💡First, I need to determine the number of men on the bus. Since two-thirds of the 48 passengers are women, I can calculate the number of women by multiplying 48 by 2/3.Next, I'll subtract the number of women from the total number of passengers to find out how many men are on the bus.After finding the total number of men, I'll calculate how many of them are standing by taking one-eighth of the total number of men.Finally, to find the number of men who are seated, I'll subtract the number of standing men from the total number of men.

💡First, I need to determine how much money each child receives based on the given ratio of 2:3:4. The total amount to be distributed is 4500.The sum of the ratio parts is 2 + 3 + 4, which equals 9 parts.Next, I'll calculate each child's share:- Child A's share is (2/9) of 4500.- Child B's share is (3/9) of 4500.- Child C's share is (4/9) of 4500.Calculating these amounts:- Child A's share = (2/9) * 4500 = 1000- Child B's share = (3/9) * 4500 = 1500- Child C's share = (4/9) * 4500 = 2000Now, focusing on Child B, who receives 1500 and invests it at an annual interest rate of 4%.To find the interest earned after one year:Interest = Principal amount * Interest rate * TimeInterest = 1500 * 4% * 1 yearInterest = 1500 * 0.04 * 1Interest = 60Finally, adding the interest to the principal amount gives the total value of Child B's share after one year:Total value = Principal amount + InterestTotal value = 1500 + 60Total value = 1560

💡Okay, so I have this problem here: I need to find the monotonic decreasing interval of the function y = e^x / x. The options are A: (-∞,1], B: (1,+∞], C: (0,1], and D: (-∞,0) and (0,1]. Hmm, let me try to figure this out step by step.First, I remember that to find where a function is increasing or decreasing, I need to look at its derivative. If the derivative is positive, the function is increasing, and if it's negative, the function is decreasing. So, I should start by finding the derivative of y with respect to x.The function is y = e^x / x. To find the derivative, I can use the quotient rule. The quotient rule says that if I have a function f(x) = g(x)/h(x), then the derivative f’(x) is [g’(x)h(x) - g(x)h’(x)] / [h(x)]². Applying that here, let me set g(x) = e^x and h(x) = x.So, g’(x) would be the derivative of e^x, which is still e^x. And h’(x) is the derivative of x, which is 1. Plugging these into the quotient rule formula, I get:f’(x) = [e^x * x - e^x * 1] / x²Simplifying the numerator, that's e^x(x - 1). So, f’(x) = e^x(x - 1) / x².Okay, so the derivative is e^x times (x - 1) divided by x squared. Now, I need to find where this derivative is negative because that's where the function is decreasing.Let me think about the sign of each part of the derivative. e^x is always positive, right? Because the exponential function is always positive for any real x. So, e^x is positive. Then, x squared is also always positive except when x is zero, but x can't be zero in the original function because we're dividing by x. So, x squared is positive for all x ≠ 0.That means the sign of the derivative f’(x) depends entirely on the sign of (x - 1). Because e^x and x² are always positive, the whole expression will be positive when (x - 1) is positive and negative when (x - 1) is negative.So, let's solve when (x - 1) is negative. That happens when x < 1. So, f’(x) is negative when x < 1. But wait, I need to consider the domain of the original function. The original function y = e^x / x is undefined at x = 0 because division by zero isn't allowed. So, the domain is all real numbers except x = 0.Therefore, the function is decreasing when x < 1, but we have to exclude x = 0. So, the decreasing intervals are (-∞, 0) and (0, 1]. Let me check the options. Option D says (-∞,0) and (0,1], which matches what I found. So, I think D is the correct answer.But just to make sure I didn't make a mistake, let me double-check my derivative. Starting again, y = e^x / x. Using the quotient rule, derivative is [e^x * x - e^x * 1] / x², which simplifies to e^x(x - 1)/x². Yep, that's correct. So, the derivative is negative when x - 1 is negative, which is x < 1. And since x can't be zero, the intervals are (-∞,0) and (0,1]. So, yeah, D is the right choice.I guess another way to think about it is to consider the behavior of the function. For x approaching negative infinity, e^x approaches zero, so the function approaches zero from the negative side because x is negative. As x approaches zero from the negative side, e^x approaches 1, so y approaches -∞. Then, as x approaches zero from the positive side, y approaches +∞. After x = 1, the function starts increasing again because the derivative becomes positive. So, the function decreases from negative infinity up to x = 0, skips x = 0, and then continues decreasing from just above zero up to x = 1. After that, it starts increasing. So, that confirms that the decreasing intervals are indeed (-∞,0) and (0,1].I think I'm confident now that D is the correct answer.

💡First, I need to understand the discount offer: for every nine windows purchased, two additional windows are given for free. This means that for every nine windows bought, the customer effectively receives eleven windows.Next, I'll calculate the cost if Dave and Doug purchase separately:- **Dave needs 10 windows**: - For every nine windows purchased, he gets two free. So, for 10 windows, he needs to buy nine windows to get two free, totaling eleven windows. However, he only needs ten, so he would pay for nine windows. - Cost for Dave: 9 windows × 100 = 900- **Doug needs 9 windows**: - For nine windows, he gets two free, but since he only needs nine, he would pay for seven windows (because two are free). - Cost for Doug: 7 windows × 100 = 700- **Total cost if purchased separately**: 900 + 700 = 1,600Now, I'll calculate the cost if they purchase together:- **Combined, they need 19 windows**: - For every nine windows purchased, they get two free. So, for 19 windows, they need to buy 17 windows (since two sets of nine windows would give four free windows, totaling 18 windows). However, they need one more window, so they would pay for 17 windows. - Cost for joint purchase: 17 windows × 100 = 1,700Finally, I'll determine the savings:- **Savings**: 1,600 (separate purchase) - 1,700 (joint purchase) = -100This indicates that purchasing together actually costs them 100 more, so there is no saving. Therefore, the savings amount is 0.

💡First, we express ( x ) in terms of ( y ), ( u ), and ( v ) using the division algorithm:[x = uy + v]where ( 0 leq v < y ).Next, we consider the expression ( x + 3uy ):[x + 3uy = (uy + v) + 3uy = 4uy + v]When ( 4uy + v ) is divided by ( y ), the term ( 4uy ) is divisible by ( y ) and leaves no remainder. The remainder is therefore determined by ( v ).Thus, the remainder when ( x + 3uy ) is divided by ( y ) is ( v ).

💡Okay, so I have this problem: I need to find the sum of all possible positive integer values of ( b ) such that the quadratic equation ( 2x^2 + 5x + b = 0 ) has rational roots. Hmm, let me think about how to approach this.First, I remember that for a quadratic equation ( ax^2 + bx + c = 0 ) to have rational roots, the discriminant must be a perfect square. The discriminant is given by ( Delta = b^2 - 4ac ). If ( Delta ) is a perfect square, then the roots are rational because the square root of a perfect square is an integer, which makes the entire expression under the square root in the quadratic formula rational.In this problem, the quadratic equation is ( 2x^2 + 5x + b = 0 ). So, comparing this to the standard form ( ax^2 + bx + c = 0 ), I can see that ( a = 2 ), ( b = 5 ), and ( c = b ) (wait, that's confusing because the coefficient ( b ) in the standard form is 5, but in this equation, the constant term is also called ( b ). Maybe I should clarify that to avoid confusion. Let me denote the constant term as ( k ) instead. So, the equation becomes ( 2x^2 + 5x + k = 0 ), where ( k ) is the positive integer we're trying to find. That might help prevent confusion with the coefficient ( b ) in the standard form.)So, with that clarification, ( a = 2 ), ( b = 5 ), and ( c = k ). Now, the discriminant ( Delta ) is ( b^2 - 4ac ), which in this case is ( 5^2 - 4 times 2 times k ). Calculating that, ( 5^2 = 25 ), and ( 4 times 2 times k = 8k ). So, ( Delta = 25 - 8k ).For the roots to be rational, ( Delta ) must be a perfect square. Also, since ( k ) is a positive integer, ( Delta ) must be non-negative because you can't take the square root of a negative number in the set of real numbers. Therefore, ( 25 - 8k geq 0 ). Solving this inequality for ( k ):( 25 - 8k geq 0 )Subtract 25 from both sides:( -8k geq -25 )Divide both sides by -8, remembering to reverse the inequality sign when dividing by a negative number:( k leq frac{25}{8} )Calculating ( frac{25}{8} ), which is approximately 3.125. Since ( k ) is a positive integer, the possible values of ( k ) are 1, 2, and 3.Now, I need to check each of these values to see if ( Delta = 25 - 8k ) is a perfect square.1. For ( k = 1 ): ( Delta = 25 - 8(1) = 25 - 8 = 17 ) 17 is not a perfect square because 4^2 = 16 and 5^2 = 25, so 17 is in between and not a perfect square.2. For ( k = 2 ): ( Delta = 25 - 8(2) = 25 - 16 = 9 ) 9 is a perfect square since 3^2 = 9.3. For ( k = 3 ): ( Delta = 25 - 8(3) = 25 - 24 = 1 ) 1 is a perfect square since 1^2 = 1.So, the values of ( k ) that make the discriminant a perfect square are 2 and 3. Therefore, the possible positive integer values of ( b ) (which we've been denoting as ( k )) are 2 and 3.The problem asks for the sum of all such possible values of ( b ). So, adding them together:( 2 + 3 = 5 )Therefore, the sum of all possible positive integer values of ( b ) is 5.Wait, let me just double-check my steps to make sure I didn't make any mistakes.1. I identified the discriminant correctly as ( 25 - 8k ).2. I set up the inequality ( 25 - 8k geq 0 ) and solved for ( k ), getting ( k leq 3.125 ), so ( k = 1, 2, 3 ).3. I checked each ( k ) value: - ( k = 1 ): ( Delta = 17 ), not a perfect square. - ( k = 2 ): ( Delta = 9 ), perfect square. - ( k = 3 ): ( Delta = 1 ), perfect square.4. Sum of valid ( k ) values: 2 + 3 = 5.Everything seems to check out. I don't think I missed any steps or made any calculation errors. So, I'm confident that the sum is indeed 5.**Final Answer**The sum of all possible positive integer values of ( b ) is boxed{5}.