| `'Okay, let\'s try to figure out this problem step by step. So, Aya walks 9 kilometers every morning and stops at a coffee shop. The time she takes includes both her walking time and the time she spends at the coffee shop, which is t minutes. \n\nFirst, when she walks at a constant speed of s km/h, the total time (including the coffee stop) is 4 hours. Then, when she increases her speed by 2 km/h (so s + 2 km/h), the total time becomes 2 hours and 24 minutes, which is 2.4 hours if I convert that to hours. The question is asking, if she walks at s + 0.5 km/h, how long will the total time be, including the t minutes at the coffee shop.\n\nHmm. So, the key here is to find out the values of s and t first, right? Because once we have those, we can compute the walking time at the new speed and then add the coffee shop time t to get the total time.\n\nLet me start by writing down the information given in equations. \n\nWhen she walks at s km/h, the time she spends walking is distance divided by speed, which is 9/s hours. But the total time is 4 hours, which includes t minutes at the coffee shop. So, the walking time plus the coffee time equals 4 hours. But we need to make sure the units are consistent. The walking time is in hours, and the coffee time is in minutes. So, we need to convert t minutes to hours by dividing by 60.\n\nSo, equation 1: 9/s + t/60 = 4.\n\nSimilarly, when she walks at (s + 2) km/h, the walking time is 9/(s + 2) hours, and the total time is 2 hours and 24 minutes. Let\'s convert 2 hours 24 minutes to hours: 24 minutes is 24/60 = 0.4 hours, so total is 2.4 hours.\n\nTherefore, equation 2: 9/(s + 2) + t/60 = 2.4.\n\nNow, we have two equations:\n\n1) 9/s + t/60 = 4\n\n2) 9/(s + 2) + t/60 = 2.4\n\nWe can subtract equation 2 from equation 1 to eliminate t/60:\n\n[9/s + t/60] - [9/(s + 2) + t/60] = 4 - 2.4\n\nSimplifying the left side: 9/s - 9/(s + 2) = 1.6\n\nSo, 9/s - 9/(s + 2) = 1.6\n\nLet me compute that:\n\n9/s - 9/(s + 2) = 1.6\n\nFactor out the 9:\n\n9[1/s - 1/(s + 2)] = 1.6\n\nCompute the expression inside the brackets:\n\n1/s - 1/(s + 2) = [ (s + 2) - s ] / [s(s + 2)] = 2 / [s(s + 2)]\n\nSo, 9 * [2 / (s(s + 2))] = 1.6\n\nMultiply 9 and 2:\n\n18 / [s(s + 2)] = 1.6\n\nSo, s(s + 2) = 18 / 1.6\n\nCalculate 18 divided by 1.6:\n\n1.6 is the same as 8/5, so 18 ÷ (8/5) = 18 * (5/8) = (90)/8 = 11.25\n\nTherefore, s(s + 2) = 11.25\n\nNow, let\'s write that as a quadratic equation:\n\ns^2 + 2s - 11.25 = 0\n\nWait, no. s(s + 2) = 11.25, so s^2 + 2s - 11.25 = 0. Let me check that.\n\nYes, expanding s(s + 2) gives s² + 2s, so the equation is s² + 2s - 11.25 = 0.\n\nTo solve this quadratic equation, we can use the quadratic formula:\n\ns = [-b ± √(b² - 4ac)] / (2a)\n\nHere, a = 1, b = 2, c = -11.25\n\nSo,\n\ns = [-2 ± √(4 + 45)] / 2\n\nWait, discriminant is b² - 4ac = (2)^2 - 4(1)(-11.25) = 4 + 45 = 49\n\nSo, square root of 49 is 7.\n\nTherefore,\n\ns = [-2 ± 7]/2\n\nWe have two solutions:\n\ns = (-2 + 7)/2 = 5/2 = 2.5\n\ns = (-2 - 7)/2 = -9/2 = -4.5\n\nBut speed can\'t be negative, so s = 2.5 km/h.\n\nOkay, so s is 2.5 km/h. Now, let\'s find t.\n\nUsing equation 1: 9/s + t/60 = 4\n\nPlug in s = 2.5:\n\n9 / 2.5 + t/60 = 4\n\nCalculate 9 divided by 2.5:\n\n9 / 2.5 = 3.6\n\nSo, 3.6 + t/60 = 4\n\nSubtract 3.6 from both sides:\n\nt/60 = 0.4\n\nMultiply both sides by 60:\n\nt = 0.4 * 60 = 24 minutes.\n\nSo, t is 24 minutes. Let\'s verify this with equation 2 to make sure.\n\nEquation 2: 9/(s + 2) + t/60 = 2.4\n\ns + 2 = 2.5 + 2 = 4.5 km/h\n\n9 / 4.5 = 2 hours\n\nt/60 = 24 / 60 = 0.4 hours\n\nSo, 2 + 0.4 = 2.4 hours, which matches. Good.\n\nNow, the question is, if she walks at s + 0.5 km/h, which is 2.5 + 0.5 = 3 km/h, how long will the total time be, including the t minutes (24 minutes) at the coffee shop.\n\nSo, the walking time at 3 km/h is 9 / 3 = 3 hours.\n\nConvert the coffee time t to hours: 24 minutes = 24/60 = 0.4 hours.\n\nTotal time is walking time + coffee time: 3 + 0.4 = 3.4 hours.\n\nConvert 3.4 hours to minutes: 3 hours is 180 minutes, 0.4 hours is 24 minutes, so total 180 + 24 = 204 minutes.\n\nWait, but let me check the question again. It says "the number of minutes the walk takes her, including the t minutes spent in the coffee shop."\n\nWait, the total time is walking time plus t minutes. So, if walking time is 3 hours, which is 180 minutes, plus t minutes which is 24 minutes, so total time is 180 + 24 = 204 minutes. So, 204 minutes is the answer.\n\nBut let me just verify once again to be sure.\n\nOriginal equations:\n\n1) 9/s + t/60 = 4\n\n2) 9/(s + 2) + t/60 = 2.4\n\nWe found s = 2.5, t = 24.\n\nAt s + 0.5 = 3 km/h, walking time is 9/3 = 3 hours = 180 minutes. Adding t = 24 minutes gives 204 minutes. That seems correct.\n\nAlternatively, in hours, 3 + 0.4 = 3.4 hours, which is 3 hours and 24 minutes, which is 204 minutes. Yep.\n\nSo, the answer should be 204 minutes.\n\n**Final Answer**\nThe total time Aya takes, including the coffee shop stop, is \\boxed{204} minutes.\n\n<|end_of_thought|>\n\n<|begin_of_solution|>\n\nAya walks 9 kilometers each morning and stops at a coffee shop. The total time includes both her walking time and the time spent at the coffee shop, denoted as \\( t \\) minutes. We need to determine the total time when she walks at \\( s + \\frac{1}{2} \\) km/h, given her walking times at speeds \\( s \\) km/h and \\( s + 2 \\) km/h.\n\nFirst, we set up the equations for the total time when walking at different speeds:\n\n1. When walking at \\( s \\) km/h:\n \\[\n \\frac{9}{s} + \\frac{t}{60} = 4\n \\]\n\n2. When walking at \\( s + 2 \\) km/h:\n \\[\n \\frac{9}{s + 2} + \\frac{t}{60} = 2.4\n \\]\n\nSubtracting the second equation from the first to eliminate \\( \\frac{t}{60} \\):\n\\[\n\\frac{9}{s} - \\frac{9}{s + 2} = 1.6\n\\]\n\nSolving this equation:\n\\[\n9 \\left( \\frac{1}{s} - \\frac{1}{s + 2} \\right) = 1.6\n\\]\n\\[\n9 \\left( \\frac{2}{s(s + 2)} \\right) = 1.6\n\\]\n\\[\n\\frac{18}{s(s + 2)} = 1.6\n\\]\n\\[\ns(s + 2) = \\frac{18}{1.6} = 11.25\n\\]\n\nSolving the quadratic equation \\( s^2 + 2s - 11.25 = 0 \\) using the quadratic formula:\n\\[\ns = \\frac{-2 \\pm \\sqrt{4 + 45}}{2} = \\frac{-2 \\pm 7}{2}\n\\]\n\\[\ns = 2.5 \\quad (\\text{since speed cannot be negative})\n\\]\n\nSubstituting \\( s = 2.5 \\) km/h into the first equation to find \\( t \\):\n\\[\n\\frac{9}{2.5} + \\frac{t}{60} = 4\n\\]\n\\[\n3.6 + \\frac{t}{60} = 4\n\\]\n\\[\n\\frac{t}{60} = 0.4\n\\]\n\\[\nt = 24 \\text{ minutes}\n\\]\n\nWhen Aya walks at \\( s + \\frac{1}{2} = 3 \\) km/h, her walking time is:\n\\[\n\\frac{9}{3} = 3 \\text{ hours} = 180 \\text{ minutes}\n\\]\n\nAdding the coffee shop time \\( t \\):\n\\[\n180 + 24 = 204 \\text{ minutes}\n\\]\n\nThus, the total time Aya takes, including the coffee shop stop, is \\(\\boxed{204}\\) minutes.\n\n<|end_of_solution|>'` |