🧠:<think>First, I need to determine how much of each ingredient Lily needs to bake enough cookies for 37 people. The original recipe makes 24 cookies and requires specific fractions of sugar, butter, and chocolate chips. To find out how much of each ingredient is needed per cookie, I'll divide the amount of each ingredient by 24. For sugar, the original recipe uses 3/4 cup. Dividing that by 24 gives (3/4) / 24 = 3/96 = 1/32 cup per cookie. Next, for butter, the original amount is 2/3 cup. Dividing by 24 results in (2/3) / 24 = 2/72 = 1/36 cup per cookie. For chocolate chips, the original amount is 1/2 cup. Dividing by 24 gives (1/2) / 24 = 1/48 cup per cookie. Now, to find the total amount needed for 37 cookies, I'll multiply each per-cookie amount by 37. For sugar: 1/32 * 37 = 37/32 cups. For butter: 1/36 * 37 = 37/36 cups. For chocolate chips: 1/48 * 37 = 37/48 cups. So, Lily needs 37/32 cups of sugar, 37/36 cups of butter, and 37/48 cups of chocolate chips to make 37 cookies.Next, if Lily accidentally used 5/4 cups of sugar instead of the required 37/32 cups, I need to calculate how much sugar she needs to remove. First, I'll convert 5/4 cups to 32nds to make the subtraction easier: 5/4 = 40/32. Then, subtract the required amount: 40/32 - 37/32 = 3/32 cups. Therefore, Lily needs to remove 3/32 cups of sugar to correct her mistake.</think>

🧠:<think>First, I'll calculate John's monthly budget allocations based on his 5,000 income. Housing takes 40%, which is 2,000. Food is 20%, amounting to 1,000. Utilities are 10%, or 500, and the same for miscellaneous. Savings are also 20%, totaling 1,000.Next, I'll address the unexpected 600 medical bill. Since John doesn't want to use his savings, he needs to reduce his food and miscellaneous budgets equally. The total reduction needed is 600, so each category will be cut by 300. This means the new food budget is 700 and the new miscellaneous budget is 200.To find the new percentages, I'll divide each adjusted amount by the total income and multiply by 100. Food will now be 14%, and miscellaneous will be 4%.For the skincare allocation, John wants to spend at least 150 without reducing his savings below 10%. His current savings are 1,000, which is 20%. To find the minimum savings percentage, I'll set up an equation where the new savings amount minus 150 equals at least 10% of his income. Solving this, I find that the minimum savings percentage must be 13%.</think>

🧠:<think>Alright, so I'm trying to figure out the coordinates of ( z_5 ) for the SID concert locations. Let me start by understanding the problem step by step.First, the centroid ( C ) is given by the average of all five complex numbers. The formula is ( C = frac{1}{5}(z_1 + z_2 + z_3 + z_4 + z_5) ). They told us that the centroid ( C ) is at (3, 4). So, in complex number terms, that would be ( 3 + 4i ).We already know four of the five complex numbers: ( z_1 = (2, 3) ), ( z_2 = (-1, 2) ), ( z_3 = (4, 1) ), and ( z_4 = (5, 6) ). I need to find ( z_5 ).Let me write down the equation for the centroid:( C = frac{1}{5}(z_1 + z_2 + z_3 + z_4 + z_5) )Multiplying both sides by 5 to get rid of the denominator:( 5C = z_1 + z_2 + z_3 + z_4 + z_5 )So, ( z_5 = 5C - (z_1 + z_2 + z_3 + z_4) )Now, let me compute each part step by step.First, calculate ( 5C ). Since ( C = 3 + 4i ), multiplying by 5 gives:( 5C = 5*(3 + 4i) = 15 + 20i )Next, I need to find the sum of ( z_1 + z_2 + z_3 + z_4 ). Let's convert each complex number to its rectangular form:- ( z_1 = 2 + 3i )- ( z_2 = -1 + 2i )- ( z_3 = 4 + 1i )- ( z_4 = 5 + 6i )Adding them together:Real parts: 2 + (-1) + 4 + 5 = 2 - 1 + 4 + 5 = (2 - 1) + (4 + 5) = 1 + 9 = 10Imaginary parts: 3 + 2 + 1 + 6 = 3 + 2 = 5; 5 + 1 = 6; 6 + 6 = 12So, the sum ( z_1 + z_2 + z_3 + z_4 = 10 + 12i )Now, subtract this sum from ( 5C ):( z_5 = (15 + 20i) - (10 + 12i) = (15 - 10) + (20i - 12i) = 5 + 8i )So, ( z_5 ) is at (5, 8). Let me double-check my calculations to make sure I didn't make a mistake.Calculating the real parts:2 (from z1) + (-1) (z2) = 11 + 4 (z3) = 55 + 5 (z4) = 10. That's correct.Imaginary parts:3 (z1) + 2 (z2) = 55 + 1 (z3) = 66 + 6 (z4) = 12. That's correct.Then, 5C is 15 + 20i, subtract 10 + 12i gives 5 + 8i. Yep, that seems right.So, the coordinates of ( z_5 ) are (5, 8).Now, moving on to the second part. We need to check if the distance from the origin to ( z_5 ) is a prime number. If not, find the closest prime number.The distance from the origin is given by the modulus of the complex number, which is ( |z| = sqrt{x^2 + y^2} ).So, for ( z_5 = (5, 8) ), the distance is:( |z_5| = sqrt{5^2 + 8^2} = sqrt{25 + 64} = sqrt{89} )Calculating ( sqrt{89} ). Hmm, I know that 9^2 is 81 and 10^2 is 100, so ( sqrt{89} ) is somewhere between 9 and 10.Let me compute it more precisely. 9^2 = 81, 9.5^2 = 90.25, which is more than 89. So, 9.4^2 = 88.36, 9.4^2 = (9 + 0.4)^2 = 81 + 7.2 + 0.16 = 88.369.4^2 = 88.369.45^2 = ?Let me compute 9.45^2:= (9 + 0.45)^2= 9^2 + 2*9*0.45 + 0.45^2= 81 + 8.1 + 0.2025= 81 + 8.1 = 89.1; 89.1 + 0.2025 = 89.3025So, 9.45^2 = 89.3025, which is just over 89. So, ( sqrt{89} ) is approximately 9.433.Wait, but 9.433^2 is approximately 89.Wait, 9.433 * 9.433:Let me compute 9 * 9 = 819 * 0.433 = 3.8970.433 * 9 = 3.8970.433 * 0.433 ≈ 0.187Adding them up:81 + 3.897 + 3.897 + 0.187 ≈ 81 + 7.794 + 0.187 ≈ 88.981, which is approximately 89. So, yeah, ( sqrt{89} ) ≈ 9.433.Now, 89 is a prime number. Wait, hold on. Wait, 89 is a prime number, right? Because it's only divisible by 1 and itself.Wait, but the distance is ( sqrt{89} ), which is approximately 9.433, but the question is about the distance, not the square of the distance.Wait, let me read the note again: "Use the Euclidean distance formula to calculate the distance from the origin to a point ( z = x + yi ): ( |z| = sqrt{x^2 + y^2} )."So, the distance is ( sqrt{89} ), which is approximately 9.433.But the problem says: "Verify whether the distance from the origin to ( z_5 ) is a prime number."Wait, hold on. Is the distance a prime number? But the distance is ( sqrt{89} ), which is an irrational number, approximately 9.433. Prime numbers are integers greater than 1. So, 9.433 is not an integer, so it's not a prime number.Wait, but hold on. Maybe the problem is referring to the squared distance? Because 89 is a prime number. But the note says to use the Euclidean distance, which is the square root.Wait, let me read the problem again:"Verify whether the distance from the origin to ( z_5 ) is a prime number. If not, determine the closest prime number to this distance."So, the distance is ( sqrt{89} ), which is approximately 9.433, which is not an integer, so it's not a prime number. Therefore, we need to find the closest prime number to 9.433.The prime numbers around 9.433 are 7, 11, and maybe 5, but 5 is too far. Let's see:Primes less than 9.433: 2, 3, 5, 7Primes greater than 9.433: 11, 13, etc.So, the closest primes are 7 and 11.Compute the distances:9.433 - 7 = 2.43311 - 9.433 = 1.567So, 11 is closer to 9.433 than 7 is.Therefore, the closest prime number is 11.Wait, but hold on. Is 89 a prime number? Yes, 89 is a prime number. So, if the distance squared is 89, which is prime, but the distance itself is sqrt(89), which is not an integer, so not prime.But the problem says: "venues whose distances from the origin are prime numbers." So, the distance must be a prime number, not the squared distance.Therefore, since sqrt(89) is not an integer, it's not prime. So, the distance is not a prime number. Therefore, we need to find the closest prime number to sqrt(89).As calculated, sqrt(89) ≈ 9.433. The closest primes are 7 and 11. 9.433 is closer to 11 (difference of ~1.567) than to 7 (difference of ~2.433). So, the closest prime is 11.Wait, but hold on. Let me think again. Is 89 a prime? Yes, 89 is a prime number. So, the squared distance is 89, which is prime, but the distance itself is sqrt(89), which is not an integer.But the problem says: "venues whose distances from the origin are prime numbers." So, the distance must be a prime number, meaning an integer prime. Since sqrt(89) is not an integer, it's not a prime number. Therefore, we need to find the closest prime to sqrt(89), which is approximately 9.433.So, the closest prime numbers are 7 and 11. Since 9.433 is closer to 11, the answer is 11.Alternatively, maybe the problem is referring to the squared distance? Let me check the problem statement again."venues whose distances from the origin are prime numbers." So, it's the distance, not the squared distance. So, it's the Euclidean distance, which is sqrt(x^2 + y^2). So, that's a real number, not necessarily integer.But prime numbers are integers. So, the distance must be an integer prime. Since sqrt(89) is not an integer, it's not a prime. Therefore, we need to find the closest prime to sqrt(89), which is approximately 9.433.So, the closest primes are 7 and 11. 9.433 - 7 = 2.433, 11 - 9.433 = 1.567. So, 11 is closer.Therefore, the closest prime number is 11.Wait, but just to make sure, let me list the primes around 9.433:Primes less than 9.433: 2, 3, 5, 7Primes greater than 9.433: 11, 13, 17, etc.So, the primes immediately surrounding 9.433 are 7 and 11.Compute the distances:9.433 - 7 = 2.43311 - 9.433 = 1.567So, 11 is closer.Therefore, the closest prime number is 11.So, to summarize:1. ( z_5 ) is at (5, 8).2. The distance from the origin to ( z_5 ) is sqrt(89) ≈ 9.433, which is not a prime number. The closest prime number is 11.I think that's it.Final Answer1. The coordinates of ( z_5 ) are boxed{(5, 8)}.2. The closest prime number to the distance from the origin to ( z_5 ) is boxed{11}.</think>

🧠:<think>Okay, so I have this problem about a model preparing for her debut. There are two sub-problems here, and I need to solve both. Let me start with Sub-problem 1.Sub-problem 1 says that the model's weekly schedule involves T hours of training, P hours of public appearances, and M hours of mentoring. The total hours available per week is 168 hours. She needs at least 8 hours of sleep per day, which totals 56 hours per week. So, the combined time for training, public appearances, and mentoring should not exceed 80% of her non-sleeping hours.First, let me figure out her non-sleeping hours. If she sleeps 56 hours a week, then her non-sleeping time is 168 - 56 = 112 hours. Okay, that makes sense.Now, the combined time for T, P, and M should not exceed 80% of her non-sleeping hours. So, 80% of 112 hours is 0.8 * 112. Let me calculate that: 0.8 * 112 = 89.6 hours. Hmm, so T + P + M ≤ 89.6 hours.But the question is asking for the maximum number of hours she can dedicate to training, public appearances, and mentoring per week. So, that would be 89.6 hours. But since we're dealing with hours, maybe we should round it to a whole number? 89.6 is approximately 90 hours. But I should check if the problem specifies whether to round or not. It just says "formulate and solve the inequality," so maybe we can leave it as 89.6.Wait, but the problem says "the maximum number of hours," so perhaps it's okay to have a decimal. But in real life, you can't really have a fraction of an hour in a schedule, but since it's a mathematical problem, maybe it's acceptable.So, the inequality is T + P + M ≤ 89.6. Therefore, the maximum number of hours she can dedicate is 89.6 hours.Wait, but let me make sure I didn't make a mistake. The total available time is 168 hours. She needs 56 hours for sleep, so 168 - 56 = 112 hours left. 80% of that is 0.8 * 112 = 89.6. Yes, that seems correct.So, the maximum combined hours for training, public appearances, and mentoring is 89.6 hours per week.Moving on to Sub-problem 2. The mentors want to maximize her learning efficiency by balancing her activities. The effectiveness E of her training is given by a quadratic equation: E = -0.5T² + 6T + 20. I need to find the number of training hours T that will maximize E, and then calculate the maximum effectiveness E.Quadratic equations have the form E = aT² + bT + c. In this case, a = -0.5, b = 6, c = 20. Since the coefficient of T² is negative (-0.5), the parabola opens downward, meaning the vertex is the maximum point. So, the maximum effectiveness occurs at the vertex of the parabola.The formula for the vertex (which gives the maximum T) is T = -b/(2a). Plugging in the values, T = -6/(2*(-0.5)) = -6/(-1) = 6. So, T = 6 hours.Now, to find the maximum effectiveness E, plug T = 6 back into the equation:E = -0.5*(6)² + 6*(6) + 20First, calculate 6 squared: 6*6 = 36Then, -0.5*36 = -18Next, 6*6 = 36So, E = -18 + 36 + 20 = (-18 + 36) + 20 = 18 + 20 = 38.Therefore, the maximum effectiveness E is 38 when she trains for 6 hours.Wait, let me double-check the calculations:E = -0.5*(6)^2 + 6*6 + 20= -0.5*36 + 36 + 20= -18 + 36 + 20= ( -18 + 36 ) + 20= 18 + 20= 38Yes, that's correct.So, summarizing:Sub-problem 1: The maximum combined hours for training, public appearances, and mentoring is 89.6 hours.Sub-problem 2: The number of training hours T that maximizes effectiveness is 6 hours, with a maximum effectiveness E of 38.I think that's all. Let me just make sure I didn't miss anything.For Sub-problem 1, the key was understanding that non-sleeping hours are 112, and 80% of that is 89.6. The inequality is T + P + M ≤ 89.6.For Sub-problem 2, recognizing it's a quadratic equation and finding the vertex gives the maximum effectiveness. Calculated T as 6 and E as 38. Seems solid.Final AnswerSub-problem 1: The maximum combined hours are boxed{89.6}.Sub-problem 2: The maximum effectiveness is achieved with boxed{6} hours of training, resulting in an effectiveness of boxed{38}.</think>