of AB and then find the distances.First, let's assign a variable to AB. Let me call AB = x. That way, I can express all the other segments in terms of x.So, AO = (4/5)x. That means the distance from A to O is 4/5 of the entire segment AB. Similarly, BP = (2/3)x, so the distance from B to P is 2/3 of AB.Since M is the midpoint, AM = MB = (1/2)x.Now, I need to figure out where exactly O and P are located on AB. Let me think about their positions.Starting from A, AO is 4/5x, so O is 4/5 of the way from A to B. Similarly, BP is 2/3x, so P is 2/3 of the way from B to A. Wait, that might be confusing. Let me clarify.If BP = (2/3)x, then the distance from B to P is 2/3 of AB. So, starting from B, moving towards A, P is 2/3 of the way. That means the distance from A to P would be AB - BP = x - (2/3)x = (1/3)x. So AP = (1/3)x.Similarly, AO = (4/5)x, so the distance from A to O is 4/5 of AB. That means the distance from O to B would be AB - AO = x - (4/5)x = (1/5)x. So OB = (1/5)x.Now, I need to find OM and PM. Let's start with OM.OM is the distance between O and M. Since M is the midpoint, its position is at (1/2)x from A. O is at (4/5)x from A. So, the distance from O to M would be the absolute difference between their positions.Wait, hold on. If O is at (4/5)x from A and M is at (1/2)x from A, then OM = |(4/5)x - (1/2)x|. Let me compute that.First, find a common denominator for 4/5 and 1/2. The common denominator is 10.So, 4/5 = 8/10 and 1/2 = 5/10. Therefore, OM = |8/10x - 5/10x| = |3/10x| = (3/10)x.But the problem states that OM = 2. So, (3/10)x = 2. Let me solve for x.Multiply both sides by 10: 3x = 20.Divide both sides by 3: x = 20/3.So, AB = 20/3 units.Now that I have AB, I can find PM.PM is the distance between P and M. Let's find the positions of P and M.M is at (1/2)x from A, which is (1/2)*(20/3) = 10/3.P is located such that AP = (1/3)x, which is (1/3)*(20/3) = 20/9.Wait, so P is at 20/9 from A, and M is at 10/3 from A. Let me convert 10/3 to ninths to make it easier to subtract.10/3 = 30/9.So, PM = |M - P| = |30/9 - 20/9| = |10/9| = 10/9.But wait, the problem says the length of PM is -10/9. Hmm, that doesn't make sense because distance can't be negative. Maybe I made a mistake in the direction.Let me think again. If P is closer to B, and M is the midpoint, then P is to the right of M if we consider A on the left and B on the right. Wait, no, if AP = 20/9 and AM = 10/3 = 30/9, then P is at 20/9 from A, which is less than 30/9. So, P is to the left of M. Therefore, PM would be M - P = 30/9 - 20/9 = 10/9. So, it's positive.But the problem states it's -10/9. Maybe the direction is considered, so if we take vectors into account, the displacement from P to M is negative if we consider the direction from A to B as positive. Wait, but in the problem statement, it's just the length, so it should be positive.Hmm, maybe I need to check my calculations again.Wait, let me go back step by step.AB = 20/3.AO = (4/5)AB = (4/5)*(20/3) = (80/15) = 16/3.So, O is at 16/3 from A.M is at 10/3 from A.So, OM = |16/3 - 10/3| = 6/3 = 2. That matches the given OM = 2.Good.Now, BP = (2/3)AB = (2/3)*(20/3) = 40/9.So, the distance from B to P is 40/9.Therefore, AP = AB - BP = 20/3 - 40/9 = (60/9 - 40/9) = 20/9.So, P is at 20/9 from A.M is at 10/3 = 30/9 from A.So, PM = |30/9 - 20/9| = 10/9.So, the length is 10/9, not negative. Maybe the negative sign is a mistake in the problem statement or it's indicating direction. But in terms of length, it should be positive.Wait, maybe the problem is using coordinates, so if we assign coordinates to the points, the position of P could be to the left of M, hence the negative sign. Let me try that.Let me assign coordinates to the points. Let me set point A at 0 and point B at x = 20/3.So, A = 0, B = 20/3.Then, M is the midpoint, so M = (0 + 20/3)/2 = 10/3 ≈ 3.333.O is at AO = 4/5 * AB = 4/5 * 20/3 = 16/3 ≈ 5.333.Wait, hold on, that can't be. If AB is 20/3 ≈ 6.666, then O is at 16/3 ≈ 5.333, which is closer to B, not A. Wait, that contradicts my earlier thought.Wait, AO is 4/5 of AB, so starting from A, moving 4/5 towards B, so O is closer to B. Similarly, BP is 2/3 of AB, so starting from B, moving 2/3 towards A, so P is closer to A.Wait, so in coordinates:A = 0, B = 20/3.O is at AO = 4/5 * AB = 4/5 * 20/3 = 16/3 ≈ 5.333.P is at BP = 2/3 * AB = 2/3 * 20/3 = 40/9 ≈ 4.444 from B. So, since B is at 20/3 ≈ 6.666, then P is at 20/3 - 40/9 = (60/9 - 40/9) = 20/9 ≈ 2.222.Wait, so P is at 20/9 ≈ 2.222, M is at 10/3 ≈ 3.333, and O is at 16/3 ≈ 5.333.So, from A to B: 0, P at ~2.222, M at ~3.333, O at ~5.333, and B at ~6.666.So, PM is the distance from P to M, which is 10/3 - 20/9 = 30/9 - 20/9 = 10/9 ≈ 1.111.So, PM is 10/9, positive. So, why does the problem say -10/9? Maybe it's a vector displacement from P to M, which would be negative if we consider the direction from A to B as positive. Because P is to the left of M, so moving from P to M is in the positive direction, so it should be positive. Hmm, maybe the negative sign is a mistake.Alternatively, maybe the problem is using a coordinate system where A is at a higher coordinate than B, so moving from A to B is negative. So, if A is at, say, 10 and B is at 0, then AB is 10 units. Then, AO = 4/5 * 10 = 8, so O is at 2. BP = 2/3 * 10 ≈ 6.666, so P is at 3.333. M is the midpoint at 5. Then, OM would be |5 - 2| = 3. But the problem says OM = 2, so that doesn't fit.Wait, maybe I need to set up a coordinate system where A is at 0 and B is at x, then express all points in terms of x.Let me try that.Let me denote AB as the segment from 0 to x on the number line.So, A = 0, B = x.Then, AO = (4/5)x, so O is at (4/5)x.BP = (2/3)x, so P is at B - BP = x - (2/3)x = (1/3)x.M is the midpoint, so M = (0 + x)/2 = x/2.Now, OM is the distance between O and M. So, O is at (4/5)x, M is at (1/2)x.So, OM = |(4/5)x - (1/2)x|.Let's compute that:First, find a common denominator for 4/5 and 1/2, which is 10.4/5 = 8/10, 1/2 = 5/10.So, OM = |8/10x - 5/10x| = |3/10x| = (3/10)x.Given that OM = 2, so (3/10)x = 2.Solving for x:Multiply both sides by 10: 3x = 20.Divide both sides by 3: x = 20/3.So, AB = 20/3.Now, let's find PM.P is at (1/3)x = (1/3)*(20/3) = 20/9.M is at x/2 = (20/3)/2 = 10/3.So, PM = |M - P| = |10/3 - 20/9|.Convert 10/3 to ninths: 10/3 = 30/9.So, PM = |30/9 - 20/9| = |10/9| = 10/9.So, the length of PM is 10/9.But the problem says it's -10/9. Hmm, maybe it's considering direction. If we take vectors, then PM would be M - P = 10/3 - 20/9 = 10/9, but if we consider the direction from P to M as positive, it's positive. If we consider the direction from M to P, it's negative. But in terms of length, it's positive.Alternatively, maybe the problem is using a coordinate system where increasing values go from B to A, so moving from B to A is positive. Then, P is at (1/3)x from A, which would be closer to B, but in that case, the coordinate system is flipped.Wait, let me think again.If we assign coordinates with A at 0 and B at x, then:- A = 0- B = x- O = (4/5)x- P = (1/3)x- M = (1/2)xSo, in this coordinate system, moving from A to B is positive.So, P is at (1/3)x, which is less than M at (1/2)x, so PM is M - P = (1/2)x - (1/3)x = (3/6 - 2/6)x = (1/6)x.Wait, but earlier I calculated PM as 10/9 when x = 20/3. Let me check:(1/6)x = (1/6)*(20/3) = 20/18 = 10/9. So, that matches.But in the coordinate system, PM is positive because M is to the right of P.Wait, so why does the problem say -10/9? Maybe it's a typo or maybe it's considering vectors with a different direction.Alternatively, maybe the problem is using a different coordinate system where A is at a higher coordinate than B, so moving from A to B is negative.Let me try that.Let me assign A at x and B at 0, so AB is from x to 0.Then, AO = (4/5)AB = (4/5)x, so O is at A - AO = x - (4/5)x = (1/5)x.BP = (2/3)AB = (2/3)x, so P is at B + BP = 0 + (2/3)x = (2/3)x.M is the midpoint, so M = (x + 0)/2 = x/2.Now, OM is the distance between O and M.O is at (1/5)x, M is at (1/2)x.So, OM = |(1/2)x - (1/5)x|.Compute that:Convert to common denominator: 1/2 = 5/10, 1/5 = 2/10.So, OM = |5/10x - 2/10x| = |3/10x| = (3/10)x.Given OM = 2, so (3/10)x = 2.Solving for x: x = (2 * 10)/3 = 20/3.So, AB = 20/3.Now, PM is the distance between P and M.P is at (2/3)x = (2/3)*(20/3) = 40/9.M is at x/2 = (20/3)/2 = 10/3 = 30/9.So, PM = |M - P| = |30/9 - 40/9| = |-10/9| = 10/9.Again, the length is 10/9. But in this coordinate system, if we consider direction from A to B as negative, then PM would be M - P = 30/9 - 40/9 = -10/9. So, the displacement vector from P to M is -10/9, indicating direction towards A.But the problem says "the length of PM is -10/9". That seems incorrect because length should be positive. Maybe the problem is referring to the vector displacement, not the length. Or perhaps it's a mistake in the sign.Alternatively, maybe the problem is using a different labeling where P is on the other side, but that doesn't make sense because BP is given.Wait, let me double-check my calculations.AB = 20/3.AO = (4/5)*(20/3) = 16/3 ≈ 5.333.BP = (2/3)*(20/3) = 40/9 ≈ 4.444.So, AP = AB - BP = 20/3 - 40/9 = 60/9 - 40/9 = 20/9 ≈ 2.222.So, P is at 20/9 from A, M is at 10/3 ≈ 3.333 from A.So, PM = 10/3 - 20/9 = 30/9 - 20/9 = 10/9.Yes, that's correct.So, the length is 10/9, positive. The negative sign might be a mistake or perhaps indicating direction in a specific coordinate system, but in terms of pure length, it's positive.Therefore, I think the answer should be 10/9, not -10/9. Maybe the problem had a typo or something.But to be thorough, let me check if I interpreted the positions correctly.AO = 4/5 AB: so from A, moving 4/5 towards B, so O is closer to B.BP = 2/3 AB: from B, moving 2/3 towards A, so P is closer to A.M is the midpoint, so it's exactly halfway.So, in coordinates from A to B:A = 0, P = 20/9 ≈ 2.222, M = 10/3 ≈ 3.333, O = 16/3 ≈ 5.333, B = 20/3 ≈ 6.666.So, P is left of M, so PM is positive if we consider from P to M.But if we consider vectors, the displacement from P to M is positive, but if we consider the displacement from M to P, it's negative.But the problem says "the length of PM is -10/9". Length should be positive, so I think it's a mistake.Alternatively, maybe the problem is using a different labeling where P is on the other side, but that contradicts BP = 2/3 AB.Wait, BP is from B, so P is on AB, closer to A.So, I think my calculations are correct, and the negative sign is incorrect.Therefore, the length of PM is 10/9.**Final Answer**The length of ( PM ) is boxed{frac{10}{9}}.