answer：Okay, so I have this problem about chairs and a table in a clubroom, and I need to figure out how many chairs there are. Let me try to break it down step by step.First, the problem says there are only chairs and a table in the clubroom. Each chair has four legs, and the table has three legs. Then, some scouts come in, each sits on their own chair, two chairs remain unoccupied, and the total number of legs in the room is 101. I need to find out how many chairs were in the clubroom.Alright, let's see. So, there are chairs and a table. Each chair has four legs, so if there are, say, C chairs, that would be 4C legs from the chairs. The table has three legs, so that's 3 legs. Now, scouts come in and sit on their own chairs. Each scout has two legs, right? So, if there are S scouts, that would be 2S legs from the scouts.But wait, the problem says that two chairs remained unoccupied. So, if there are C chairs in total, and two are unoccupied, that means C - 2 chairs are occupied by scouts. So, the number of scouts should be equal to the number of occupied chairs, which is C - 2.So, putting it all together, the total number of legs in the room would be the legs from the chairs plus the legs from the table plus the legs from the scouts. That is:Total legs = (Legs from chairs) + (Legs from table) + (Legs from scouts)Which translates to:101 = 4C + 3 + 2SBut since the number of scouts S is equal to the number of occupied chairs, which is C - 2, we can substitute S with C - 2:101 = 4C + 3 + 2(C - 2)Let me simplify this equation step by step.First, expand the term 2(C - 2):2(C - 2) = 2C - 4So, substituting back into the equation:101 = 4C + 3 + 2C - 4Now, combine like terms:4C + 2C = 6C3 - 4 = -1So, the equation becomes:101 = 6C - 1Now, add 1 to both sides to isolate the term with C:101 + 1 = 6C102 = 6CNow, divide both sides by 6 to solve for C:C = 102 / 6C = 17So, there are 17 chairs in the clubroom.Wait, let me double-check to make sure I didn't make any mistakes.If there are 17 chairs, then the legs from the chairs are 17 * 4 = 68 legs.The table has 3 legs, so total legs from furniture are 68 + 3 = 71 legs.Now, the number of scouts is 17 - 2 = 15 scouts.Each scout has 2 legs, so 15 * 2 = 30 legs.Adding that to the furniture legs: 71 + 30 = 101 legs.Yes, that matches the total given in the problem. So, my answer seems correct.I think I've got it. There are 17 chairs in the clubroom.

answer：Okay, so I have this equation: (x - 6)(2x + 8) + x² - 15x + 56 = 0. I need to find the solutions p and q, and then calculate (p + 3)(q + 3). Hmm, that sounds a bit complicated, but let me try to break it down step by step.First, I think I should simplify the equation. It has two parts: (x - 6)(2x + 8) and then x² - 15x + 56. Maybe I can expand the first part and then combine like terms with the second part. Let me try that.Expanding (x - 6)(2x + 8): I'll use the distributive property, which is also known as the FOIL method for binomials. So, First, Outer, Inner, Last.First: x * 2x = 2x²Outer: x * 8 = 8xInner: -6 * 2x = -12xLast: -6 * 8 = -48Now, combining those terms: 2x² + 8x - 12x - 48. Let me combine the like terms 8x and -12x. That gives me 2x² - 4x - 48.Okay, so the first part simplifies to 2x² - 4x - 48. Now, let's add the second part of the equation, which is x² - 15x + 56. So, combining both parts:2x² - 4x - 48 + x² - 15x + 56.Let me combine like terms again. For the x² terms: 2x² + x² = 3x².For the x terms: -4x - 15x = -19x.For the constant terms: -48 + 56 = 8.So, putting it all together, the equation simplifies to 3x² - 19x + 8 = 0.Alright, now I have a quadratic equation: 3x² - 19x + 8 = 0. I need to find the solutions p and q. Since it's a quadratic equation, I can use the quadratic formula, which is x = [-b ± √(b² - 4ac)] / (2a), where a = 3, b = -19, and c = 8.Let me plug in those values:x = [-(-19) ± √((-19)² - 4*3*8)] / (2*3)x = [19 ± √(361 - 96)] / 6x = [19 ± √265] / 6So, the solutions are p = [19 + √265]/6 and q = [19 - √265]/6.Now, I need to find (p + 3)(q + 3). Let me write that out:(p + 3)(q + 3) = ([19 + √265]/6 + 3) * ([19 - √265]/6 + 3)Hmm, that looks a bit messy, but maybe I can simplify each part first before multiplying.Let's handle the first term: [19 + √265]/6 + 3. To add these, I need a common denominator. 3 can be written as 18/6. So:[19 + √265]/6 + 18/6 = [19 + √265 + 18]/6 = [37 + √265]/6Similarly, the second term: [19 - √265]/6 + 3. Again, 3 is 18/6:[19 - √265]/6 + 18/6 = [19 - √265 + 18]/6 = [37 - √265]/6So now, (p + 3)(q + 3) becomes:([37 + √265]/6) * ([37 - √265]/6)Oh, this looks like a difference of squares. Remember, (a + b)(a - b) = a² - b². So, applying that here:([37]² - [√265]²) / (6 * 6) = (1369 - 265) / 36Calculating the numerator: 1369 - 265 = 1104So, 1104 / 36. Let me simplify that fraction. Both numerator and denominator are divisible by 12.1104 ÷ 12 = 9236 ÷ 12 = 3So, 92/3. That's the simplified form.Wait, let me double-check my calculations to make sure I didn't make any mistakes.First, expanding (x - 6)(2x + 8): 2x² + 8x - 12x - 48. Combine like terms: 2x² - 4x - 48. Then adding x² -15x +56: 3x² -19x +8. That seems correct.Quadratic formula: x = [19 ± √(361 - 96)] /6. 361 - 96 is 265, so √265. That looks right.Then, adding 3 to each solution: [19 + √265]/6 + 3 = [19 + √265 + 18]/6 = [37 + √265]/6. Similarly for the other term. Then multiplying them: (37 + √265)(37 - √265) = 37² - (√265)² = 1369 - 265 = 1104. Divided by 36, which simplifies to 92/3. Yeah, that all checks out.I think I did everything correctly. So, the final answer should be 92/3.**Final Answer**boxed{dfrac{92}{3}}

answer：Okay, so I have this problem here about Lexie's mom trying to place her watercolor paintings equally in some rooms in the house. Lexie has 32 watercolor paintings, and they want to put 8 paintings in each room. The question is asking how many rooms there are in the house.Alright, let's break this down. First, I need to understand what's being asked. We have a total number of paintings, which is 32, and we want to distribute them equally into rooms, with each room holding 8 paintings. So, the main goal is to find out how many rooms are needed to fit all 32 paintings, given that each room can hold 8 paintings.Hmm, so this seems like a division problem. Division is about splitting a number into equal parts. In this case, we're splitting 32 paintings into groups of 8. So, the number of rooms would be the result of dividing 32 by 8.Let me write that down to visualize it better: 32 divided by 8. In mathematical terms, that's 32 ÷ 8. Now, I need to calculate what 32 divided by 8 equals.I know that 8 times 4 is 32 because 8 + 8 + 8 + 8 equals 32. So, if I have 8 paintings in each room, and I have 4 rooms, that would account for all 32 paintings. Therefore, 32 divided by 8 equals 4.Wait, let me double-check that to make sure I didn't make a mistake. If I multiply 8 by 4, do I get back to 32? Yes, because 8 times 4 is indeed 32. That confirms that dividing 32 by 8 gives us 4 rooms.Is there another way to think about this? Maybe using multiplication tables or visualizing it with objects. If I imagine 32 paintings laid out and I group them into sets of 8, how many sets would I have? Let's see: the first room gets 8 paintings, the second room another 8, the third room another 8, and the fourth room the last 8. That's 4 rooms in total.Alternatively, I could think about it as repeated subtraction. Starting with 32 paintings, if I subtract 8 paintings for each room, how many times can I do that before I reach zero? Let's try:32 minus 8 is 24 (that's one room),24 minus 8 is 16 (that's two rooms),16 minus 8 is 8 (that's three rooms),8 minus 8 is 0 (that's four rooms).So, I subtracted 8 four times to get to zero, which means there are 4 rooms.Another way to approach this is by using the concept of factors. Since 8 is a factor of 32, dividing 32 by 8 will give us a whole number, which is the number of rooms. The factors of 32 are numbers that can divide 32 without leaving a remainder. The factors of 32 are 1, 2, 4, 8, 16, and 32. Since we're grouping into sets of 8, the corresponding factor pair is 4 and 8, meaning 4 rooms.I could also use the distributive property or other arithmetic properties to verify this, but in this case, simple division seems sufficient and straightforward.Just to ensure I'm not missing anything, let's consider if there's any possibility of having a remainder. If the total number of paintings wasn't perfectly divisible by the number of paintings per room, we would have some leftover paintings, which would imply that we might need an additional room. However, since 32 divided by 8 is exactly 4 with no remainder, we don't have any leftover paintings, so 4 rooms are sufficient.In summary, by dividing the total number of paintings (32) by the number of paintings per room (8), we determine that there are 4 rooms needed to equally distribute all the paintings.

answer：First, I need to determine the price of the first variety of tea per kilogram. The problem states that three varieties of tea are mixed in the ratio 1:1:2, and the resulting mixture is priced at Rs. 153 per kilogram. The third variety is priced at Rs. 175.5 per kilogram, and the second variety is priced at Rs. 135 per kilogram.I'll start by calculating the total weight of the mixture based on the given ratio. Since the ratio is 1:1:2, the total weight is 1 + 1 + 2 = 4 kilograms.Next, I'll calculate the total cost of the mixture by multiplying the average price per kilogram by the total weight: 153 * 4 = 612 Rs.Now, I'll express the total cost of the mixture as the sum of the costs of each variety:- Cost of the first variety: 1 * P (where P is the price per kilogram)- Cost of the second variety: 1 * 135 Rs- Cost of the third variety: 2 * 175.5 RsSetting up the equation:P + 135 + 351 = 612Combining the known values:P + 486 = 612Finally, solving for P:P = 612 - 486P = 126Therefore, the price of the first variety of tea per kilogram is Rs. 126.

answer：Okay, so I have this function f(x) defined piecewise. It's x minus 3 when x is less than 5, and x squared when x is greater than or equal to 5. The problem is asking me to compute the sum of f inverse from -8 to 25. That is, f inverse of -8 plus f inverse of -7 and so on up to f inverse of 25. First, I need to understand what f inverse is. Since f is piecewise, its inverse will also be piecewise. So, I should probably find the inverse functions for each piece separately.Starting with the first piece: f(x) = x - 3 when x < 5. To find the inverse, I can set y = x - 3 and solve for x. That gives x = y + 3. So, the inverse function for this part is g inverse(y) = y + 3. But I need to consider the domain of this inverse. Since the original function f(x) = x - 3 is defined for x < 5, the range of this part is y < 5 - 3, which is y < 2. So, g inverse is valid for y < 2.Now, the second piece: f(x) = x squared when x >= 5. To find the inverse, set y = x squared and solve for x. That gives x = sqrt(y). But since x is greater than or equal to 5, we only take the positive square root. So, the inverse function for this part is h inverse(y) = sqrt(y). The domain for this inverse is the range of the original function, which is y >= 5 squared, so y >= 25.Wait, hold on. The original function f(x) = x squared when x >= 5, so the range is y >= 25. Therefore, h inverse(y) is only valid for y >= 25.So, putting it together, the inverse function f inverse(y) is:- y + 3 when y < 2- sqrt(y) when y >= 25But wait, what about the values of y between 2 and 25? The original function f(x) isn't defined for those y values because the first piece only goes up to y < 2 and the second piece starts at y >= 25. So, f inverse isn't defined for y between 2 and 25. But the problem is asking for the sum from y = -8 to y = 25. Hmm, does that mean we only include the terms where f inverse is defined?Let me check the original function f(x). For x < 5, f(x) = x - 3, which gives y < 2. For x >= 5, f(x) = x squared, which gives y >= 25. So, there's a gap in the range of f(x) between 2 and 25. That means f inverse isn't defined for y between 2 and 25. So, in the sum from y = -8 to y = 25, we only include the terms where y < 2 and y >= 25.So, the sum is f inverse(-8) + f inverse(-7) + ... + f inverse(1) + f inverse(25). Because for y = 2 to y = 24, f inverse isn't defined, so those terms are excluded.Now, let's compute each part separately.First, for y from -8 to 1 (since y < 2), f inverse(y) = y + 3. So, f inverse(-8) = -8 + 3 = -5, f inverse(-7) = -7 + 3 = -4, and so on, up to f inverse(1) = 1 + 3 = 4.So, the sum from y = -8 to y = 1 is the sum of the integers from -5 to 4. Let me list them out: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4. That's 10 terms.Wait, let me count: from -5 to 4 inclusive, that's 10 numbers. The sum of an arithmetic series can be calculated as (number of terms)/2 times (first term + last term). So, that's 10/2 * (-5 + 4) = 5 * (-1) = -5.Wait, but let me double-check. Adding them up: (-5) + (-4) + (-3) + (-2) + (-1) + 0 + 1 + 2 + 3 + 4. Let's pair them: (-5 + 4) = -1, (-4 + 3) = -1, (-3 + 2) = -1, (-2 + 1) = -1, (-1 + 0) = -1. So, that's 5 pairs each summing to -1, so total is -5. Okay, that matches.Next, for y = 25, f inverse(25) = sqrt(25) = 5.So, the total sum is the sum from y = -8 to y = 1, which is -5, plus f inverse(25) which is 5. So, -5 + 5 = 0.Wait, that seems too straightforward. Let me make sure I didn't miss anything.First, confirming the inverse functions:- For y < 2, f inverse(y) = y + 3. Correct.- For y >= 25, f inverse(y) = sqrt(y). Correct.The range of f(x) is (-infty, 2) union [25, infty). So, f inverse is only defined for y < 2 and y >= 25. Therefore, in the sum from y = -8 to y = 25, only y = -8, -7, ..., 1 and y = 25 are included. The rest are undefined, so they are excluded.Calculating the sum from y = -8 to y = 1: that's 10 terms, sum is -5. Then adding f inverse(25) = 5, total is 0.Wait, but the problem says "Compute the value of f^{-1}(-8) + f^{-1}(-7) + dots + f^{-1}(24) + f^{-1}(25)". So, does that mean we include all integers from -8 to 25, but f inverse is only defined for some of them? So, we have to sum only the defined terms.Yes, that's correct. So, the sum is from y = -8 to y = 25, but only including the terms where f inverse is defined, which are y < 2 and y >= 25. So, y = -8, -7, ..., 1 and y = 25.Therefore, the sum is indeed -5 + 5 = 0.Wait, but in the initial calculation, I thought the sum from -5 to 4 is -5, but actually, the terms are f inverse(-8) to f inverse(1), which are -5 to 4, summing to -5. Then adding f inverse(25) = 5, total is 0.Yes, that seems correct.But let me think again. Maybe I made a mistake in the number of terms. From y = -8 to y = 1, that's 10 terms: -8, -7, -6, -5, -4, -3, -2, -1, 0, 1. Each of these corresponds to f inverse(y) = y + 3, so:f inverse(-8) = -5f inverse(-7) = -4f inverse(-6) = -3f inverse(-5) = -2f inverse(-4) = -1f inverse(-3) = 0f inverse(-2) = 1f inverse(-1) = 2f inverse(0) = 3f inverse(1) = 4So, the terms are: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4.Adding these up:(-5) + (-4) + (-3) + (-2) + (-1) + 0 + 1 + 2 + 3 + 4Let's pair them:(-5 + 4) = -1(-4 + 3) = -1(-3 + 2) = -1(-2 + 1) = -1(-1 + 0) = -1So, 5 pairs each summing to -1, total is -5.Then, f inverse(25) = 5.So, total sum is -5 + 5 = 0.Yes, that seems correct.Wait, but the problem says "Compute the value of f^{-1}(-8) + f^{-1}(-7) + dots + f^{-1}(24) + f^{-1}(25)". So, does that mean we include all integers from -8 to 25, but f inverse is only defined for some of them? So, we have to sum only the defined terms.Yes, that's correct. So, the sum is indeed -5 + 5 = 0.But wait, in the initial problem statement, it's written as f^{-1}(-8) + f^{-1}(-7) + dots + f^{-1}(24) + f^{-1}(25). So, the ellipsis (...) suggests that we include all integers in between, but since f inverse is undefined for y between 2 and 24, those terms are excluded. So, the sum is only over y = -8 to y =1 and y=25.Therefore, the total sum is 0.Wait, but in the initial thought process, I thought the sum was 0, but the assistant's initial answer was 5. Hmm, maybe I made a mistake.Wait, no, the assistant's initial answer was 5, but after thinking through, I realized it's 0. Let me check again.Wait, no, in the initial problem, the assistant's answer was 5, but in my detailed thought process, I concluded it's 0. So, which one is correct?Wait, let's recast the problem.The function f(x) is defined as:f(x) = x - 3 for x < 5f(x) = x^2 for x >= 5So, f is invertible because it's strictly increasing on both pieces. For x <5, f(x) = x -3 is linear with slope 1, so it's increasing. For x >=5, f(x) = x^2 is also increasing since x >=5, so the function is continuous at x=5? Wait, no, at x=5, f(5) = 5^2 =25, but for x approaching 5 from the left, f(x) approaches 5 -3 =2. So, there's a jump discontinuity at x=5, from 2 to 25. Therefore, the range of f is (-infty, 2) union [25, infty). So, f inverse is defined on (-infty, 2) and [25, infty).Therefore, in the sum from y = -8 to y =25, we have to include only y values where f inverse is defined, which are y <2 and y >=25. So, y = -8, -7, ...,1 and y=25.So, the sum is f inverse(-8) + f inverse(-7) + ... + f inverse(1) + f inverse(25).As calculated before, f inverse(-8) = -5, f inverse(-7) = -4, ..., f inverse(1)=4, and f inverse(25)=5.So, the sum is (-5) + (-4) + (-3) + (-2) + (-1) + 0 +1 +2 +3 +4 +5.Wait, hold on, I think I made a mistake earlier. Because f inverse(25) is 5, so when adding to the previous sum, which was -5, adding 5 gives 0. But wait, in the detailed list, the terms from f inverse(-8) to f inverse(1) are -5 to 4, which sum to -5, and then adding f inverse(25)=5, total is 0.But wait, in the list, f inverse(-8) to f inverse(1) are 10 terms: -5, -4, -3, -2, -1, 0,1,2,3,4. Sum is -5.Then f inverse(25)=5. So, total sum is -5 +5=0.But wait, in the initial problem, the assistant's answer was 5, but in my detailed thought process, I concluded it's 0. So, which one is correct?Wait, perhaps I made a mistake in the number of terms. Let me recount.From y = -8 to y =1, that's 10 terms: -8, -7, -6, -5, -4, -3, -2, -1, 0,1.Each of these y's maps to f inverse(y) = y +3, so:f inverse(-8) = -5f inverse(-7) = -4f inverse(-6) = -3f inverse(-5) = -2f inverse(-4) = -1f inverse(-3) = 0f inverse(-2) =1f inverse(-1)=2f inverse(0)=3f inverse(1)=4So, the terms are: -5, -4, -3, -2, -1, 0,1,2,3,4.Summing these: (-5) + (-4) + (-3) + (-2) + (-1) +0 +1 +2 +3 +4.Let's add them step by step:Start with 0.Add -5: total -5Add -4: total -9Add -3: total -12Add -2: total -14Add -1: total -15Add 0: total -15Add 1: total -14Add 2: total -12Add 3: total -9Add 4: total -5.Wait, that's different from before. So, the sum is -5.Then, adding f inverse(25)=5, total sum is -5 +5=0.Wait, but earlier I thought the sum was -5, but when I paired them, I got -5 as well. So, the total sum is 0.But in the initial problem, the assistant's answer was 5, which is incorrect. So, the correct answer should be 0.Wait, but let me double-check the pairing method.The terms are: -5, -4, -3, -2, -1, 0,1,2,3,4.Pairing them as follows:(-5 +4) = -1(-4 +3) = -1(-3 +2) = -1(-2 +1) = -1(-1 +0) = -1So, 5 pairs, each summing to -1, total is -5.Then, adding f inverse(25)=5, total is 0.Yes, that's correct.Therefore, the final answer is 0.But wait, in the initial problem, the assistant's answer was 5, but in my detailed thought process, I concluded it's 0. So, the correct answer is 0.Wait, but let me think again. Maybe I made a mistake in the pairing.Wait, the terms are: -5, -4, -3, -2, -1, 0,1,2,3,4.If I pair them as (-5 +4), (-4 +3), (-3 +2), (-2 +1), (-1 +0), that's 5 pairs, each summing to -1, so total is -5.Yes, that's correct.Then, adding f inverse(25)=5, total is 0.Therefore, the answer is 0.But wait, in the initial problem, the assistant's answer was 5, which is incorrect. So, the correct answer is 0.Wait, but let me think about the function f(x) again. For x <5, f(x)=x-3, which is invertible, and for x >=5, f(x)=x^2, which is also invertible since it's increasing for x >=5.So, f inverse(y) is y +3 for y <2, and sqrt(y) for y >=25.Therefore, in the sum from y=-8 to y=25, we include y=-8 to y=1 and y=25.So, the sum is f inverse(-8) + f inverse(-7) + ... + f inverse(1) + f inverse(25).As calculated, the sum is 0.Therefore, the final answer is 0.

answer：To find the expression for ( f(x) ) given that ( f(x+1) = x^2 + 2x + 2 ), I can use substitution. Let me set ( t = x + 1 ), which means ( x = t - 1 ). Substituting ( x = t - 1 ) into the given equation:[f(t) = (t - 1)^2 + 2(t - 1) + 2]Expanding the terms:[(t - 1)^2 = t^2 - 2t + 1][2(t - 1) = 2t - 2]Adding these together with the constant term:[f(t) = t^2 - 2t + 1 + 2t - 2 + 2]Simplifying the expression:[f(t) = t^2 + 1]Therefore, replacing ( t ) with ( x ) gives the final expression for ( f(x) ):[f(x) = x^2 + 1]