answer：First, I need to determine the total amount of butter the woman needs for her dessert recipe. She has two purchasing options:1. **Single Package:** - Cost: 7 - Quantity: 16 ounces2. **Combination with Coupon:** - 8-ounce package: 4 - Two 4-ounce packages at a 50% discount: - Original price per 4-ounce package: 2 - Discounted price per 4-ounce package: 1 - Total for two 4-ounce packages: 2 - **Total Cost:** 4 + 2 = 6Since the lowest price she can pay is 6, she chooses the second option. Now, calculating the total amount of butter she gets with the second option:- 8 ounces (from the 8-ounce package)- 8 ounces (from the two 4-ounce packages)**Total Butter:** 8 ounces + 8 ounces = 16 ouncesTherefore, the woman needs **16 ounces** of butter for her dessert recipe.

answer：Okay, so I have this problem where there's a regular polygon with m sides, and it's exactly enclosed by m+1 regular polygons each with n sides. They all fit perfectly without any overlaps or gaps. I need to find the value of n when m is 6. The options are A) 3, B) 5, C) 6, D) 7, and E) 12.First, let me try to visualize this. There's a regular hexagon, which has 6 sides, and around it, there are 7 regular polygons each with n sides. These 7 polygons fit perfectly around the hexagon, so their arrangement must complement the angles of the hexagon.I remember that the sum of the interior angles of a regular polygon with m sides is given by (m-2)*180 degrees. So, for a hexagon, that would be (6-2)*180 = 4*180 = 720 degrees. Therefore, each interior angle of the hexagon is 720/6 = 120 degrees.Now, since the hexagon is enclosed by these 7 n-sided polygons, the way they fit around it must relate to the angles at each vertex. I think the key here is to consider the angles at which these polygons meet around the hexagon.Each vertex of the hexagon is 120 degrees. Now, when the n-sided polygons are placed around the hexagon, they must fit into the space around each vertex. Since there are 7 polygons, I might think that each contributes some angle at each vertex of the hexagon.Wait, but actually, each polygon is placed around the hexagon, so perhaps each polygon is adjacent to one side of the hexagon. But since there are 7 polygons and only 6 sides, maybe one of them is placed at a corner? Hmm, that might complicate things.Alternatively, perhaps each polygon is placed such that it shares a side with the hexagon, but since there are 7 polygons and only 6 sides, one polygon must be placed differently. Maybe it's placed at a corner where two sides meet.But I'm not sure. Maybe I should think about the angles more carefully. Each polygon has an interior angle, and when they fit around the hexagon, their angles must complement the hexagon's angles.Wait, the hexagon has an interior angle of 120 degrees, so the angle between two adjacent polygons around the hexagon would be 180 - 120 = 60 degrees. Is that right?No, actually, when you place polygons around another polygon, the angles at the vertices where they meet must add up to 360 degrees around a point. But in this case, the hexagon is in the center, so the angles of the surrounding polygons must fit around the hexagon's angles.Wait, maybe I should think about the exterior angles. The exterior angle of a regular polygon is 360/m degrees. For the hexagon, that's 360/6 = 60 degrees. So each exterior angle is 60 degrees.Now, if the surrounding polygons are fitting around the hexagon, their angles must match up with the hexagon's exterior angles. Since there are 7 polygons, each contributing an angle at each vertex of the hexagon, maybe the sum of their angles at each vertex is 60 degrees.But wait, each vertex of the hexagon is surrounded by two sides of the hexagon and one side of a surrounding polygon. So the angle contributed by the surrounding polygon at each vertex must be 60 degrees.Wait, no, the angle at the vertex is 120 degrees for the hexagon, so the surrounding polygons must fit into the remaining space. But since they are regular polygons, their angles must match up.I'm getting a bit confused. Maybe I should think about the angle each surrounding polygon contributes at each vertex of the hexagon.If the hexagon has an interior angle of 120 degrees, then the angle between two adjacent surrounding polygons at that vertex would be 180 - 120 = 60 degrees. So each surrounding polygon contributes an angle of 60 degrees at each vertex.But wait, each surrounding polygon is a regular n-sided polygon, so its interior angle is (n-2)*180/n degrees. But if the angle it contributes at the vertex is 60 degrees, then:(n-2)*180/n = 60Solving for n:(n-2)*180 = 60n180n - 360 = 60n180n - 60n = 360120n = 360n = 3Wait, that would make n=3, which is a triangle. But the answer options include 3 as option A. But I'm not sure if this is correct because there are 7 polygons around the hexagon, and if each contributes a 60-degree angle, then 7*60=420, which is more than 360, which doesn't make sense.Hmm, maybe I made a mistake. Let me think again.Each vertex of the hexagon is 120 degrees. The surrounding polygons must fit around this angle. So the total angle around the vertex is 360 degrees, so the surrounding polygons must contribute 360 - 120 = 240 degrees at each vertex.But since there are 7 polygons, each contributing an angle at each vertex, but actually, each polygon is placed at a side of the hexagon, so each polygon contributes an angle at two vertices of the hexagon.Wait, maybe I'm overcomplicating it. Let's think about the arrangement.If there are 7 n-sided polygons around a hexagon, each polygon shares a side with the hexagon. But since the hexagon has only 6 sides, one polygon must share a vertex instead. So, 6 polygons share a side with the hexagon, and one polygon shares a vertex.But I'm not sure. Maybe it's better to think about the angles.Each polygon has an interior angle of (n-2)*180/n. When they are placed around the hexagon, their angles must fit into the space around the hexagon's vertices.At each vertex of the hexagon, the angle is 120 degrees, so the surrounding polygons must fit into the remaining 240 degrees (since 360 - 120 = 240). But since there are 7 polygons, each contributing an angle at each vertex, but actually, each polygon is placed at a side, so each polygon contributes an angle at two vertices.Wait, maybe each polygon is placed such that it shares a side with the hexagon, and thus contributes an angle at each end of that side. So each polygon contributes two angles at two vertices of the hexagon.But since there are 7 polygons, and 6 sides, one polygon must be placed differently. Maybe it's placed at a corner where two sides meet, contributing an angle at that corner.This is getting complicated. Maybe I should look for another approach.I remember that when regular polygons fit around a point, their angles must add up to 360 degrees. So, if the hexagon is in the center, and the surrounding polygons are placed around it, their angles at the center must add up to 360 degrees.But the hexagon itself has angles, so maybe the surrounding polygons' angles plus the hexagon's angles at each vertex must add up to 360 degrees.Wait, no. The hexagon is in the center, so the surrounding polygons are placed around it, each sharing a side with the hexagon. At each vertex where the hexagon and two surrounding polygons meet, the angles must add up to 360 degrees.So, at each vertex of the hexagon, the angle is 120 degrees, and the two surrounding polygons each contribute an angle. So, 120 + 2*(angle of surrounding polygon) = 360.Therefore, 2*(angle of surrounding polygon) = 360 - 120 = 240.So, angle of surrounding polygon = 120 degrees.But the angle of a regular n-sided polygon is (n-2)*180/n.So, (n-2)*180/n = 120Solving for n:(n-2)*180 = 120n180n - 360 = 120n180n - 120n = 36060n = 360n = 6So, n=6, which is option C.Wait, that makes sense. Each surrounding polygon is a hexagon, and their angles at each vertex are 120 degrees, which complements the hexagon's 120-degree angle to make 360 degrees around each vertex.But wait, if each surrounding polygon is a hexagon, then how do they fit around the central hexagon? If each surrounding polygon is a hexagon, they would each share a side with the central hexagon, and their other sides would connect to other surrounding polygons.But since there are 7 surrounding polygons, and each is a hexagon, they would need to fit around the central hexagon. But a regular hexagon has 6 sides, so arranging 7 hexagons around it might not fit perfectly.Wait, maybe I made a mistake. If each surrounding polygon is a hexagon, then each contributes a 120-degree angle at each vertex, and since there are 7 polygons, the total angle around the central hexagon would be 7*120 = 840 degrees, which is way more than 360 degrees.That can't be right. So, my previous approach must be wrong.Let me think again. Maybe the angle contributed by each surrounding polygon is not 120 degrees, but something else.If the central polygon is a hexagon with interior angle 120 degrees, then the angle between two surrounding polygons at that vertex would be 180 - 120 = 60 degrees.So, each surrounding polygon contributes an angle of 60 degrees at each vertex.But if each surrounding polygon is a regular n-sided polygon, then its interior angle is (n-2)*180/n.But if the angle it contributes at the vertex is 60 degrees, then:(n-2)*180/n = 60Solving for n:(n-2)*180 = 60n180n - 360 = 60n180n - 60n = 360120n = 360n = 3So, n=3, which is a triangle. But then, if each surrounding polygon is a triangle, how do they fit around the hexagon?Each triangle would share a side with the hexagon, and their other sides would connect to other triangles. But since there are 7 triangles, and each triangle has 3 sides, it's unclear how they would fit around the hexagon without overlapping or leaving gaps.Wait, maybe the angle I'm considering is not the interior angle of the surrounding polygon, but the exterior angle.The exterior angle of a regular polygon is 360/n degrees. So, if the surrounding polygons contribute an exterior angle of 60 degrees at each vertex, then:360/n = 60n = 360/60 = 6So, n=6 again.But earlier, I thought that would lead to too many degrees around the vertex. Wait, maybe I was confusing interior and exterior angles.If the surrounding polygons contribute an exterior angle of 60 degrees at each vertex, then the interior angle would be 180 - 60 = 120 degrees, which is the same as the hexagon's interior angle. That might make sense.So, each surrounding polygon is a hexagon, contributing an exterior angle of 60 degrees at each vertex, which complements the hexagon's interior angle of 120 degrees to make 180 degrees along the side.But wait, around each vertex, the total angle should be 360 degrees. The hexagon contributes 120 degrees, and the two surrounding polygons contribute 60 degrees each, so 120 + 60 + 60 = 240 degrees, which is less than 360 degrees. That doesn't add up.I'm getting confused again. Maybe I need to think about the arrangement differently.Perhaps the surrounding polygons are placed such that each shares a vertex with the hexagon, rather than a side. So, each surrounding polygon is placed at a corner of the hexagon, contributing an angle there.In that case, the angle contributed by each surrounding polygon at the hexagon's vertex would be 120 degrees, as the hexagon's interior angle is 120 degrees, and the surrounding polygon's angle would fit into the remaining space.Wait, but if the surrounding polygon's angle is 120 degrees, then:(n-2)*180/n = 120Solving for n:(n-2)*180 = 120n180n - 360 = 120n60n = 360n=6Again, n=6. But then, how do 7 hexagons fit around a central hexagon? It seems like it would require more space.Wait, maybe the surrounding polygons are placed such that each shares a side with the hexagon, but since there are 7 polygons, one of them must share a vertex instead. So, 6 polygons share a side with the hexagon, and one shares a vertex.In that case, the 6 polygons sharing a side would each contribute an angle at their shared vertex, and the 7th polygon would contribute an angle at the vertex where two sides of the hexagon meet.So, at each vertex where a surrounding polygon shares a side with the hexagon, the angle contributed by the surrounding polygon would be 60 degrees (since the hexagon's exterior angle is 60 degrees). Therefore, the surrounding polygon's interior angle at that vertex would be 180 - 60 = 120 degrees.So, (n-2)*180/n = 120Solving for n:(n-2)*180 = 120n180n - 360 = 120n60n = 360n=6Again, n=6.But then, the 7th polygon, which shares a vertex with the hexagon, would need to contribute an angle at that vertex. The hexagon's interior angle is 120 degrees, so the surrounding polygon would need to contribute an angle such that 120 + angle = 360 degrees.Wait, no, at that vertex, the hexagon's angle is 120 degrees, and the surrounding polygon contributes an angle, so the total should be 360 degrees.Therefore, angle of surrounding polygon = 360 - 120 = 240 degrees.But that's impossible because the interior angle of a regular polygon can't be 240 degrees. The maximum interior angle for a regular polygon is less than 180 degrees.So, this approach must be wrong.Maybe the surrounding polygons are placed such that each shares a side with the hexagon, and their other sides connect to other surrounding polygons. Since there are 7 surrounding polygons, each sharing a side with the hexagon, but the hexagon only has 6 sides, one of the surrounding polygons must share a side with two other surrounding polygons.Wait, that might make sense. So, 6 surrounding polygons share a side with the hexagon, and the 7th polygon shares a side with two of the surrounding polygons, effectively forming a sort of star shape.In that case, the angles at the vertices where the surrounding polygons meet would need to add up appropriately.But I'm not sure. Maybe I should look for a different approach.I remember that in tessellation problems, the angles around a point must add up to 360 degrees. So, if the central polygon is a hexagon with interior angles of 120 degrees, then the surrounding polygons must fit into the space around it such that their angles plus the hexagon's angles add up to 360 degrees at each vertex.But since the hexagon is in the center, the surrounding polygons are placed around it, so at each vertex where the hexagon and two surrounding polygons meet, the angles must add up to 360 degrees.So, the hexagon's interior angle is 120 degrees, and the two surrounding polygons each contribute an angle. Therefore:120 + 2*(angle of surrounding polygon) = 360So, 2*(angle of surrounding polygon) = 240Therefore, angle of surrounding polygon = 120 degrees.So, each surrounding polygon has an interior angle of 120 degrees, which means:(n-2)*180/n = 120Solving for n:(n-2)*180 = 120n180n - 360 = 120n60n = 360n=6So, n=6 again.But then, how do 7 hexagons fit around a central hexagon? It seems like it would require more space, but maybe it's possible in a specific arrangement.Alternatively, maybe the surrounding polygons are placed such that each shares a vertex with the hexagon, rather than a side. In that case, the angle contributed by each surrounding polygon at the hexagon's vertex would be 60 degrees, as the hexagon's exterior angle is 60 degrees.So, if each surrounding polygon contributes an angle of 60 degrees at the hexagon's vertex, then:(n-2)*180/n = 60Solving for n:(n-2)*180 = 60n180n - 360 = 60n120n = 360n=3So, n=3, which is a triangle.But then, how do 7 triangles fit around a hexagon? Each triangle would share a vertex with the hexagon, and their other sides would connect to other triangles. But 7 triangles would require more space than the hexagon's 6 vertices can provide.Wait, maybe the triangles are placed such that each shares a side with the hexagon, but since there are 7 triangles, one of them must share a side with two other triangles, creating a sort of overlap.But I'm not sure if that's possible without gaps or overlaps.I'm going in circles here. Let me try to summarize.If the surrounding polygons are placed such that each shares a side with the hexagon, then each contributes an interior angle of 120 degrees at the hexagon's vertex, leading to n=6.If they are placed such that each shares a vertex with the hexagon, contributing an angle of 60 degrees, leading to n=3.But both solutions seem to have issues with fitting 7 polygons around the hexagon.Wait, maybe the key is that there are 7 polygons, so the arrangement is such that each polygon is placed at a side, but one polygon is placed at a corner, effectively contributing two angles at one vertex.So, 6 polygons share a side with the hexagon, contributing an angle of 120 degrees each, and the 7th polygon shares a vertex, contributing two angles of 60 degrees each.But then, the total angles at the vertex where the 7th polygon is placed would be 120 (hexagon) + 60 + 60 = 240, which is less than 360, so that doesn't work.Alternatively, the 7th polygon might contribute an angle of 120 degrees at that vertex, making the total 120 + 120 = 240, still less than 360.I'm stuck. Maybe I should look for another approach.I remember that in regular tessellations, only triangles, squares, and hexagons can tessellate the plane without gaps or overlaps. Since the surrounding polygons are regular, maybe n has to be one of these numbers.Given the options, n could be 3, 5, 6, 7, or 12. So, 3, 5, 6, 7, 12.If n=3, triangles, as I thought earlier, but fitting 7 triangles around a hexagon seems difficult.If n=5, pentagons, their interior angle is 108 degrees. If each contributes 108 degrees at the vertex, then 120 + 2*108 = 336, which is less than 360, so that doesn't work.If n=6, hexagons, each contributes 120 degrees, so 120 + 2*120 = 360, which works.If n=7, heptagons, their interior angle is (7-2)*180/7 ≈ 128.57 degrees. So, 120 + 2*128.57 ≈ 377.14, which is more than 360, so that doesn't work.If n=12, dodecagons, their interior angle is (12-2)*180/12 = 150 degrees. So, 120 + 2*150 = 420, which is way more than 360.So, the only value that works is n=6, because 120 + 2*120 = 360 degrees.Therefore, the answer is C) 6.

answer：First, I need to determine the total number of highlighters by adding the quantities of each color.I will start by listing the number of highlighters for each color:- Pink: 15- Yellow: 12- Blue: 9- Green: 7- Purple: 6Next, I will add these numbers together step by step:15 (pink) + 12 (yellow) = 2727 + 9 (blue) = 3636 + 7 (green) = 4343 + 6 (purple) = 49After performing the addition, the total number of highlighters is 49.

answer：Okay, so I have this geometry problem here about triangle XYZ. Let me try to visualize it first. The triangle has vertices X, Y, and Z. They tell me that the lengths of sides XY and XZ are 6 inches and 8 inches respectively. Also, there's a median from vertex X to the midpoint of side YZ, and this median is 5 inches long. I need to find the length of side YZ.Hmm, okay. So, let me recall what a median is. A median in a triangle is a line segment joining a vertex to the midpoint of the opposite side. So, in this case, the median XM goes from vertex X to the midpoint M of side YZ. That means YZ is split into two equal parts by M, so YM equals MZ.Now, I remember there's a formula that relates the lengths of the sides of a triangle to the length of a median. I think it's called Apollonius's theorem. Let me try to remember how it goes. I think it states that the sum of the squares of any two sides of a triangle is equal to twice the square of the median to the third side plus twice the square of the half third side. Wait, that sounds a bit complicated. Let me write it down to make it clearer.Apollonius's theorem says:[ XY^2 + XZ^2 = 2XM^2 + 2left(frac{YZ}{2}right)^2 ]Let me check if I got that right. So, the squares of the two sides (XY and XZ) are equal to twice the square of the median (XM) plus twice the square of half the third side (YZ/2). Yeah, that seems right.Let me plug in the values I have into this formula. XY is 6 inches, so XY squared is 36. XZ is 8 inches, so XZ squared is 64. The median XM is 5 inches, so XM squared is 25. And YZ is what I'm trying to find, so let's call that length 'a'. Therefore, half of YZ is a/2, and its square is (a/2)^2.So substituting these into the theorem:[ 36 + 64 = 2(25) + 2left(frac{a}{2}right)^2 ]Let me compute the left side first. 36 plus 64 is 100. On the right side, 2 times 25 is 50. Then, 2 times (a/2)^2 is 2*(a²/4) which simplifies to a²/2. So, putting it all together:[ 100 = 50 + frac{a^2}{2} ]Hmm, okay, so I can subtract 50 from both sides to isolate the term with 'a':[ 100 - 50 = frac{a^2}{2} ][ 50 = frac{a^2}{2} ]To solve for a², I can multiply both sides by 2:[ 100 = a^2 ]So, taking the square root of both sides gives:[ a = sqrt{100} ][ a = 10 ]Wait, so YZ is 10 inches long? Let me double-check my steps to make sure I didn't make a mistake.1. I used Apollonius's theorem correctly: XY² + XZ² = 2XM² + 2(YZ/2)².2. Plugged in the values: 6² + 8² = 2*5² + 2*(a/2)².3. Calculated 36 + 64 = 100 and 2*25 = 50.4. Subtracted 50 from 100 to get 50 = a²/2.5. Multiplied both sides by 2 to get a² = 100, so a = 10.Everything seems to check out. So, I think the length of YZ is indeed 10 inches.Just to visualize it another way, maybe I can think of triangle XYZ with sides XY=6, XZ=8, and median XM=5. If I consider the coordinates, perhaps placing point X at the origin, point Y somewhere on the x-axis, and point Z somewhere in the plane. But that might complicate things more. Alternatively, I can think of the triangle as two smaller triangles created by the median XM. Each of these smaller triangles would have sides of 5, 6, and half of YZ, and 5, 8, and half of YZ. But I think the Apollonius's theorem approach was more straightforward.Another way to approach this might be using vectors or coordinate geometry, but since I already have a solution using Apollonius's theorem, and it makes sense, I think I can be confident that the length of YZ is 10 inches.So, the answer should be 10 inches, which corresponds to option B.**Final Answer**boxed{B}

answer：Okay, so I'm trying to solve this problem where a bike and a truck both took 9 hours to cover 72 miles each. I need to find the difference between their speeds. Hmm, let's see. First, I remember that speed is calculated by dividing the distance traveled by the time taken. So, for both the bike and the truck, I can use the formula:[ text{Speed} = frac{text{Distance}}{text{Time}} ]Alright, so for the bike, the distance is 72 miles and the time is 9 hours. Let me calculate that:[ text{Speed of bike} = frac{72 text{ miles}}{9 text{ hours}} = 8 text{ mph} ]Okay, that seems straightforward. Now, doing the same for the truck:[ text{Speed of truck} = frac{72 text{ miles}}{9 text{ hours}} = 8 text{ mph} ]Wait a minute, both the bike and the truck have the same speed? That's interesting. So, if they both traveled the same distance in the same amount of time, their speeds must be the same. But just to make sure I didn't make a mistake, let me double-check my calculations. For the bike:[ frac{72}{9} = 8 ]And for the truck:[ frac{72}{9} = 8 ]Yep, both are 8 mph. So, the difference between their speeds would be:[ 8 text{ mph} - 8 text{ mph} = 0 text{ mph} ]That means there's no difference in their speeds. They were moving at the same speed. I guess that makes sense because if two objects cover the same distance in the same amount of time, their speeds must be equal. It's kind of logical when you think about it. So, I think I've got it right.

answer：First, I need to understand what the associative property of addition entails. It states that the way numbers are grouped when adding does not affect the sum. In other words, ( (a + b) + c = a + (b + c) ).Next, I'll examine each option provided to determine which one exclusively uses the associative property.**Option A:** ( 34 + 47 = 47 + 35 ) Here, the positions of the numbers have been swapped. This demonstrates the commutative property of addition, which states that the order of the numbers does not affect the sum. Therefore, this option does not solely rely on the associative property.**Option B:** ( 62 + (47 + 68) = (62 + 68) + 47 ) In this case, both the grouping of the numbers and their order have been changed. This involves both the associative and commutative properties of addition. Since it uses more than just the associative property, this option is not the correct answer.**Option C:** ( 57 + 24 + 76 = 57 + (24 + 76) ) Here, the order of the numbers remains the same, but the grouping has been altered. This is a direct application of the associative property of addition, as it only changes the way the numbers are grouped without altering their order.**Option D:** There is no information provided for this option, so it cannot be evaluated.Based on this analysis, the only option that exclusively uses the associative property of addition is **Option C**.