answer：Okay, so I'm trying to figure out how long it will take Vasya to run up and down an escalator that's moving upwards. The problem says that Vasya runs twice as fast down as he does up. When the escalator isn't working, it takes him 6 minutes to go up and down. When the escalator is moving down, it takes him 13.5 minutes. I need to find out how long it will take him when the escalator is moving up.First, I think I need to define some variables. Let me call the length of the escalator "L." I guess it doesn't matter what unit I use for L since it will cancel out in the end. Let me also define Vasya's speed going up as "v." Then, since he runs twice as fast down, his speed going down would be "2v."When the escalator isn't working, it's just Vasya running up and down. So, the time it takes him to go up is L divided by his upward speed, which is L/v. Similarly, the time to go down is L divided by his downward speed, which is L/(2v). The total time is the sum of these two, and that's given as 6 minutes. So, I can write the equation:L/v + L/(2v) = 6Let me simplify this equation. Combining the terms, I get:(2L + L)/(2v) = 63L/(2v) = 6Then, solving for L/v:3L/(2v) = 6Multiply both sides by 2v:3L = 12vDivide both sides by 3:L = 4vSo, L is 4 times his upward speed. That might be useful later.Now, let's consider the case when the escalator is moving down. The escalator has its own speed, let's call it "e." When the escalator is moving down, Vasya is running up against the escalator's movement and running down with it.When he's going up, his effective speed relative to the ground is his running speed minus the escalator's speed. So, his upward speed becomes (v - e). When he's going down, his effective speed is his running speed plus the escalator's speed, so (2v + e).The time to go up is L/(v - e) and the time to go down is L/(2v + e). The total time is given as 13.5 minutes. So, the equation is:L/(v - e) + L/(2v + e) = 13.5But from earlier, we know that L = 4v. So, substituting that in:4v/(v - e) + 4v/(2v + e) = 13.5Let me simplify this equation. First, factor out the 4v:4v [1/(v - e) + 1/(2v + e)] = 13.5Let me compute the expression inside the brackets:1/(v - e) + 1/(2v + e) = [ (2v + e) + (v - e) ] / [ (v - e)(2v + e) ]= (3v) / [ (v - e)(2v + e) ]So, plugging this back into the equation:4v * (3v) / [ (v - e)(2v + e) ] = 13.512v² / [ (v - e)(2v + e) ] = 13.5Let me write this as:12v² = 13.5 * (v - e)(2v + e)Let me expand the right side:(v - e)(2v + e) = 2v² + ve - 2ve - e² = 2v² - ve - e²So, the equation becomes:12v² = 13.5*(2v² - ve - e²)Let me divide both sides by 13.5 to make it simpler:12v² / 13.5 = 2v² - ve - e²Simplify 12/13.5:12 / 13.5 = 8/9So,(8/9)v² = 2v² - ve - e²Let me bring all terms to one side:2v² - ve - e² - (8/9)v² = 0(2 - 8/9)v² - ve - e² = 0(10/9)v² - ve - e² = 0Multiply through by 9 to eliminate fractions:10v² - 9ve - 9e² = 0This is a quadratic equation in terms of e. Let me write it as:9e² + 9ve - 10v² = 0Wait, actually, I think I made a sign error when moving terms. Let me double-check:From:12v² = 13.5*(2v² - ve - e²)Divide both sides by 13.5:12v² / 13.5 = 2v² - ve - e²Which is:(8/9)v² = 2v² - ve - e²Then, moving all terms to the left:(8/9)v² - 2v² + ve + e² = 0Which is:(8/9 - 18/9)v² + ve + e² = 0(-10/9)v² + ve + e² = 0Multiply through by 9:-10v² + 9ve + 9e² = 0Multiply both sides by -1:10v² - 9ve - 9e² = 0Yes, that's correct.So, quadratic equation in e:10v² - 9ve - 9e² = 0Let me write it as:9e² + 9ve - 10v² = 0Wait, no, that would be:-9e² + 9ve + 10v² = 0Wait, I'm getting confused. Let me just use the standard quadratic form.Let me write it as:9e² + 9ve - 10v² = 0Wait, no:From 10v² - 9ve - 9e² = 0It's:-9e² -9ve +10v² =0Multiply both sides by -1:9e² +9ve -10v² =0Yes, that's correct.So, quadratic equation in e:9e² +9ve -10v² =0Let me solve for e using quadratic formula.e = [ -9v ± sqrt( (9v)^2 - 4*9*(-10v²) ) ] / (2*9)Compute discriminant:(9v)^2 -4*9*(-10v²) =81v² +360v²=441v²So,e = [ -9v ± sqrt(441v²) ] / 18e = [ -9v ±21v ] /18So, two solutions:e = (-9v +21v)/18 =12v/18=2v/3e=(-9v -21v)/18=-30v/18=-5v/3Since speed can't be negative, we take e=2v/3So, the escalator speed is (2/3)v.Now, we need to find the time when the escalator is moving up. So, when the escalator is moving up, Vasya is going up with the escalator and down against it.When going up, his effective speed is (v + e). When going down, his effective speed is (2v - e).So, time to go up: L/(v + e)Time to go down: L/(2v - e)Total time: L/(v + e) + L/(2v - e)Again, we know L=4v, so substitute:4v/(v + e) +4v/(2v - e)Substitute e=2v/3:4v/(v + 2v/3) +4v/(2v -2v/3)Simplify denominators:v +2v/3=5v/32v -2v/3=4v/3So,4v/(5v/3) +4v/(4v/3)=4v*(3/(5v)) +4v*(3/(4v))= (12v)/(5v) + (12v)/(4v)=12/5 +12/4=2.4 +3=5.4 minutesConvert to seconds:5.4*60=324 secondsSo, the answer is 324 seconds.

answer：Okay, so I need to figure out how many books Mike bought at the yard sale. Let's see, he started with 35 books and now has 91 books. Hmm, so I think I need to find out the difference between these two numbers. First, I'll write down the numbers:- Starting number of books: 35- Current number of books: 91Now, to find out how many books he bought, I should subtract the starting number from the current number. That makes sense because if he had 35 and now has 91, the difference will tell me how many he added.So, I'll set up the subtraction:91 - 35Let me do the subtraction step by step to make sure I don't make a mistake. Starting from the units place:1 (from 91) minus 5 (from 35) – wait, I can't subtract 5 from 1. So I need to borrow from the tens place. The 9 in the tens place becomes 8, and the 1 in the units place becomes 11. Now, 11 minus 5 is 6.Next, the tens place:8 (after borrowing) minus 3 is 5.So, putting it together, 91 minus 35 equals 56.Therefore, Mike bought 56 books at the yard sale.

answer：First, I need to determine the number of lollipops Joann ate on the first day. Let's denote this number as ( a ).According to the problem, each subsequent day she ate 8 more lollipops than the previous day. Therefore, the number of lollipops eaten each day can be expressed as:- Day 1: ( a )- Day 2: ( a + 8 )- Day 3: ( a + 16 )- Day 4: ( a + 24 )- Day 5: ( a + 32 )Next, I'll calculate the total number of lollipops eaten over the five days by summing these expressions:[a + (a + 8) + (a + 16) + (a + 24) + (a + 32) = 5a + 80]The problem states that the total number of lollipops eaten is 150. So, I'll set up the equation:[5a + 80 = 150]Solving for ( a ):[5a = 150 - 80 5a = 70 a = 14]Now that I know the number of lollipops eaten on the first day (( a = 14 )), I can find the number eaten on the third day:[a + 16 = 14 + 16 = 30]Therefore, Joann ate 30 lollipops on the third day.

answer：Alright, so I've got this problem here where I need to find the intersection of two sets, M and N. Let me read it again to make sure I understand.M is defined as all x such that -1 is less than x, which is less than 5. So, M is the open interval from -1 to 5. That seems straightforward enough.N is a bit trickier. It's defined as all x such that x times (x minus 4) is greater than zero. Hmm, okay, so I need to solve this inequality: x(x - 4) > 0.I remember that to solve quadratic inequalities, it's helpful to find the critical points where the expression equals zero. So, let's set x(x - 4) equal to zero and solve for x.x(x - 4) = 0This gives us two solutions: x = 0 and x = 4. These are the points where the expression changes sign, so they divide the number line into intervals that I can test.The intervals are:1. From negative infinity to 02. From 0 to 43. From 4 to positive infinityNow, I need to test each interval to see where the product x(x - 4) is positive.Let's start with the first interval: negative infinity to 0. Let's pick a test point, say x = -1.Plugging into the expression: (-1)(-1 - 4) = (-1)(-5) = 5, which is positive. So, the expression is positive in this interval.Next, the interval from 0 to 4. Let's choose x = 2.Plugging in: 2(2 - 4) = 2(-2) = -4, which is negative. So, the expression is negative here.Lastly, the interval from 4 to positive infinity. Let's pick x = 5.Plugging in: 5(5 - 4) = 5(1) = 5, which is positive. So, the expression is positive in this interval as well.Therefore, the solution set for N is the union of the intervals where the expression is positive: (-∞, 0) and (4, ∞).Now, I need to find the intersection of M and N. M is (-1, 5), and N is (-∞, 0) ∪ (4, ∞). So, the intersection will be the overlap between these two sets.Let's break it down:- The overlap between (-1, 5) and (-∞, 0) is (-1, 0).- The overlap between (-1, 5) and (4, ∞) is (4, 5).So, combining these two overlapping intervals, the intersection M ∩ N is (-1, 0) ∪ (4, 5).Looking back at the options provided:A: (-1,0)B: (-1,0)∪(4,5)C: (0,4)D: (4,5)The correct answer should be option B, which matches our solution.I think I've got this right. I just need to make sure I didn't make any mistakes in solving the inequality or in finding the intersection. Let me double-check.For the inequality x(x - 4) > 0, the critical points are indeed 0 and 4. Testing intervals around these points correctly shows that the expression is positive in (-∞, 0) and (4, ∞). Then, intersecting this with M, which is (-1, 5), gives us the two intervals (-1, 0) and (4, 5). Yep, that seems correct.So, I'm confident that option B is the right answer.

answer：Okay, so I've got this problem here: there are 4 friends playing a video game online, and then 5 more players join them. Each player has a certain number of lives, and altogether, they have 27 lives. The question is asking how many lives each player has.Alright, let's break this down step by step. First, I need to figure out how many players there are in total. There are initially 4 friends, and then 5 more players join. So, I guess I need to add those together. That would be 4 plus 5, which equals 9 players in total.Now, the total number of lives is 27. So, if there are 9 players and 27 lives, I think I need to divide the total number of lives by the number of players to find out how many lives each player has. That makes sense because if the lives are distributed equally among all players, dividing should give me the number of lives per player.So, let's do that calculation: 27 divided by 9. Hmm, 9 times 3 is 27, so 27 divided by 9 is 3. That means each player has 3 lives.Wait a minute, is there any other way to interpret this problem? Could it be that the initial 4 friends have a different number of lives than the 5 new players? The problem says "each player had a certain number of lives," which makes me think that all players have the same number of lives. If they had different numbers, the problem would probably specify that.Also, the problem doesn't mention anything about lives being added or removed when the new players joined, so it's safe to assume that the total number of lives is just distributed equally among all players from the start.Let me double-check my calculations to make sure I didn't make a mistake. 4 friends plus 5 players is indeed 9 players. 27 divided by 9 is 3. Yep, that seems correct.Is there any other information I might be missing? The problem doesn't mention anything about lives being shared or anything like that, so I think my initial approach is the right one.In summary, there are 9 players in total, and they have 27 lives altogether. Dividing 27 by 9 gives me 3 lives per player. So, each player has 3 lives.

answer：Alright, so I need to add these two fractions: 3/7 and 9/12. Hmm, okay, let's see. I remember that when adding fractions, they need to have the same denominator. So, first, maybe I should simplify the fractions if possible.Looking at 9/12, I think I can simplify that. Both 9 and 12 are divisible by 3, right? So if I divide both numerator and denominator by 3, that should simplify it. Let me do that: 9 divided by 3 is 3, and 12 divided by 3 is 4. So, 9/12 simplifies to 3/4. Cool, that makes it easier.Now, I have 3/7 and 3/4. I need to add these two. To add fractions, they must have the same denominator. So, I need to find a common denominator for 7 and 4. The denominators are 7 and 4, and I need the least common denominator (LCD) for these two numbers. The multiples of 7 are 7, 14, 21, 28, 35, and so on. The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, etc. I see that 28 is the smallest common multiple. So, 28 will be our common denominator.Now, I need to convert both fractions to have 28 as the denominator. Starting with 3/7: to get from 7 to 28, I multiply by 4. So, I need to multiply both the numerator and denominator by 4. That gives me (3*4)/(7*4) = 12/28. Okay, that's the first fraction converted.Next, 3/4: to get from 4 to 28, I multiply by 7. So, I multiply both the numerator and denominator by 7. That gives me (3*7)/(4*7) = 21/28. Great, now both fractions have the same denominator.Now, I can add them together: 12/28 + 21/28. Since the denominators are the same, I just add the numerators. 12 + 21 equals 33. So, the sum is 33/28.I should check if this fraction can be simplified further. Let's see, 33 and 28. The factors of 33 are 1, 3, 11, 33, and the factors of 28 are 1, 2, 4, 7, 14, 28. The only common factor is 1, so 33/28 is already in its simplest form.Just to make sure I didn't make any mistakes, let me recap the steps:1. Simplified 9/12 to 3/4.2. Found the LCD of 7 and 4, which is 28.3. Converted both fractions to have 28 as the denominator: 12/28 and 21/28.4. Added the numerators: 12 + 21 = 33.5. Confirmed that 33/28 cannot be simplified further.Everything seems to check out. So, the sum of 3/7 and 9/12 is 33/28.