answer：Okay, so I have this problem where I need to find the value of ( m ) such that the smallest number in the decomposition of ( m^3 ) is 73. The decomposition rule is given through some examples, so I need to figure out the pattern or rule they're using.First, let me look at the examples provided:For squares:- ( 2^2 = 1 + 3 )- ( 3^2 = 1 + 3 + 5 )- ( 4^2 = 1 + 3 + 5 + 7 )- ( 5^2 = 1 + 3 + 5 + 7 + 9 )So, it seems like for ( m^2 ), they're summing the first ( m ) odd numbers starting from 1. That makes sense because the sum of the first ( m ) odd numbers is indeed ( m^2 ). For example, ( 1 + 3 + 5 + 7 + 9 = 25 = 5^2 ).Now, looking at the cubes:- ( 2^3 = 3 + 5 )- ( 3^3 = 7 + 9 + 11 )- ( 4^3 = 13 + 15 + 17 + 19 )Hmm, this is interesting. For ( 2^3 ), they're using two consecutive odd numbers starting from 3. For ( 3^3 ), they're using three consecutive odd numbers starting from 7. For ( 4^3 ), they're using four consecutive odd numbers starting from 13.I need to figure out the pattern here. Let's see:- For ( m = 2 ), the decomposition starts at 3.- For ( m = 3 ), it starts at 7.- For ( m = 4 ), it starts at 13.Let me list these starting numbers:- ( m = 2 ): 3- ( m = 3 ): 7- ( m = 4 ): 13Looking at the differences between these starting numbers:- From ( m = 2 ) to ( m = 3 ): 7 - 3 = 4- From ( m = 3 ) to ( m = 4 ): 13 - 7 = 6Hmm, the differences are increasing by 2 each time. So, 4, 6, which suggests the next difference might be 8, making the next starting number 13 + 8 = 21 for ( m = 5 ). Let me check if that makes sense.If ( m = 5 ), then the decomposition should be five consecutive odd numbers starting from 21. So, ( 21 + 23 + 25 + 27 + 29 ). Let's add these up:21 + 23 = 4444 + 25 = 6969 + 27 = 9696 + 29 = 125And ( 5^3 = 125 ), so that checks out. Okay, so the starting number for ( m^3 ) seems to follow a pattern where each starting number increases by 2 more than the previous increase.But maybe there's a formula for the starting number. Let me see if I can find a general expression.Looking at the starting numbers:- ( m = 2 ): 3- ( m = 3 ): 7- ( m = 4 ): 13Let me see if these can be expressed in terms of ( m ).For ( m = 2 ): 3 = 2^2 - 2 + 1 = 4 - 2 + 1 = 3For ( m = 3 ): 7 = 3^2 - 3 + 1 = 9 - 3 + 1 = 7For ( m = 4 ): 13 = 4^2 - 4 + 1 = 16 - 4 + 1 = 13Oh, that seems to fit! So, the starting number for ( m^3 ) is ( m^2 - m + 1 ).Let me verify this with ( m = 5 ):Starting number = ( 5^2 - 5 + 1 = 25 - 5 + 1 = 21 ). Which matches what I thought earlier. Perfect.So, if the smallest number in the decomposition of ( m^3 ) is 73, then:( m^2 - m + 1 = 73 )Now, I need to solve this quadratic equation for ( m ).Let me write it down:( m^2 - m + 1 = 73 )Subtract 73 from both sides:( m^2 - m + 1 - 73 = 0 )Simplify:( m^2 - m - 72 = 0 )Now, I need to solve ( m^2 - m - 72 = 0 ).I can use the quadratic formula:( m = frac{-b pm sqrt{b^2 - 4ac}}{2a} )Here, ( a = 1 ), ( b = -1 ), ( c = -72 ).Plugging in:( m = frac{-(-1) pm sqrt{(-1)^2 - 4(1)(-72)}}{2(1)} )Simplify:( m = frac{1 pm sqrt{1 + 288}}{2} )( m = frac{1 pm sqrt{289}}{2} )( sqrt{289} = 17 ), so:( m = frac{1 pm 17}{2} )This gives two solutions:1. ( m = frac{1 + 17}{2} = frac{18}{2} = 9 )2. ( m = frac{1 - 17}{2} = frac{-16}{2} = -8 )Since ( m ) is a natural number (positive integer), we discard the negative solution.Therefore, ( m = 9 ).Let me double-check this result.If ( m = 9 ), then the starting number should be:( 9^2 - 9 + 1 = 81 - 9 + 1 = 73 )Which is exactly the smallest number given in the problem. So, that's correct.Additionally, let's verify the decomposition for ( m = 9 ):The decomposition should be nine consecutive odd numbers starting from 73.So, the numbers are: 73, 75, 77, 79, 81, 83, 85, 87, 89.Let's add them up:73 + 75 = 148148 + 77 = 225225 + 79 = 304304 + 81 = 385385 + 83 = 468468 + 85 = 553553 + 87 = 640640 + 89 = 729And ( 9^3 = 729 ), so that checks out perfectly.Therefore, I'm confident that ( m = 9 ) is the correct answer.**Final Answer**The value of ( m ) is boxed{9}.

answer：Okay, so I have this problem here about a geometric sequence. The sequence is {1, 1/2, 1/4, 1/8, ...}, and I need to find the number of terms, n, such that the sum of the first n terms is 31/16. Hmm, let's see.First, I remember that a geometric sequence is one where each term is multiplied by a common ratio to get the next term. In this case, the first term is 1, and each subsequent term is half of the previous one. So the common ratio, r, is 1/2. That makes sense because 1 times 1/2 is 1/2, then 1/2 times 1/2 is 1/4, and so on.Now, the problem is about the sum of the first n terms. I recall there's a formula for the sum of the first n terms of a geometric sequence. Let me try to remember it. I think it's something like S_n = a_1 * (1 - r^n) / (1 - r), where a_1 is the first term, r is the common ratio, and n is the number of terms. Let me write that down:S_n = a_1 * (1 - r^n) / (1 - r)In this problem, a_1 is 1, and r is 1/2. So plugging those values in, the formula becomes:S_n = 1 * (1 - (1/2)^n) / (1 - 1/2)Simplifying the denominator, 1 - 1/2 is 1/2. So now the formula is:S_n = (1 - (1/2)^n) / (1/2)Dividing by 1/2 is the same as multiplying by 2, so:S_n = 2 * (1 - (1/2)^n)Which simplifies to:S_n = 2 - 2*(1/2)^nWait, let me check that again. If I have (1 - (1/2)^n) divided by 1/2, that's the same as multiplying by 2, so:2*(1 - (1/2)^n) = 2 - 2*(1/2)^nBut 2*(1/2)^n is the same as (2)*(1/2)^n. Since 2 is 2^1, and (1/2)^n is 2^(-n), so multiplying them gives 2^(1 - n). So:2 - 2*(1/2)^n = 2 - 2^(1 - n)Alternatively, I can write 2*(1/2)^n as (1/2)^(n - 1). Wait, is that correct? Let me see:2*(1/2)^n = 2*(2^(-n)) = 2^(1 - n). Alternatively, since 2 is 2^1, so 2^1 * 2^(-n) = 2^(1 - n). So yes, that's correct.But maybe it's simpler to keep it as 2*(1/2)^n for now.So, the sum S_n is 2 - 2*(1/2)^n, and we're told that this sum equals 31/16. So:2 - 2*(1/2)^n = 31/16Let me write that equation down:2 - 2*(1/2)^n = 31/16I need to solve for n. Let's rearrange this equation step by step.First, subtract 2 from both sides:-2*(1/2)^n = 31/16 - 2But 2 is 32/16, so:-2*(1/2)^n = 31/16 - 32/16 = -1/16So:-2*(1/2)^n = -1/16Multiply both sides by -1 to eliminate the negative signs:2*(1/2)^n = 1/16Now, divide both sides by 2:(1/2)^n = (1/16)/2 = 1/32Wait, that doesn't seem right. Let me check my steps again.Wait, 2*(1/2)^n = 1/16So, (1/2)^n = (1/16)/2 = 1/32Yes, that's correct.But 1/32 is equal to (1/2)^5 because (1/2)^5 = 1/32.So, (1/2)^n = (1/2)^5Therefore, n must be 5.Wait, let me make sure I didn't make a mistake in my calculations.Starting again:Sum of first n terms: S_n = 2 - 2*(1/2)^nGiven S_n = 31/16So:2 - 2*(1/2)^n = 31/16Subtract 2 from both sides:-2*(1/2)^n = 31/16 - 2Convert 2 to sixteenths: 2 = 32/16So:-2*(1/2)^n = 31/16 - 32/16 = -1/16Multiply both sides by -1:2*(1/2)^n = 1/16Divide both sides by 2:(1/2)^n = 1/32Since 1/32 is (1/2)^5, n must be 5.Yes, that seems correct.Alternatively, I can think about the sum another way. The sum of an infinite geometric series with |r| < 1 is S = a_1 / (1 - r). In this case, S = 1 / (1 - 1/2) = 2. So the sum approaches 2 as n approaches infinity. Since 31/16 is less than 2, it makes sense that n is a finite number, specifically 5.Let me verify by calculating the sum for n=5:First term: 1Second term: 1 + 1/2 = 3/2Third term: 3/2 + 1/4 = 7/4Fourth term: 7/4 + 1/8 = 15/8Fifth term: 15/8 + 1/16 = 31/16Yes, that's exactly the sum given. So n=5 is correct.I think I've got it. The number of terms needed is 5.

answer：First, I need to find the value of ( a ) using the given average of the data set. The average is calculated by dividing the sum of all values by the number of values. [frac{a + 5 + 6 + 7 + 7 + 8 + 11 + 12}{8} = 8]Simplifying the equation:[frac{a + 56}{8} = 8 implies a + 56 = 64 implies a = 8]With ( a = 8 ), the data set becomes ( 5, 6, 7, 7, 8, 8, 11, 12 ). To find the median, I need to determine the middle value(s) of the ordered data set. Since there are 8 numbers, the median is the average of the 4th and 5th numbers:[frac{7 + 8}{2} = 7.5]Therefore, the median of the data set is ( 7.5 ), which corresponds to choice ( C ).

answer：Okay, so I have this problem where I need to find the maximum value of the expression (x^4 y + x^3 y + x^2 y + xy + xy^2 + xy^3 + xy^4) given that (x + y = 5). Hmm, that seems a bit complicated at first glance, but maybe I can simplify it somehow.First, I notice that every term in the expression has an (xy) factor. Maybe I can factor that out to make things easier. Let me try that:[x^4 y + x^3 y + x^2 y + xy + xy^2 + xy^3 + xy^4 = xy(x^3 + x^2 + x + 1 + y + y^2 + y^3)]Okay, so now the expression is factored into (xy) times a bunch of other terms. Let me see if I can group those terms in a way that might be easier to handle. I notice that (x^3 + y^3) is a common expression, as well as (x^2 + y^2), and so on. Maybe I can rewrite the expression inside the parentheses as:[x^3 + y^3 + x^2 + y^2 + x + y + 1]That seems a bit more manageable. Now, since I know that (x + y = 5), maybe I can express some of these higher powers in terms of (x + y) and (xy). I remember that for symmetric expressions, we can use identities involving (x + y) and (xy).Let me denote (p = xy). That might make things simpler. So, (p = xy). Now, I know that (x + y = 5), so I can use that to express other terms in terms of (p).First, let's find (x^2 + y^2). I remember that:[(x + y)^2 = x^2 + 2xy + y^2]So, substituting (x + y = 5) and (xy = p):[25 = x^2 + 2p + y^2][x^2 + y^2 = 25 - 2p]Alright, so (x^2 + y^2 = 25 - 2p). Good.Next, let's find (x^3 + y^3). I recall that:[(x + y)^3 = x^3 + 3x^2 y + 3xy^2 + y^3]Which can be rewritten as:[(x + y)^3 = x^3 + y^3 + 3xy(x + y)]So, substituting (x + y = 5) and (xy = p):[125 = x^3 + y^3 + 3p cdot 5][125 = x^3 + y^3 + 15p][x^3 + y^3 = 125 - 15p]Great, so (x^3 + y^3 = 125 - 15p).Now, let's go back to the expression inside the parentheses:[x^3 + y^3 + x^2 + y^2 + x + y + 1]Substituting the expressions we found:[(125 - 15p) + (25 - 2p) + 5 + 1]Let me compute that step by step:First, (125 - 15p + 25 - 2p):Combine the constants: (125 + 25 = 150)Combine the (p) terms: (-15p - 2p = -17p)So that gives (150 - 17p)Then, add the remaining terms: (5 + 1 = 6)So total expression becomes:(150 - 17p + 6 = 156 - 17p)So, the entire expression we're trying to maximize is:[xy (x^3 + y^3 + x^2 + y^2 + x + y + 1) = p (156 - 17p)]Which simplifies to:[156p - 17p^2]Hmm, so we have a quadratic in terms of (p). To find the maximum value, we can treat this as a quadratic function (f(p) = -17p^2 + 156p). Since the coefficient of (p^2) is negative, the parabola opens downward, meaning the vertex is the maximum point.The vertex of a parabola (f(p) = ap^2 + bp + c) is at (p = -frac{b}{2a}). In this case, (a = -17) and (b = 156), so:[p = -frac{156}{2 times -17} = frac{156}{34} = frac{78}{17}]So, the maximum occurs when (p = frac{78}{17}). Now, we can plug this back into the quadratic to find the maximum value:[fleft(frac{78}{17}right) = -17left(frac{78}{17}right)^2 + 156 times frac{78}{17}]Let me compute this step by step.First, compute (left(frac{78}{17}right)^2):[left(frac{78}{17}right)^2 = frac{78^2}{17^2} = frac{6084}{289}]Then, multiply by -17:[-17 times frac{6084}{289} = -frac{17 times 6084}{289}]But notice that 17 is a factor of 289 (since 289 = 17^2), so:[-frac{17 times 6084}{17 times 17} = -frac{6084}{17}]Now, compute the second term:[156 times frac{78}{17} = frac{156 times 78}{17}]Again, 156 divided by 17 is 9.176... Wait, but maybe we can simplify this fraction.Let me see: 156 divided by 17 is 9 with a remainder of 3 (since 17*9=153, 156-153=3). So, 156 = 17*9 + 3. Therefore,[frac{156 times 78}{17} = frac{(17 times 9 + 3) times 78}{17} = 9 times 78 + frac{3 times 78}{17}]Compute 9*78: 9*70=630, 9*8=72, so 630+72=702.Compute 3*78: 234. Then, 234 divided by 17: 17*13=221, so 234-221=13. So, 234/17 = 13 + 13/17.Therefore, total is 702 + 13 + 13/17 = 715 + 13/17.Wait, that seems a bit messy. Maybe I should compute it as:156 divided by 17 is 9.176, but perhaps better to compute 156*78 first.156*78: Let's compute 150*78 + 6*78.150*78: 150*70=10500, 150*8=1200, so total 10500+1200=11700.6*78=468.So, total 11700 + 468 = 12168.Then, 12168 divided by 17: Let's divide 12168 by 17.17*700=11900.12168 - 11900 = 268.17*15=255.268 - 255=13.So, 12168/17=700 + 15 + 13/17=715 + 13/17.So, 156*(78/17)=715 + 13/17.Therefore, putting it all together:[fleft(frac{78}{17}right) = -frac{6084}{17} + 715 + frac{13}{17}]Convert 715 to seventeenths: 715 = 715*(17/17) = (715*17)/17.Compute 715*17: Let's compute 700*17=11900, 15*17=255, so total 11900+255=12155.So, 715 = 12155/17.Similarly, 13/17 is just 13/17.So, total expression:[-frac{6084}{17} + frac{12155}{17} + frac{13}{17}]Combine the numerators:-6084 + 12155 + 13 = (12155 - 6084) + 13 = 6071 + 13 = 6084.So, total is 6084/17.Therefore, the maximum value is 6084/17.Wait, that's interesting. So, the maximum value is 6084 divided by 17, which is 358.470588... But since the problem asks for the maximum value, and it's a fraction, we can leave it as 6084/17.But let me double-check my calculations to make sure I didn't make any mistakes.First, when I factored out (xy), I got (xy(x^3 + y^3 + x^2 + y^2 + x + y + 1)). Then, I expressed (x^3 + y^3) and (x^2 + y^2) in terms of (p = xy). That gave me 125 - 15p and 25 - 2p, respectively. Adding those together with (x + y = 5) and the constant 1, I got 156 - 17p. So, the expression becomes (p(156 - 17p)), which is a quadratic in (p).Then, I found the vertex of the quadratic, which is at (p = 78/17). Plugging that back into the quadratic, I got 6084/17. That seems correct.But just to be thorough, let me check if (p = 78/17) is achievable given that (x + y = 5). Since (x) and (y) are real numbers, the discriminant of the quadratic equation (t^2 - 5t + p = 0) must be non-negative.The discriminant is (25 - 4p). Plugging in (p = 78/17):[25 - 4*(78/17) = 25 - 312/17 = (425 - 312)/17 = 113/17]Which is positive, so yes, real roots exist. Therefore, (p = 78/17) is achievable, and the maximum value is indeed 6084/17.So, I think that's the answer.**Final Answer**The maximum value is boxed{dfrac{6084}{17}}.

answer：Okay, so I have this problem about a summer school with n students, where n is greater than 4. The problem has two conditions about friendships among the students. Let me try to understand and break it down.First, condition (i) says that if two students are friends, then they don't have any common friends. Hmm, that means if Alice and Bob are friends, there's no other student who is friends with both Alice and Bob. So, their friendship is unique in that sense.Condition (ii) states that if two students are not friends, then they have exactly two common friends. So, if Alice and Bob aren't friends, there are precisely two other students who are friends with both Alice and Bob.The problem has two parts. The first part asks me to prove that 8n - 7 must be a perfect square. The second part is to determine the smallest possible value of n.Alright, let's tackle the first part. I think modeling this situation as a graph might help. In graph theory, each student can be represented as a vertex, and an edge between two vertices indicates that those two students are friends.So, let's consider a graph G where each vertex represents a student, and edges represent friendships. Now, condition (i) translates to: if two vertices are adjacent (friends), they have no common neighbors. That is, their adjacency doesn't share any common vertices. Condition (ii) translates to: if two vertices are non-adjacent (not friends), they have exactly two common neighbors. So, every pair of non-friend vertices has exactly two mutual friends.This seems like a specific kind of graph with certain regularity conditions. Maybe it's a strongly regular graph? I remember that strongly regular graphs have parameters (n, k, λ, μ), where n is the number of vertices, k is the degree of each vertex, λ is the number of common neighbors for adjacent vertices, and μ is the number of common neighbors for non-adjacent vertices.In our case, for adjacent vertices, λ = 0 because they don't have any common friends. For non-adjacent vertices, μ = 2 because they have exactly two common friends. So, this graph would be a strongly regular graph with parameters (n, k, 0, 2).I think there's a formula that relates these parameters. Let me recall. For a strongly regular graph, the following equation must hold:k(k - λ - 1) = (n - k - 1)μPlugging in our values, λ = 0 and μ = 2, we get:k(k - 0 - 1) = (n - k - 1) * 2Simplifying:k(k - 1) = 2(n - k - 1)Let me write that down:k^2 - k = 2n - 2k - 2Bring all terms to one side:k^2 - k - 2n + 2k + 2 = 0Simplify:k^2 + k - 2n + 2 = 0So, we have a quadratic equation in terms of k:k^2 + k + (2 - 2n) = 0To solve for k, we can use the quadratic formula:k = [-b ± sqrt(b^2 - 4ac)] / (2a)Here, a = 1, b = 1, c = (2 - 2n). So,k = [-1 ± sqrt(1 - 4*(1)*(2 - 2n))]/2Simplify the discriminant:sqrt(1 - 8 + 8n) = sqrt(8n - 7)So,k = [-1 ± sqrt(8n - 7)] / 2Since k is the degree of each vertex, it must be a positive integer. Therefore, sqrt(8n - 7) must be an integer, say m. So,sqrt(8n - 7) = mWhich implies,8n - 7 = m^2So, 8n - 7 must be a perfect square. That proves the first part.Now, moving on to the second part: determining the smallest possible value of n. Since n > 4, we need to find the smallest integer n such that 8n - 7 is a perfect square.Let me write that equation again:8n - 7 = m^2So,8n = m^2 + 7n = (m^2 + 7)/8Since n must be an integer, (m^2 + 7) must be divisible by 8. So, m^2 ≡ 1 mod 8 because 7 ≡ -1 mod 8, so m^2 ≡ 1 mod 8.I remember that squares modulo 8 can only be 0, 1, or 4. So, m^2 ≡ 1 mod 8 implies that m must be odd because even numbers squared are 0 or 4 mod 8.Let me test small odd integers for m:- m = 1: 1^2 + 7 = 8; n = 8/8 = 1. But n must be greater than 4, so discard.- m = 3: 9 + 7 = 16; n = 16/8 = 2. Still too small.- m = 5: 25 + 7 = 32; n = 32/8 = 4. Still n > 4 is required.- m = 7: 49 + 7 = 56; n = 56/8 = 7. Okay, n=7 is greater than 4.Let me check if n=7 is possible. So, in this case, m=7, n=7.Going back to the quadratic equation for k:k = [-1 ± 7]/2We discard the negative solution because degree can't be negative:k = ( -1 + 7 ) / 2 = 6/2 = 3So, each vertex has degree 3.Is there a strongly regular graph with parameters (7, 3, 0, 2)? Yes, I think that's the complement of the cycle graph C7, but wait, actually, the complement of C7 is also a strongly regular graph.Wait, let me recall. The complement of a strongly regular graph is also strongly regular. The parameters would be (n, n - k - 1, n - 2 - 2k + μ, n - 2k + λ). Hmm, maybe I should look it up, but I think for n=7, k=3, λ=0, μ=2, it's the complement of the cycle graph C7, which is known as the complement of the 7-cycle.Alternatively, it's the complement of the 7-vertex cycle graph, which is actually the 7-vertex strongly regular graph with these parameters. So, yes, it exists.Therefore, n=7 is the smallest possible value.Wait, just to make sure, are there any smaller n? We saw that for m=1,3,5, n=1,2,4 which are too small. The next odd m is 7, giving n=7. So, yes, 7 is the smallest.So, summarizing:1. We model the problem as a strongly regular graph with parameters (n, k, 0, 2). Using the formula for strongly regular graphs, we derive that 8n - 7 must be a perfect square.2. To find the smallest n > 4, we solve 8n - 7 = m^2 for integers m and n, finding that the smallest such n is 7.**Final Answer**The smallest possible value of ( n ) is boxed{7}.

answer：Okay, so I have this problem here about orange juice in a vat and pouring it into five glasses equally. Let me try to understand what it's asking. Alright, the vat has 153 pints of orange juice. I need to pour this into five glasses, and each glass should have the same amount. So, basically, I need to divide the total amount of orange juice by the number of glasses to find out how much each glass will have.Hmm, division seems to be the key here. I remember that when you want to split something equally among a certain number of groups, you use division. So, if I have 153 pints and I want to split it into 5 glasses, I should divide 153 by 5.Let me write that down: 153 ÷ 5. Okay, now I need to perform this division. I think I can do this step by step. First, I'll see how many times 5 goes into 15. Well, 5 times 3 is 15, so that's straightforward. Then, I bring down the next digit, which is 3. Now, I have 3 left. 5 doesn't go into 3, so I guess I need to add a decimal point and a zero to continue the division.So, now I have 30. I'll see how many times 5 goes into 30. That's exactly 6 times because 5 times 6 is 30. So, putting it all together, 153 divided by 5 is 30.6. Wait, does that make sense? Let me double-check. If I multiply 30.6 by 5, I should get back to 153. Let's see: 30 times 5 is 150, and 0.6 times 5 is 3. So, 150 plus 3 is indeed 153. Okay, that checks out.But I'm a bit confused about the decimal point. Why do we add a zero after the decimal? I think it's because when we're dividing and the divisor doesn't go into the remaining number evenly, we add a decimal and continue the division by bringing down zeros to get a more precise answer. So, in this case, after dividing 15 by 5 to get 3, we have a remainder of 3. Adding a decimal and a zero makes it 30, which 5 can divide into 6 times. That gives us the 0.6 part.I also wonder if there's another way to approach this problem, maybe by breaking it down into smaller parts. For example, I could think of 153 as 150 plus 3. Then, dividing each part by 5 separately. So, 150 divided by 5 is 30, and 3 divided by 5 is 0.6. Adding those together gives me 30.6 pints per glass. That seems to be the same result as before, so it reinforces that 30.6 is the correct answer.Another thing I'm thinking about is whether the answer should be a whole number or if it's okay to have a decimal. In real-life scenarios, when pouring liquids, it's common to have measurements with decimals, especially if the total doesn't divide evenly. So, 30.6 pints per glass makes sense in this context.I also want to visualize this. Imagine I have five glasses and I'm pouring orange juice into each one equally. If I pour 30 pints into each glass, that's 150 pints total, which is less than 153. So, I have 3 pints left. To distribute this equally, I need to add 0.6 pints to each glass, making it 30.6 pints per glass. That way, all five glasses have the same amount, and I've used up all 153 pints.I think I've covered all the steps and checked my work to make sure the answer is correct. It's always good to verify by multiplying the result by the number of glasses to ensure it equals the original amount. In this case, 30.6 times 5 equals 153, so that confirms the answer is accurate.