💡First, I need to determine the distance the train covers at its current average speed of ( V ) km/h in 50 minutes. Since speed is in km/h and time is in minutes, I'll convert 50 minutes to hours by dividing by 60. This gives me ( frac{50}{60} ) hours. The distance ( D ) can then be expressed as ( D = V times frac{50}{60} ).Next, I know that to reduce the journey time to 40 minutes, the train must travel at 60 km/h. Converting 40 minutes to hours, I get ( frac{40}{60} ) hours. Using the same distance formula, the distance ( D ) can also be expressed as ( D = 60 times frac{40}{60} ).Since the distance ( D ) remains the same in both scenarios, I can set the two expressions for ( D ) equal to each other: ( V times frac{50}{60} = 60 times frac{40}{60} ).To solve for ( V ), I'll multiply both sides of the equation by 60 to eliminate the fractions: ( V times 50 = 60 times 40 ). This simplifies to ( V = frac{60 times 40}{50} ).Finally, calculating the value gives me ( V = frac{2400}{50} = 48 ) km/h. Therefore, the current average speed of the train is 48 km/h.

💡First, I need to determine the one-time sitting fee for Sam's Picture Emporium. I'll start by calculating the total cost for 12 sheets of pictures at both companies.For John's Photo World:- The cost per sheet is 2.75.- The one-time sitting fee is 125.- The total cost for 12 sheets is calculated as: [ 2.75 times 12 + 125 = 33 + 125 = 158 ]For Sam's Picture Emporium:- The cost per sheet is 1.50.- Let ( S ) represent the one-time sitting fee.- The total cost for 12 sheets is: [ 1.50 times 12 + S = 18 + S ]Since the total costs are equal for 12 sheets:[158 = 18 + S]To find ( S ), I'll subtract 18 from both sides:[S = 158 - 18 = 140]Therefore, the one-time sitting fee for Sam's Picture Emporium is 140.

💡First, I need to determine the number of cellphones Miley bought. Each cellphone costs 800, and there's a 5% discount for buying more than one.I'll start by calculating the total cost without the discount by multiplying the number of cellphones, ( n ), by 800.Next, I'll apply the 5% discount to the total cost. A 5% discount means Miley pays 95% of the total cost.Setting up the equation, the discounted total cost equals 1520. So, ( 0.95 times 800n = 1520 ).Solving for ( n ), I'll divide both sides by 760 to find that ( n = 2 ).Therefore, Miley bought 2 cellphones.

💡First, I need to determine how much money John spends on the 15 packets of chips. Since each packet of chips costs 2, the total cost for 15 packets is 15 multiplied by 2, which equals 30.Next, I'll calculate how much money John has left after purchasing the chips. His total budget is 45, and he spent 30 on chips. Subtracting these amounts gives 15 remaining for corn chips.John can buy 10 packets of corn chips with the remaining 15. To find the price per packet of corn chips, I'll divide the total amount he can spend on corn chips by the number of packets he can buy. Dividing 15 by 10 packets results in 1.50 per packet.Therefore, the price per packet of corn chips is 1.50.

💡Okay, so I have this problem here about a function y = f(x) that's an odd function with a period of 2, defined on all real numbers. I need to find f(2017) and choose the correct answer from the options given: A is -2017, B is 0, C is 1, and D is 2017.Hmm, let me recall what it means for a function to be odd and periodic. An odd function satisfies the property f(-x) = -f(x) for all x. That means the graph of the function is symmetric about the origin. On the other hand, a periodic function with period T satisfies f(x + T) = f(x) for all x. In this case, the period is 2, so f(x + 2) = f(x) for all x.So, the function f is both odd and periodic with period 2. I need to find f(2017). Let me think about how to approach this.First, since the function has a period of 2, I can use this property to simplify f(2017). The idea is to subtract multiples of the period from 2017 until I get a number within one period, say between 0 and 2, or maybe between -1 and 1, depending on how I approach it.Let me calculate how many times 2 goes into 2017. So, 2017 divided by 2 is 1008.5. That means 2017 = 2 * 1008 + 1. So, 2017 is 1008 full periods plus 1. Therefore, f(2017) = f(1) because adding or subtracting full periods doesn't change the value of the function.So now, I just need to find f(1). But wait, the function is also odd. So, f(-x) = -f(x). Let me see if I can use this property to find f(1).Since f is periodic with period 2, f(1) = f(1 + 2k) for any integer k. But I don't know if that helps directly. Maybe I can relate f(1) to f(-1) using the odd function property.So, f(-1) = -f(1). But since the function is periodic, f(-1) is equal to f(-1 + 2) because adding the period 2 doesn't change the value. So, f(-1) = f(1). Therefore, f(-1) = f(1).But from the odd function property, f(-1) = -f(1). So, putting these together: f(1) = -f(1). That implies that 2f(1) = 0, so f(1) = 0.Therefore, f(2017) = f(1) = 0.Wait, let me double-check that. So, f is odd, so f(-x) = -f(x). Also, f is periodic with period 2, so f(x + 2) = f(x). So, f(-1) = f(-1 + 2) = f(1). But f(-1) is also equal to -f(1) because it's odd. So, f(1) = -f(1), which means f(1) must be zero. That seems correct.So, f(2017) = f(1) = 0. Therefore, the answer should be B: 0.Is there another way to think about this? Maybe by considering specific examples of functions that are both odd and periodic with period 2. For example, the sine function is odd and periodic, but its period is 2π, not 2. So, maybe a scaled version? Let's see, if I have a function like sin(πx), then the period would be 2 because sin(πx + 2π) = sin(π(x + 2)) = sin(πx + 2π) = sin(πx). So, sin(πx) is an odd function with period 2.Let's compute f(1) for this function. sin(π*1) = sin(π) = 0. Similarly, sin(π*(-1)) = sin(-π) = -sin(π) = 0. So, in this case, f(1) = 0. That's consistent with what I found earlier.Another example: suppose f(x) is a triangle wave that's odd and periodic with period 2. At x = 1, which is the midpoint of the period, the function would cross zero. So, f(1) = 0 again.Therefore, regardless of the specific function, as long as it's odd and periodic with period 2, f(1) must be zero. Hence, f(2017) is zero.I think that's solid. I can't think of a case where f(1) wouldn't be zero given these properties. So, yeah, the answer is B: 0.**Final Answer**boxed{B}

💡First, I need to understand the given ratio of blue to yellow to green to red disks, which is 3:7:8:4. This means that for every 3 blue disks, there are 7 yellow, 8 green, and 4 red disks.Next, I'll calculate the total number of parts in the ratio by adding them together: 3 + 7 + 8 + 4 = 22 parts.Since the total number of disks in the bag is 176, I'll determine how many disks each part represents by dividing the total number of disks by the total number of parts: 176 ÷ 22 = 8 disks per part.Now, I'll find the number of green and blue disks:- Green disks: 8 parts × 8 disks per part = 64 disks- Blue disks: 3 parts × 8 disks per part = 24 disksFinally, I'll calculate how many more green disks there are than blue disks by subtracting the number of blue disks from the number of green disks: 64 - 24 = 40.Therefore, there are 40 more green disks than blue disks in the bag.