💡Okay, so I have this problem about two jars with marbles. Each jar has the same number of marbles, and each marble is either blue or green. The ratio of blue to green marbles in Jar 1 is 7:3, and in Jar 2, it's 6:4. The total number of green marbles in both jars combined is 80. I need to find how many more blue marbles are in Jar 1 than in Jar 2.Hmm, let me break this down. First, both jars have the same total number of marbles. Let's call that number N for each jar. So, Jar 1 has N marbles, and Jar 2 also has N marbles. In Jar 1, the ratio of blue to green marbles is 7:3. That means for every 7 blue marbles, there are 3 green marbles. So, the total parts of the ratio are 7 + 3 = 10 parts. Therefore, the number of green marbles in Jar 1 is (3/10) of N, and the number of blue marbles is (7/10) of N.Similarly, in Jar 2, the ratio of blue to green marbles is 6:4. That simplifies to 3:2, but I'll keep it as 6:4 for consistency. So, the total parts here are 6 + 4 = 10 parts as well. Therefore, the number of green marbles in Jar 2 is (4/10) of N, and the number of blue marbles is (6/10) of N.The problem states that the total number of green marbles in both jars is 80. So, the green marbles from Jar 1 plus the green marbles from Jar 2 equal 80. Let me write that as an equation:(3/10)N + (4/10)N = 80Combining the terms on the left side:(7/10)N = 80To find N, I can multiply both sides by 10/7:N = 80 * (10/7) = 800/7 ≈ 114.2857Wait, that's a fractional number of marbles, which doesn't make sense because marbles are whole objects. Hmm, maybe I made a mistake somewhere.Let me check my steps again. I set up the ratios correctly, right? Jar 1 has 3/10 green marbles, Jar 2 has 4/10 green marbles. Adding them gives 7/10 of N equals 80. So, N should be 800/7, which is approximately 114.2857. But since the number of marbles must be a whole number, maybe I need to reconsider.Perhaps the ratios are given in such a way that the total number of marbles in each jar is a multiple of 10. Let me think. If Jar 1 has a ratio of 7:3, then the total marbles must be a multiple of 10. Similarly, Jar 2 has a ratio of 6:4, which is also a multiple of 10. So, N must be a multiple of 10.But according to my calculation, N is 800/7, which is not a multiple of 10. That suggests that maybe my initial approach is incorrect.Wait, perhaps I should let the number of marbles in each jar be 10k, where k is an integer. That way, the ratios will hold perfectly. Let me try that.Let’s denote the number of marbles in each jar as 10k. Then, in Jar 1, the number of green marbles is 3k, and in Jar 2, the number of green marbles is 4k. The total green marbles are 3k + 4k = 7k. According to the problem, 7k = 80. So, k = 80/7 ≈ 11.4286.Again, this is not an integer. Hmm, this is confusing. Maybe the problem allows for fractional marbles? But that doesn't make sense in reality.Wait, perhaps the total number of marbles in each jar isn't necessarily the same multiple. Let me think differently. Let me denote the number of marbles in Jar 1 as N1 and in Jar 2 as N2. The problem says they have the same number of marbles, so N1 = N2 = N.In Jar 1, green marbles are (3/10)N, and in Jar 2, green marbles are (4/10)N. So, total green marbles are (3/10)N + (4/10)N = (7/10)N = 80. Therefore, N = 80 * (10/7) = 800/7 ≈ 114.2857.But since N must be an integer, maybe the problem is designed in such a way that we can still proceed with this fractional number, even though it's not practical in real life. Let me proceed with N = 800/7.So, the number of blue marbles in Jar 1 is (7/10)N = (7/10)*(800/7) = 80.Similarly, the number of blue marbles in Jar 2 is (6/10)N = (6/10)*(800/7) = (4800)/70 = 480/7 ≈ 68.5714.Now, the difference in the number of blue marbles between Jar 1 and Jar 2 is 80 - 68.5714 ≈ 11.4286.Since the answer choices are integers, I need to round this to the nearest whole number, which is 11.Wait, but 11 is one of the options, option A. So, is the answer 11?But let me double-check my calculations. Maybe I made a mistake in setting up the equations.Total green marbles: (3/10)N + (4/10)N = (7/10)N = 80 => N = 800/7.Blue marbles in Jar 1: (7/10)*(800/7) = 80.Blue marbles in Jar 2: (6/10)*(800/7) = (4800)/70 = 480/7 ≈ 68.5714.Difference: 80 - 68.5714 ≈ 11.4286, which is approximately 11.So, the answer should be 11, which is option A.But wait, another way to think about it: maybe the total number of marbles in each jar is 10k, and the total green marbles are 7k = 80, so k = 80/7, which is approximately 11.4286. But since k must be an integer, perhaps the problem is designed to have k as a fraction, and we just proceed with that.Alternatively, maybe I should consider that the total number of marbles in each jar is the same, but not necessarily a multiple of 10. Let me try that approach.Let’s denote the number of marbles in each jar as N.In Jar 1, the ratio of blue to green is 7:3, so the number of green marbles is (3/10)N.In Jar 2, the ratio of blue to green is 6:4, which simplifies to 3:2, so the number of green marbles is (2/5)N.Total green marbles: (3/10)N + (2/5)N = (3/10 + 4/10)N = (7/10)N = 80.So, N = 80 * (10/7) = 800/7 ≈ 114.2857.Again, same result. So, the number of blue marbles in Jar 1 is (7/10)N = 80, and in Jar 2 is (6/10)N = 68.5714.Difference is approximately 11.4286, so 11.Therefore, the answer is 11, option A.

💡First, I need to determine the total number of girls after the additional 465 girls joined the school. Initially, there are 632 girls, so adding 465 gives:632 girls + 465 girls = 1097 girlsNext, I know that there are 687 more girls than boys in the school. Let B represent the number of boys. According to the problem:1097 girls - B boys = 687To find the number of boys, I rearrange the equation:B boys = 1097 girls - 687Calculating this gives:B boys = 410Therefore, there are 410 boys in the school.

💡Alright, so I have this problem here: I need to find the coefficient of the third term when expanding the expression (1 - x)(1 + 2x)^5 and then arranging it in ascending powers of x. Hmm, okay. Let me think about how to approach this.First, I remember that when you have an expression like (a + b)^n, you can expand it using the binomial theorem. The binomial theorem says that (a + b)^n equals the sum from k=0 to n of C(n, k) * a^(n - k) * b^k, where C(n, k) is the combination of n things taken k at a time.In this case, the expression is (1 + 2x)^5. So, applying the binomial theorem here, I can write this as the sum from k=0 to 5 of C(5, k) * (1)^(5 - k) * (2x)^k. Simplifying that, since 1 raised to any power is just 1, it becomes the sum from k=0 to 5 of C(5, k) * (2x)^k.So, expanding that out, the terms would be:- For k=0: C(5, 0) * (2x)^0 = 1 * 1 = 1- For k=1: C(5, 1) * (2x)^1 = 5 * 2x = 10x- For k=2: C(5, 2) * (2x)^2 = 10 * 4x^2 = 40x^2- For k=3: C(5, 3) * (2x)^3 = 10 * 8x^3 = 80x^3- For k=4: C(5, 4) * (2x)^4 = 5 * 16x^4 = 80x^4- For k=5: C(5, 5) * (2x)^5 = 1 * 32x^5 = 32x^5So, putting it all together, (1 + 2x)^5 expands to 1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5.Now, the original expression is (1 - x) multiplied by this expansion. So, I need to multiply each term in the expansion by (1 - x). Let me write that out:(1 - x)(1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5)To multiply these, I'll distribute each term in the first parenthesis to each term in the second parenthesis. That means I'll have:1*(1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5) - x*(1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5)Let me compute each part separately.First, multiplying by 1:1*1 = 11*10x = 10x1*40x^2 = 40x^21*80x^3 = 80x^31*80x^4 = 80x^41*32x^5 = 32x^5So, that part is just the same as the expansion: 1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5.Now, subtracting x times each term:x*1 = xx*10x = 10x^2x*40x^2 = 40x^3x*80x^3 = 80x^4x*80x^4 = 80x^5x*32x^5 = 32x^6So, the second part is: x + 10x^2 + 40x^3 + 80x^4 + 80x^5 + 32x^6.Now, subtracting this from the first part:(1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5) - (x + 10x^2 + 40x^3 + 80x^4 + 80x^5 + 32x^6)Let me write this out term by term:1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5 - x - 10x^2 - 40x^3 - 80x^4 - 80x^5 - 32x^6Now, let's combine like terms.Starting with the constant term:1Next, the x terms:10x - x = 9xNext, the x^2 terms:40x^2 - 10x^2 = 30x^2Next, the x^3 terms:80x^3 - 40x^3 = 40x^3Next, the x^4 terms:80x^4 - 80x^4 = 0x^4Next, the x^5 terms:32x^5 - 80x^5 = -48x^5Finally, the x^6 term:-32x^6So, putting it all together, the expanded expression is:1 + 9x + 30x^2 + 40x^3 + 0x^4 - 48x^5 - 32x^6Simplifying, we can write it as:1 + 9x + 30x^2 + 40x^3 - 48x^5 - 32x^6Wait, but the problem mentions arranging it in ascending powers of x. So, the terms are already in ascending powers, starting from x^0 up to x^6.Now, the question asks for the coefficient of the third term. Let's count the terms:1st term: 1 (constant term, x^0)2nd term: 9x (x^1)3rd term: 30x^2 (x^2)So, the third term is 30x^2, and its coefficient is 30.Wait, but let me double-check my calculations to make sure I didn't make a mistake.Starting from the expansion:(1 + 2x)^5 = 1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5Multiplying by (1 - x):1*(1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5) = 1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5Minus x*(1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5) = x + 10x^2 + 40x^3 + 80x^4 + 80x^5 + 32x^6Subtracting:1 + 10x + 40x^2 + 80x^3 + 80x^4 + 32x^5 - x - 10x^2 - 40x^3 - 80x^4 - 80x^5 - 32x^6Combine like terms:1 + (10x - x) + (40x^2 - 10x^2) + (80x^3 - 40x^3) + (80x^4 - 80x^4) + (32x^5 - 80x^5) - 32x^6Which simplifies to:1 + 9x + 30x^2 + 40x^3 + 0x^4 - 48x^5 - 32x^6Yes, that looks correct. So, the third term is indeed 30x^2, so the coefficient is 30.I think that's the answer. Let me just make sure I didn't skip any steps or make any arithmetic errors.Wait, another way to approach this might be to find the first few terms without expanding everything. Since we only need the coefficient of the third term, maybe we can find it by considering the product up to x^2.Let me try that.We have (1 - x)(1 + 2x)^5. Let's denote (1 + 2x)^5 as A + Bx + Cx^2 + Dx^3 + Ex^4 + Fx^5, where A, B, C, D, E, F are the coefficients.Then, multiplying by (1 - x):(1 - x)(A + Bx + Cx^2 + Dx^3 + Ex^4 + Fx^5) = A + Bx + Cx^2 + Dx^3 + Ex^4 + Fx^5 - Ax - Bx^2 - Cx^3 - Dx^4 - Ex^5 - Fx^6Now, collecting terms up to x^2:Constant term: Ax term: Bx - Ax = (B - A)xx^2 term: Cx^2 - Bx^2 = (C - B)x^2So, the third term (x^2 term) is (C - B)x^2, so its coefficient is C - B.From the expansion of (1 + 2x)^5, we have:A = C(5, 0)*(2x)^0 = 1B = C(5, 1)*(2x)^1 = 5*2 = 10C = C(5, 2)*(2x)^2 = 10*4 = 40So, C - B = 40 - 10 = 30Therefore, the coefficient of the third term is 30.Yes, that's consistent with my earlier result. So, I'm confident that the answer is 30.**Final Answer**The coefficient of the third term is boxed{30}.

💡Okay, so I need to find the maximum value of the function f(x) = |x + 2017| - |x - 2016|. Hmm, absolute value functions can sometimes be tricky because they change their behavior depending on whether the expression inside is positive or negative. I remember that to handle absolute values, it's often helpful to break the problem into different cases based on the critical points where the expressions inside the absolute values change sign.First, let me identify those critical points. The expressions inside the absolute values are (x + 2017) and (x - 2016). These will change sign when x + 2017 = 0 and x - 2016 = 0, respectively. Solving these:1. x + 2017 = 0 ⇒ x = -20172. x - 2016 = 0 ⇒ x = 2016So, the critical points are at x = -2017 and x = 2016. These points divide the real number line into three intervals:1. x < -20172. -2017 ≤ x < 20163. x ≥ 2016I'll need to analyze each interval separately to determine the behavior of f(x) in each case.**Case 1: x < -2017**In this interval, both (x + 2017) and (x - 2016) are negative because x is less than both -2017 and 2016. Therefore, the absolute values can be rewritten without the absolute value signs by taking the negative of the expressions inside.So, |x + 2017| = -(x + 2017) and |x - 2016| = -(x - 2016). Therefore, f(x) becomes:f(x) = -(x + 2017) - (-(x - 2016)) = -x - 2017 + x - 2016 = (-x + x) + (-2017 - 2016) = 0 - 4033 = -4033So, for all x < -2017, f(x) is a constant -4033.**Case 2: -2017 ≤ x < 2016**In this interval, (x + 2017) is non-negative because x is greater than or equal to -2017, but (x - 2016) is still negative because x is less than 2016. Therefore, |x + 2017| = x + 2017 and |x - 2016| = -(x - 2016).So, f(x) becomes:f(x) = (x + 2017) - (-(x - 2016)) = x + 2017 + x - 2016 = 2x + (2017 - 2016) = 2x + 1This is a linear function with a slope of 2, which means it's increasing as x increases. Therefore, in this interval, the maximum value of f(x) occurs at the right endpoint, which is x = 2016.Let me compute f(2016):f(2016) = 2*(2016) + 1 = 4032 + 1 = 4033So, at x = 2016, f(x) = 4033.**Case 3: x ≥ 2016**In this interval, both (x + 2017) and (x - 2016) are positive because x is greater than or equal to 2016, which is greater than -2017. Therefore, both absolute values can be removed without changing the sign:f(x) = (x + 2017) - (x - 2016) = x + 2017 - x + 2016 = (x - x) + (2017 + 2016) = 0 + 4033 = 4033So, for all x ≥ 2016, f(x) is a constant 4033.**Putting it all together:**- For x < -2017, f(x) = -4033- For -2017 ≤ x < 2016, f(x) increases from f(-2017) to f(2016) = 4033- For x ≥ 2016, f(x) = 4033Therefore, the maximum value of f(x) is 4033, which occurs for all x ≥ 2016.I should double-check my calculations to make sure I didn't make any mistakes.In Case 1, when x < -2017, both expressions inside the absolute values are negative, so taking the negative of both gives f(x) = -4033. That seems correct.In Case 2, when -2017 ≤ x < 2016, f(x) simplifies to 2x + 1, which is indeed increasing. Plugging in x = 2016 gives 2*2016 + 1 = 4033, which matches with Case 3.In Case 3, when x ≥ 2016, both expressions are positive, so f(x) simplifies to 4033, which is consistent.Therefore, my conclusion is that the maximum value of f(x) is 4033.

💡Okay, so I have this problem here: If ( x ) is real and ( 4y^2 + 4xy + x + 6 = 0 ), then I need to find the complete set of values of ( x ) for which ( y ) is real. The options are given from A to E.First, I remember that for a quadratic equation in terms of ( y ), the discriminant must be non-negative for ( y ) to be real. That makes sense because if the discriminant is negative, the solutions would be complex, not real. So, I need to treat this equation as a quadratic in ( y ) and find the discriminant.Let me write the equation again: ( 4y^2 + 4xy + (x + 6) = 0 ). Comparing this to the standard quadratic form ( ay^2 + by + c = 0 ), I can see that ( a = 4 ), ( b = 4x ), and ( c = x + 6 ).Now, the discriminant ( D ) is given by ( D = b^2 - 4ac ). Plugging in the values, I get:[D = (4x)^2 - 4 times 4 times (x + 6)]Calculating each part:- ( (4x)^2 = 16x^2 )- ( 4 times 4 = 16 )- ( 16 times (x + 6) = 16x + 96 )So, putting it all together:[D = 16x^2 - (16x + 96) = 16x^2 - 16x - 96]Hmm, I can factor out a 16 from the first two terms:[D = 16(x^2 - x) - 96]But maybe it's better to factor the entire expression. Let me see:[16x^2 - 16x - 96 = 16(x^2 - x - 6)]Now, I need to factor ( x^2 - x - 6 ). Looking for two numbers that multiply to -6 and add to -1. Hmm, 2 and -3 work because 2 * (-3) = -6 and 2 + (-3) = -1.So, ( x^2 - x - 6 = (x - 3)(x + 2) ).Therefore, the discriminant becomes:[D = 16(x - 3)(x + 2)]For ( y ) to be real, the discriminant must be greater than or equal to zero:[16(x - 3)(x + 2) geq 0]Since 16 is a positive constant, it doesn't affect the inequality's direction. So, we can simplify to:[(x - 3)(x + 2) geq 0]Now, I need to find the values of ( x ) for which this product is non-negative. To do this, I'll analyze the critical points where each factor is zero: ( x = 3 ) and ( x = -2 ). These points divide the number line into three intervals:1. ( x < -2 )2. ( -2 < x < 3 )3. ( x > 3 )I'll test each interval:1. **For ( x < -2 )**: Let's pick ( x = -3 ). - ( x - 3 = -6 ) (negative) - ( x + 2 = -1 ) (negative) - Product: ( (-6)(-1) = 6 ) (positive)2. **For ( -2 < x < 3 )**: Let's pick ( x = 0 ). - ( x - 3 = -3 ) (negative) - ( x + 2 = 2 ) (positive) - Product: ( (-3)(2) = -6 ) (negative)3. **For ( x > 3 )**: Let's pick ( x = 4 ). - ( x - 3 = 1 ) (positive) - ( x + 2 = 6 ) (positive) - Product: ( (1)(6) = 6 ) (positive)So, the product ( (x - 3)(x + 2) ) is non-negative in the intervals ( x leq -2 ) and ( x geq 3 ). At the critical points ( x = -2 ) and ( x = 3 ), the product is zero, which satisfies the inequality ( geq 0 ).Therefore, the complete set of values of ( x ) for which ( y ) is real is ( x leq -2 ) or ( x geq 3 ).Looking back at the options:A. ( x leq -2 ) or ( x geq 3 )B. ( x leq 2 ) or ( x geq 3 )C. ( x leq -3 ) or ( x geq 2 )D. ( -3 leq x leq 2 )E. ( -2 leq x leq 3 )So, the correct answer is option A.

💡Okay, so I have this problem with three complex numbers: z₁ = -1 + 2i, z₂ = 1 - i, and z₃ = 3 - 2i. These correspond to points A, B, and C respectively. The question says that the vector OC is equal to x times vector OA plus y times vector OB. I need to find the value of x + y.Hmm, let me break this down. First, complex numbers can be represented as points in the complex plane, right? So z₁ = -1 + 2i corresponds to the point A(-1, 2), z₂ = 1 - i corresponds to point B(1, -1), and z₃ = 3 - 2i corresponds to point C(3, -2). So, vectors OA, OB, and OC are just the position vectors from the origin to points A, B, and C. That means vector OA is (-1, 2), vector OB is (1, -1), and vector OC is (3, -2). The equation given is vector OC = x * vector OA + y * vector OB. So, in terms of coordinates, that would translate to:(3, -2) = x*(-1, 2) + y*(1, -1)Which means:3 = -x + yand-2 = 2x - ySo now I have a system of two equations with two variables, x and y. I need to solve this system to find the values of x and y, and then add them together to get x + y.Let me write down the equations:1) 3 = -x + y2) -2 = 2x - yHmm, maybe I can solve this using substitution or elimination. Let me try elimination because the coefficients look manageable.If I add equation 1 and equation 2 together, the y terms might cancel out.Adding equation 1: 3 = -x + yand equation 2: -2 = 2x - ySo adding the left sides: 3 + (-2) = 1Adding the right sides: (-x + y) + (2x - y) = (-x + 2x) + (y - y) = x + 0 = xSo, 1 = xOkay, so x is 1. Now, plug this back into one of the original equations to find y. Let's use equation 1:3 = -x + ySubstitute x = 1:3 = -1 + yAdd 1 to both sides:3 + 1 = y4 = ySo y is 4. Therefore, x is 1 and y is 4, so x + y is 1 + 4 = 5.Wait, let me double-check to make sure I didn't make a mistake. Let's substitute x = 1 and y = 4 into equation 2:-2 = 2x - yPlug in x = 1 and y = 4:-2 = 2*1 - 4-2 = 2 - 4-2 = -2Yep, that works. So the solution is correct.So, the value of x + y is 5.**Final Answer**The value of ( x + y ) is boxed{5}.