answer：Okay, so I have this problem here about an isosceles triangle with two sides measuring 3 and 7, and I need to find the perimeter. The options are A: 13, B: 17, C: 13 or 17, and D: 13 or 10. Hmm, let me think about this step by step.First, I remember that an isosceles triangle has two sides that are equal in length. So, in this case, we have two sides given: 3 and 7. That means either the two equal sides are both 3, or both 7, right? Or maybe one of them is the base, and the other two are equal. Wait, no, actually, in an isosceles triangle, the two equal sides are called the legs, and the third side is the base. So, depending on which sides are equal, the perimeter will change.Let me consider the two possibilities:1. The two equal sides are both 3. So, the sides would be 3, 3, and 7. Then, the perimeter would be 3 + 3 + 7, which is 13. But wait, I need to make sure that this actually forms a valid triangle. For a triangle to be valid, the sum of the lengths of any two sides must be greater than the length of the third side. So, let's check: - 3 + 3 > 7? That's 6 > 7, which is not true. So, this configuration doesn't work because it doesn't satisfy the triangle inequality theorem. Therefore, the sides cannot be 3, 3, and 7.2. The two equal sides are both 7. So, the sides would be 7, 7, and 3. Let's calculate the perimeter: 7 + 7 + 3 = 17. Now, let's check if this forms a valid triangle: - 7 + 7 > 3? That's 14 > 3, which is true. - 7 + 3 > 7? That's 10 > 7, which is also true. - 7 + 3 > 7? Same as above, 10 > 7, which is true. So, all the conditions are satisfied, and this configuration works. Therefore, the perimeter is 17.Wait, but the options include both 13 and 17 as possible answers. But I just figured out that 13 doesn't work because the sides 3, 3, and 7 don't form a valid triangle. So, why is 13 an option? Maybe I'm missing something.Let me double-check. If the two equal sides are 3, then the third side is 7. But as I saw earlier, 3 + 3 is not greater than 7, so that's not a valid triangle. Therefore, 13 can't be the perimeter because such a triangle doesn't exist. So, only 17 is possible.But then why does option C say "13 or 17"? Maybe there's another way to interpret the problem. Is it possible that the two given sides are not necessarily the equal sides? Wait, the problem says it's an isosceles triangle with two sides measuring 3 and 7. So, does that mean that exactly two sides are 3 and 7, and the third side is something else? Or does it mean that the triangle has sides of 3 and 7, and it's isosceles, so the third side must be equal to one of them?I think it's the latter. Since it's an isosceles triangle, it must have at least two equal sides. So, if two sides are 3 and 7, the third side must be equal to either 3 or 7. Therefore, the possible side lengths are either 3, 3, 7 or 7, 7, 3. As we saw earlier, 3, 3, 7 doesn't work, but 7, 7, 3 does. So, the only valid perimeter is 17.But then why is 13 an option? Maybe I'm misunderstanding the problem. Let me read it again: "Given an isosceles triangle with two sides measuring 3 and 7 respectively, the perimeter of this isosceles triangle is ( )." So, it's saying that two sides are 3 and 7, and it's isosceles. So, the third side must be equal to one of these. So, either the sides are 3, 3, 7 or 7, 7, 3. As we saw, only 7, 7, 3 works, giving a perimeter of 17.But wait, maybe the problem is not specifying that the two given sides are the equal sides. Maybe it's just saying that the triangle has sides of 3 and 7, and it's isosceles, so the third side could be either 3 or 7. But in that case, the perimeter would be either 3 + 3 + 7 = 13 or 7 + 7 + 3 = 17. However, as we saw, 3 + 3 + 7 doesn't form a valid triangle, so only 17 is possible.But the options include 13 or 17, so maybe the problem is considering both possibilities, even though one is invalid. Or perhaps the problem is assuming that the two given sides are the equal sides, which would mean that the third side is different. Wait, no, in an isosceles triangle, the two equal sides are called the legs, and the third side is the base. So, if the two legs are 3 and 7, that would mean the triangle has sides 3, 7, and something else. But that doesn't make sense because in an isosceles triangle, the two legs are equal. So, if two sides are 3 and 7, and it's isosceles, then the third side must be equal to one of them.I think the confusion comes from whether the two given sides are the equal sides or not. If the two given sides are the equal sides, then the third side is different. But if the two given sides are not the equal sides, then the third side must be equal to one of them. Wait, but in an isosceles triangle, there are exactly two equal sides, so if two sides are given as 3 and 7, they can't both be equal unless 3 = 7, which they aren't. Therefore, one of them must be the equal side, and the other is the base.So, either:- The two equal sides are 3, and the base is 7. But as we saw, this doesn't work because 3 + 3 is not greater than 7.- The two equal sides are 7, and the base is 3. This works because 7 + 7 > 3, 7 + 3 > 7, and 7 + 3 > 7.Therefore, only the second case is valid, giving a perimeter of 17.But then why is 13 an option? Maybe the problem is considering that the two given sides could be the base and one of the equal sides, but in that case, the third side would have to be equal to the base or the other equal side. Wait, no, if the base is 3, then the other two sides are both 7, giving a perimeter of 17. If the base is 7, then the other two sides are both 3, but that doesn't form a valid triangle, so it's not possible.Therefore, the only valid perimeter is 17, which is option B.Wait, but the original problem didn't specify whether the two given sides are the equal sides or not. It just says "two sides measuring 3 and 7 respectively." So, maybe it's possible that the two given sides are the equal sides, but that would mean 3 = 7, which is not possible. Therefore, the two given sides cannot both be equal sides. Therefore, one of them must be the base, and the other must be one of the equal sides. So, if the base is 3, then the other two sides are both 7, giving a perimeter of 17. If the base is 7, then the other two sides are both 3, but that doesn't work because 3 + 3 is not greater than 7. Therefore, only 17 is possible.So, the answer should be B: 17.But wait, the options include C: 13 or 17. Maybe the problem is considering that the two given sides could be any two sides, not necessarily the equal ones. So, if the triangle has sides 3, 7, and something else, and it's isosceles, then the third side could be either 3 or 7. But as we saw, if the third side is 3, then the sides are 3, 3, 7, which doesn't form a valid triangle. If the third side is 7, then the sides are 7, 7, 3, which does form a valid triangle. Therefore, only 17 is possible.So, why is 13 an option? Maybe the problem is assuming that the two given sides are the equal sides, but that would mean 3 = 7, which is impossible. Therefore, the only valid perimeter is 17.Wait, perhaps the problem is not specifying that the two given sides are the equal sides, but just that the triangle has two sides of 3 and 7, and it's isosceles. So, the third side could be either 3 or 7, but only 7 works. Therefore, the perimeter is 17.So, I think the correct answer is B: 17.But I'm still confused why 13 is an option. Maybe the problem is considering that the two given sides are not necessarily the equal sides, but just two sides, and the third side could be equal to one of them, even if it doesn't form a valid triangle. But that doesn't make sense because in geometry, a triangle must satisfy the triangle inequality theorem.Alternatively, maybe the problem is considering that the two given sides are the equal sides, but that would require 3 = 7, which is impossible. Therefore, the only valid configuration is when the two equal sides are 7, and the base is 3, giving a perimeter of 17.So, I think the answer is B: 17.

answer：Okay, so Jesse has 21 bananas, and he wants to share them equally among his 3 friends. I need to figure out how many bananas each friend gets. Alright, let's break this down step by step.First, I know that sharing equally means dividing the total number of bananas by the number of friends. So, I have 21 bananas and 3 friends. That means I need to divide 21 by 3. Hmm, division can sometimes be tricky, but I think I can handle this.Let me write it out: 21 ÷ 3. I remember that division is the opposite of multiplication. So, if I think about what number multiplied by 3 gives me 21, that should be the answer. Let me see... 3 times 7 is 21. Yeah, that's right. So, 21 divided by 3 is 7.Wait, just to make sure I'm not making a mistake, let me visualize it. If Jesse has 21 bananas and he gives them out one by one to each friend, how many would each get? Let's count: Friend 1 gets 1, Friend 2 gets 2, Friend 3 gets 3, then back to Friend 1 for 4, Friend 2 for 5, Friend 3 for 6, and so on until all 21 are distributed. If I keep track, each friend should end up with 7 bananas. Yeah, that matches what I got earlier.Another way to think about it is by using multiplication tables. I know that 3 times 7 is 21, so dividing 21 by 3 should logically give me 7. That seems consistent.Is there another method to confirm this? Maybe using repeated subtraction. If I start with 21 and subtract 3 repeatedly until I reach zero, the number of times I subtract 3 will be the answer. Let's try:21 - 3 = 18 (1 time)18 - 3 = 15 (2 times)15 - 3 = 12 (3 times)12 - 3 = 9 (4 times)9 - 3 = 6 (5 times)6 - 3 = 3 (6 times)3 - 3 = 0 (7 times)So, I subtracted 3 seven times to get from 21 to 0. That means 21 divided by 3 is indeed 7.Just to ensure I'm not overlooking anything, let's consider if there's any remainder. If 21 divided by 3 leaves a remainder, that would mean the bananas can't be perfectly shared. But since 3 times 7 is exactly 21, there's no remainder. Each friend gets a whole number of bananas, which is 7.I could also use the concept of grouping. If I have 21 bananas and I want to make 3 equal groups, how many bananas would be in each group? By grouping them, I can see that each group would have 7 bananas. This is another way to confirm that the division is correct.Thinking about real-life scenarios, if I have 21 candies and I want to distribute them equally among 3 children, each child would get 7 candies. It's the same principle here with bananas. This practical example helps reinforce that the answer is correct.I might also consider using visual aids, like drawing 21 bananas and then circling them into 3 groups. Counting the bananas in each group would show that each has 7. This hands-on method can be especially helpful for visual learners to understand the concept of division.Additionally, using number lines can be a useful tool. Starting at 0, I can make jumps of 3 until I reach 21. Each jump represents one banana given to a friend. Counting the number of jumps would tell me how many bananas each friend gets. So, starting at 0, jumping to 3 (1st banana), then to 6 (2nd), 9 (3rd), 12 (4th), 15 (5th), 18 (6th), and finally to 21 (7th). That's 7 jumps, meaning each friend gets 7 bananas.It's also worth noting that division is essentially splitting a number into equal parts. In this case, splitting 21 into 3 equal parts gives us 7 each. This understanding is fundamental in mathematics and is used in various real-world applications, such as dividing resources, sharing costs, or even cooking and baking where ingredients need to be portioned out.To further solidify my understanding, I can think about the inverse operation. If each friend gets 7 bananas, then multiplying 7 by 3 should give me back the total number of bananas Jesse had. 7 times 3 is indeed 21, which confirms that the division was performed correctly.In summary, through multiple methods—direct division, multiplication tables, repeated subtraction, grouping, real-life examples, visual aids, number lines, and inverse operations—I have consistently arrived at the conclusion that each of Jesse's 3 friends would receive 7 bananas. This thorough approach ensures that my answer is accurate and reliable.

answer：Okay, so I'm trying to solve this problem about a fruit wholesaler who sells apples. There are three parts to the problem, and I need to figure out each one step by step. Let me start by understanding what each part is asking.First, part (1) asks for the functional relationship between the average daily sales ( y ) (in boxes) and the selling price ( x ) (in yuan per box). I know that at a price of 50 yuan, 90 boxes are sold on average. Also, for every 1 yuan increase in price, the average daily sales decrease by 3 boxes. So, this seems like a linear relationship because the change in sales is constant for each unit change in price.Let me think about how to model this. If the price is ( x ) yuan, and the base price is 50 yuan, then the increase in price is ( x - 50 ) yuan. For each yuan increase, sales decrease by 3 boxes, so the total decrease in sales would be ( 3(x - 50) ) boxes. Therefore, the average daily sales ( y ) can be expressed as:[y = 90 - 3(x - 50)]Simplifying this equation:[y = 90 - 3x + 150][y = -3x + 240]So, that should be the functional relationship for part (1). Let me double-check. If ( x = 50 ), then ( y = -3(50) + 240 = -150 + 240 = 90 ), which matches the given information. If the price increases by 1 yuan to 51, then ( y = -3(51) + 240 = -153 + 240 = 87 ), which is a decrease of 3 boxes. That seems correct.Moving on to part (2), which asks for the functional relationship between the average daily sales profit ( w ) (in yuan) and the selling price ( x ). I know that profit is calculated as (selling price - cost price) multiplied by the number of boxes sold. The cost price per box is 40 yuan, so the profit per box is ( x - 40 ) yuan. The total profit ( w ) would then be:[w = (x - 40) times y]From part (1), we have ( y = -3x + 240 ). Substituting this into the profit equation:[w = (x - 40)(-3x + 240)]Let me expand this:[w = x(-3x + 240) - 40(-3x + 240)][w = -3x^2 + 240x + 120x - 9600][w = -3x^2 + 360x - 9600]So, the profit function is a quadratic equation. Let me verify this. If ( x = 50 ), then:[w = -3(50)^2 + 360(50) - 9600][w = -3(2500) + 18000 - 9600][w = -7500 + 18000 - 9600][w = 900]Which makes sense because at 50 yuan, selling 90 boxes, the profit is ( (50 - 40) times 90 = 10 times 90 = 900 ) yuan. So, that checks out.Now, part (3) asks for the selling price per box that will maximize the profit, given that the price cannot exceed 55 yuan. Since the profit function is a quadratic equation, it has a maximum or minimum value at its vertex. The general form of a quadratic equation is ( ax^2 + bx + c ), and the vertex occurs at ( x = -frac{b}{2a} ).In our profit function ( w = -3x^2 + 360x - 9600 ), ( a = -3 ) and ( b = 360 ). Plugging these into the vertex formula:[x = -frac{360}{2(-3)} = -frac{360}{-6} = 60]So, the vertex is at ( x = 60 ). Since the coefficient of ( x^2 ) is negative (( a = -3 )), the parabola opens downward, meaning the vertex is a maximum point. Therefore, the maximum profit occurs at ( x = 60 ) yuan.However, the problem states that the selling price must not exceed 55 yuan. This means we cannot set the price at 60 yuan; the maximum allowed is 55 yuan. Therefore, we need to check if the maximum profit within the allowed range (up to 55 yuan) is at 55 yuan or if it's somewhere else.Since the vertex is at 60 yuan, which is outside the allowed range, the maximum profit within the range ( x leq 55 ) will occur at the highest allowed price, which is 55 yuan. Let me confirm this by calculating the profit at 55 yuan and comparing it to the profit at a slightly lower price, say 54 yuan.Calculating profit at 55 yuan:[w = -3(55)^2 + 360(55) - 9600][w = -3(3025) + 19800 - 9600][w = -9075 + 19800 - 9600][w = 1125]Calculating profit at 54 yuan:[w = -3(54)^2 + 360(54) - 9600][w = -3(2916) + 19440 - 9600][w = -8748 + 19440 - 9600][w = 1092]So, the profit at 55 yuan is 1125 yuan, and at 54 yuan, it's 1092 yuan. This shows that as we increase the price towards 55 yuan, the profit increases. Therefore, the maximum profit within the allowed range is indeed at 55 yuan.Just to be thorough, let me check the profit at 50 yuan again:[w = 900 text{ yuan}]And at 55 yuan, it's 1125 yuan, which is higher. So, increasing the price from 50 to 55 yuan increases the profit, and since we can't go beyond 55 yuan, 55 yuan is the optimal price for maximum profit.I think that covers all parts of the problem. I've derived the sales function, the profit function, and determined the optimal price within the given constraint.

answer：First, I need to solve the equation (frac{2}{3}m = frac{5}{6}n) to find the relationship between (m) and (n).I'll start by isolating (frac{m}{n}). To do this, I can divide both sides of the equation by (frac{2}{3}m):[frac{frac{5}{6}n}{frac{2}{3}m} = frac{5}{6} div frac{2}{3} = frac{5}{6} times frac{3}{2} = frac{15}{12} = frac{5}{4}]So, (frac{m}{n} = frac{5}{4}).Next, I need to find (frac{m - n}{n}). I can rewrite this expression as:[frac{m - n}{n} = frac{m}{n} - frac{n}{n} = frac{m}{n} - 1]Substituting the value of (frac{m}{n}) that I found earlier:[frac{m}{n} - 1 = frac{5}{4} - 1 = frac{5}{4} - frac{4}{4} = frac{1}{4}]Therefore, the value of (frac{m - n}{n}) is (frac{1}{4}).

answer：Alright, so I've got this problem about a right-angled triangle, and I need to figure out by how much I should shorten both legs equally to achieve two different outcomes: one where the area is halved, and another where the legs become the sides of a rectangle with the same perimeter as the original triangle. Hmm, okay, let's break this down step by step.First, let's visualize the original right-angled triangle. It has two legs, let's call them 'a' and 'b', and a hypotenuse 'c'. The area of this triangle is (1/2)*a*b, and the perimeter is a + b + c. Now, I need to shorten both legs by the same amount, let's call this amount 'x', so the new legs become (a - x) and (b - x). Starting with part (a), where the goal is to have a right-angled triangle with half the area. So, the new area should be (1/2)*(a - x)*(b - x) = (1/2)*[(1/2)*a*b] = (1/4)*a*b. Wait, no, actually, the original area is (1/2)*a*b, so half of that would be (1/4)*a*b. So, setting up the equation:(1/2)*(a - x)*(b - x) = (1/4)*a*bMultiplying both sides by 2 to eliminate the fraction:(a - x)*(b - x) = (1/2)*a*bExpanding the left side:a*b - a*x - b*x + x^2 = (1/2)*a*bSubtracting (1/2)*a*b from both sides:a*b - a*x - b*x + x^2 - (1/2)*a*b = 0Simplifying:(1/2)*a*b - a*x - b*x + x^2 = 0Hmm, this is a quadratic equation in terms of x. Let's rearrange it:x^2 - (a + b)*x + (1/2)*a*b = 0To solve for x, I can use the quadratic formula:x = [(a + b) ± sqrt((a + b)^2 - 4*(1/2)*a*b)] / 2Simplifying inside the square root:sqrt(a^2 + 2*a*b + b^2 - 2*a*b) = sqrt(a^2 + b^2)So, x = [ (a + b) ± sqrt(a^2 + b^2) ] / 2Now, sqrt(a^2 + b^2) is the hypotenuse 'c', so:x = [ (a + b) ± c ] / 2Since we're dealing with lengths, x must be positive and less than both a and b. So, we take the smaller value:x = [ (a + b) - c ] / 2That makes sense because if we subtract the hypotenuse from the sum of the legs, we get twice the inradius of the triangle, which is the radius of the circle inscribed within the triangle. So, x is equal to the inradius.Moving on to part (b), where we need to shorten the legs to form a rectangle with the same perimeter as the original triangle. The perimeter of the original triangle is a + b + c. The perimeter of the rectangle will be 2*(length + width). If we shorten each leg by 'y', the new legs become (a - y) and (b - y), which will be the length and width of the rectangle.So, setting the perimeters equal:a + b + c = 2*(a - y) + 2*(b - y)Simplifying the right side:2*a - 2*y + 2*b - 2*y = 2*a + 2*b - 4*ySo, the equation becomes:a + b + c = 2*a + 2*b - 4*ySubtracting a + b from both sides:c = a + b - 4*ySolving for y:4*y = a + b - cy = (a + b - c) / 4But from part (a), we know that (a + b - c)/2 is the inradius 'r'. So, y = r / 2.Therefore, to form a rectangle with the same perimeter, we need to shorten each leg by half the inradius.Wait, let me double-check that. If y = (a + b - c)/4, and from part (a), x = (a + b - c)/2, then indeed y is half of x. That seems consistent.So, summarizing:For part (a), the legs should be shortened by the inradius 'r', which is (a + b - c)/2.For part (b), the legs should be shortened by half the inradius, which is (a + b - c)/4.I think that makes sense. The inradius relates to the area and the perimeter, so it's logical that it plays a role in both parts of the problem. I should probably verify with some example values to ensure this works.Let's take a simple right-angled triangle, say with legs 3 and 4, so the hypotenuse is 5.For part (a):x = (3 + 4 - 5)/2 = (2)/2 = 1So, shortening each leg by 1 gives new legs of 2 and 3. The area of the original triangle is (1/2)*3*4 = 6, and the area of the new triangle is (1/2)*2*3 = 3, which is indeed half. Good.For part (b):y = (3 + 4 - 5)/4 = 2/4 = 0.5So, shortening each leg by 0.5 gives new legs of 2.5 and 3.5. The original perimeter is 3 + 4 + 5 = 12. The perimeter of the rectangle is 2*(2.5 + 3.5) = 2*6 = 12, which matches. Perfect.Okay, so the formulas check out with an example. I think I've got it right.

answer：First, I need to understand the weightings of the two tests. The first test is worth 40% of the total grade, and the second test is worth 60%.Brittany scored 78 on her first test. To find the weighted contribution of this score, I multiply 78 by 0.40, which gives me 31.2.Let’s denote Brittany’s score on the second test as ( G ). The weighted contribution of the second test is ( 0.60 times G ).The overall weighted average after both tests is 81. This means that the sum of the weighted contributions of both tests equals 81:[31.2 + 0.60G = 81]To find ( G ), I subtract 31.2 from both sides of the equation:[0.60G = 81 - 31.2][0.60G = 49.8]Next, I divide both sides by 0.60 to solve for ( G ):[G = frac{49.8}{0.60}][G = 83]Therefore, Brittany scored 83 on her second test.