answer：First, identify the two attractions that must be visited consecutively: Haunted Mansion (HM) and Space Mountain (SM). These two can be treated as a single unit.Next, determine the total number of units to arrange. Since HM and SM are combined into one unit, along with the other four attractions, there are 5 units in total.Calculate the number of ways to arrange these 5 units, which is 5 factorial (5!).Since HM must come immediately before SM, there is only 1 way to arrange these two within their combined unit.Finally, multiply the number of arrangements of the 5 units by the number of arrangements within the combined unit to get the total number of valid orders.

answer：First, I recognize that Sarah has a total of 23 albums in her collection, and 15 of these are shared with Emily. To find out how many albums are unique to Sarah, I subtract the shared albums from her total:23 - 15 = 8 albums unique to Sarah.Next, it's given that Emily has 8 albums that are not in Sarah's collection. These are unique to Emily.Finally, to determine the total number of albums that are in either Sarah's or Emily's collection but not both, I add the unique albums from both collections:8 (unique to Sarah) + 8 (unique to Emily) = 16 albums.

answer：Alright, so I've got this problem about runners X, Y, and Z racing over a distance d. It says that X can beat Y by 25 yards, Y can beat Z by 15 yards, and X can beat Z by 35 yards. I need to figure out what d is. The options are 70, 75, 80, or 85 yards. Hmm, okay.First, I think I need to understand what it means when it says X beats Y by 25 yards. I guess that means when X finishes the race (which is distance d), Y has only run d - 25 yards. Similarly, when Y finishes the race, Z has only run d - 15 yards. And when X finishes, Z has run d - 35 yards.So, maybe I can set up some equations based on their speeds. If they're running at uniform speeds, then speed is distance over time. So, if I can figure out the time it takes for each runner to finish the race, I can relate their speeds.Let me denote the speeds of X, Y, and Z as v_x, v_y, and v_z respectively. When X finishes the race, Y has run d - 25 yards. The time it takes X to finish is d / v_x. In that same time, Y runs d - 25 yards, so:d / v_x = (d - 25) / v_ySimilarly, when Y finishes the race, Z has run d - 15 yards. The time it takes Y to finish is d / v_y. In that time, Z runs d - 15 yards:d / v_y = (d - 15) / v_zAnd when X finishes the race, Z has run d - 35 yards. So, the time it takes X to finish is d / v_x, and in that time, Z runs d - 35 yards:d / v_x = (d - 35) / v_zOkay, so now I have three equations:1. d / v_x = (d - 25) / v_y2. d / v_y = (d - 15) / v_z3. d / v_x = (d - 35) / v_zI need to solve these equations to find d. Maybe I can express v_y and v_z in terms of v_x and then substitute.From equation 1:v_y = (d - 25) / d * v_xFrom equation 2:v_z = (d - 15) / d * v_yBut I can substitute v_y from equation 1 into this:v_z = (d - 15) / d * (d - 25) / d * v_xSo, v_z = [(d - 15)(d - 25)] / d^2 * v_xNow, from equation 3:v_z = (d - 35) / d * v_xSo, setting the two expressions for v_z equal:[(d - 15)(d - 25)] / d^2 * v_x = (d - 35) / d * v_xI can cancel out v_x from both sides:[(d - 15)(d - 25)] / d^2 = (d - 35) / dMultiply both sides by d^2 to eliminate denominators:(d - 15)(d - 25) = d(d - 35)Now, let's expand both sides:Left side: (d - 15)(d - 25) = d^2 - 25d - 15d + 375 = d^2 - 40d + 375Right side: d(d - 35) = d^2 - 35dSo, set them equal:d^2 - 40d + 375 = d^2 - 35dSubtract d^2 from both sides:-40d + 375 = -35dAdd 40d to both sides:375 = 5dDivide both sides by 5:d = 75So, the distance d is 75 yards. Let me check if this makes sense.If d is 75 yards, then when X finishes, Y has run 75 - 25 = 50 yards. So, the ratio of their speeds is v_x / v_y = 75 / 50 = 3/2.Similarly, when Y finishes 75 yards, Z has run 75 - 15 = 60 yards. So, v_y / v_z = 75 / 60 = 5/4.Then, the ratio of X's speed to Z's speed should be (3/2) * (5/4) = 15/8.Now, when X runs 75 yards, Z should run (8/15)*75 = 40 yards. So, X beats Z by 75 - 40 = 35 yards, which matches the problem statement.Everything checks out. So, d is 75 yards.

answer：First, I need to determine the interest rate for the Rs. 1400 loan. I know the total interest from both loans is Rs. 390 over 3.9 years.I'll start by calculating the interest from the Rs. 1000 loan at 3% per year. Using the simple interest formula:Interest = Principal × Rate × TimeInterest on Rs. 1000 = 1000 × 0.03 × 3.9 = Rs. 117Next, I'll find the interest from the Rs. 1400 loan by subtracting the interest from the Rs. 1000 loan from the total interest:Interest on Rs. 1400 = 390 - 117 = Rs. 273Now, I'll use the simple interest formula again to find the interest rate for the Rs. 1400 loan:Interest = Principal × Rate × Time273 = 1400 × (R/100) × 3.9Solving for R:R = (273 / (1400 × 3.9)) × 100R = (273 / 5460) × 100R = 0.05 × 100R = 5%Therefore, the interest rate for the Rs. 1400 loan is 5% per year.

answer：First, I need to determine how many cakes are left after Carol eats some from each day's batch. For the first day, Sara baked 7 cakes and Carol ate 4, leaving 3 cakes. On the second day, 12 cakes were baked and 6 were eaten, leaving 6 cakes. The third day had 8 cakes baked with 3 eaten, leaving 5 cakes. On the fourth day, 10 cakes were baked and 2 were eaten, leaving 8 cakes. Finally, on the fifth day, 15 cakes were baked and 3 were eaten, leaving 12 cakes.Next, I'll calculate the frosting needed for the remaining cakes based on the frosting rate per cake for each day.For the first day's remaining 3 cakes, with 2 cans per cake, that's 3 * 2 = 6 cans.For the second day's 6 cakes, using 3 cans per cake, it's 6 * 3 = 18 cans.The third day's 5 cakes require 4 cans each, so 5 * 4 = 20 cans.The fourth day's 8 cakes need 3 cans each, totaling 8 * 3 = 24 cans.Lastly, the fifth day's 12 cakes require 2 cans each, amounting to 12 * 2 = 24 cans.Adding all these together: 6 + 18 + 20 + 24 + 24 equals 92 cans of frosting in total.

answer：Okay, so I've got this problem here: Define an operation a bowtie b as a plus the square root of (b plus the square root of (b plus the square root of (b plus ...))). And it says that 5 bowtie x equals 12. I need to find the value of x. Hmm, that seems a bit complicated, but let me try to break it down step by step.First, let me make sure I understand the operation. The bowtie operation takes two numbers, a and b, and adds a to an infinite nested square root expression that starts with b. So, 5 bowtie x would be 5 plus the square root of (x plus the square root of (x plus the square root of (x plus ...))). And this whole thing equals 12. So, I can write that as:5 + sqrt(x + sqrt(x + sqrt(x + ...))) = 12Alright, so if I subtract 5 from both sides, I get:sqrt(x + sqrt(x + sqrt(x + ...))) = 7Now, this expression inside the square root is an infinite nested radical. I remember that sometimes these kinds of expressions can be simplified by recognizing that they repeat themselves. Let me denote the entire infinite nested radical as some variable, say y. So, let me set:y = sqrt(x + sqrt(x + sqrt(x + ...)))But wait, the expression inside the square root is the same as y itself, right? Because it's the same infinite sequence of square roots. So, substituting that in, I get:y = sqrt(x + y)Now, I can square both sides to eliminate the square root:y^2 = x + ySo, rearranging terms:y^2 - y - x = 0Now, from earlier, I know that y is equal to 7 because sqrt(x + sqrt(x + ...)) equals 7. So, substituting y = 7 into the equation:7^2 - 7 - x = 0Calculating 7 squared is 49, so:49 - 7 - x = 0Which simplifies to:42 - x = 0Therefore, solving for x:x = 42Wait, that seems straightforward. Let me double-check to make sure I didn't make any mistakes. Starting from the beginning:5 bowtie x = 12Which translates to:5 + sqrt(x + sqrt(x + ...)) = 12Subtracting 5:sqrt(x + sqrt(x + ...)) = 7Let y = sqrt(x + sqrt(x + ...)), so y = sqrt(x + y)Squaring both sides:y^2 = x + ySubstituting y = 7:49 = x + 7So, x = 42Yep, that checks out. I think I did it right. So, the value of x is 42.Just to be thorough, let me plug x = 42 back into the original expression to see if it equals 12.So, 5 bowtie 42 = 5 + sqrt(42 + sqrt(42 + sqrt(42 + ...)))We already determined that the nested radical equals 7, so:5 + 7 = 12Which matches the given equation. So, everything seems consistent. I don't see any errors in my reasoning. Therefore, I'm confident that x is indeed 42.**Final Answer**The value of ( x ) is boxed{42}.