answer：Alright, so I've got this problem here about two construction teams, Team A and Team B, building different sections of a road. Team A has to build a 9-kilometer section, and Team B has to build a 12-kilometer section. The problem says that Team B builds 1 kilometer more per month than Team A and finishes half a month earlier than Team A. I need to figure out which equation correctly represents this situation if Team A builds x kilometers per month.Okay, let's break this down step by step. First, Team A's rate is x kilometers per month. Since Team B builds 1 kilometer more per month than Team A, Team B's rate must be x + 1 kilometers per month. That makes sense.Now, I need to find out how long it takes each team to finish their respective sections. Time is equal to the amount of work divided by the rate, right? So for Team A, the time to build 9 kilometers would be 9 divided by x, which is 9/x months. Similarly, for Team B, the time to build 12 kilometers would be 12 divided by (x + 1), so that's 12/(x + 1) months.The problem also states that Team B finishes half a month earlier than Team A. So, the time it takes Team A minus the time it takes Team B should equal half a month. In equation form, that would be:Time for Team A - Time for Team B = 1/2 monthSubstituting the expressions I found earlier, that becomes:9/x - 12/(x + 1) = 1/2Hmm, that looks like option A. But wait, let me double-check to make sure I didn't mix up the teams. Team B is faster, so their time should be less than Team A's time. If I subtract Team B's time from Team A's time, I should get a positive number, which is 1/2 month. So, yes, 9/x - 12/(x + 1) = 1/2 seems correct.Just to be thorough, let me look at the other options:Option B: 12/(x + 1) - 9/x = 1/2This would mean Team B's time minus Team A's time equals half a month, which would imply Team B takes longer, which contradicts the problem statement. So B is out.Option C: 9/(x + 1) - 12/x = 1/2This seems to mix up the rates. Team A's rate is x, so their time should be 9/x, not 9/(x + 1). Similarly, Team B's time should be 12/(x + 1), not 12/x. So C doesn't make sense.Option D: 12/x - 9/(x + 1) = 1/2Again, this mixes up the rates. Team B's rate is x + 1, so their time should be 12/(x + 1), not 12/x. Team A's time is 9/x, not 9/(x + 1). So D is also incorrect.Therefore, after checking all the options, option A is the correct equation that represents the given situation.

answer：First, I need to determine the total number of letters in the word. The word is "BA₁B₁A₂B₂A₃," which consists of 6 letters.Next, I should identify how many of these letters are identical. There are three A's (A₁, A₂, A₃) and two B's (B₁, B₂). Even though the A's and B's are labeled with subscripts, they are considered distinct because of these subscripts.Since all the letters are unique due to their subscripts, this problem becomes a permutation of 6 distinct items.The formula for the number of permutations of ( n ) distinct items is ( n! ).Applying this formula to the 6 letters:[6! = 720]Therefore, there are 720 different ways to arrange the letters of the word.

answer：First, I need to determine the mean of the five numbers. The mean is calculated by dividing the sum of the numbers by the number of numbers. Given that the sum of the five numbers is one-third, I can find the mean by dividing one-third by five.Next, I need to compute the sum of this mean and the original sum. This involves adding the mean, which I have just calculated, to the initial sum of one-third.Finally, I will express the result as a common fraction to provide the final answer.

answer：Okay, so I need to find all values of ( p ) such that the numbers ( 9p + 10 ), ( 3p ), and ( |p - 8| ) form the first, second, and third terms of a geometric progression. Hmm, let me recall what a geometric progression is. In a geometric progression, each term after the first is found by multiplying the previous term by a constant called the common ratio. So, if I have three terms ( a ), ( b ), and ( c ) in a geometric progression, then ( b = a times r ) and ( c = b times r ), where ( r ) is the common ratio. This also means that ( b^2 = a times c ). So, applying this to the given terms, I can set up the equation ( (3p)^2 = (9p + 10) times |p - 8| ). That simplifies to ( 9p^2 = (9p + 10) times |p - 8| ). Now, the absolute value function ( |p - 8| ) complicates things a bit because it can split into two cases: one where ( p - 8 ) is non-negative, and another where it's negative. Let me consider the first case where ( p geq 8 ). In this case, ( |p - 8| = p - 8 ). Substituting this into the equation, I get ( 9p^2 = (9p + 10)(p - 8) ). Expanding the right side, I have ( 9p^2 = 9p^2 - 72p + 10p - 80 ), which simplifies to ( 9p^2 = 9p^2 - 62p - 80 ). Subtracting ( 9p^2 ) from both sides gives ( 0 = -62p - 80 ). Solving for ( p ), I get ( 62p = -80 ), so ( p = -frac{80}{62} = -frac{40}{31} ). But wait, in this case, we assumed ( p geq 8 ), and ( -frac{40}{31} ) is definitely not greater than or equal to 8. So, this solution doesn't work here. Alright, moving on to the second case where ( p < 8 ). Here, ( |p - 8| = 8 - p ). Plugging this into the equation, I have ( 9p^2 = (9p + 10)(8 - p) ). Expanding the right side, it becomes ( 9p^2 = 72p - 9p^2 + 80 - 10p ). Combining like terms, that's ( 9p^2 = 62p + 80 - 9p^2 ). Bringing all terms to one side, I get ( 18p^2 - 62p - 80 = 0 ). Now, I need to solve this quadratic equation. Using the quadratic formula ( p = frac{-b pm sqrt{b^2 - 4ac}}{2a} ), where ( a = 18 ), ( b = -62 ), and ( c = -80 ). Plugging in these values, I get ( p = frac{62 pm sqrt{(-62)^2 - 4 times 18 times (-80)}}{2 times 18} ). Calculating the discriminant, ( (-62)^2 = 3844 ), and ( 4 times 18 times (-80) = -5760 ). So, the discriminant becomes ( 3844 + 5760 = 9604 ). The square root of 9604 is 98. So, the solutions are ( p = frac{62 + 98}{36} ) and ( p = frac{62 - 98}{36} ). Calculating these, ( frac{62 + 98}{36} = frac{160}{36} = frac{40}{9} ), and ( frac{62 - 98}{36} = frac{-36}{36} = -1 ). Now, I need to check if these solutions satisfy the condition ( p < 8 ). Both ( frac{40}{9} ) (which is approximately 4.44) and ( -1 ) are indeed less than 8, so they are valid solutions. Therefore, the values of ( p ) that satisfy the given condition are ( p = -1 ) and ( p = frac{40}{9} ).

answer：Okay, so Mitch is trying to bake a cake, and he needs 3 cups of flour and 2 cups of sugar. But here's the thing—he doesn't have a one-cup scoop. Instead, he only has a 1/3 cup scoop. Hmm, that's a bit tricky, but I think I can figure it out.First, let's tackle the flour. He needs 3 cups of flour, and his scoop is 1/3 cup. So, how many scoops does he need to make up 3 cups? Well, if one scoop is 1/3 cup, then to get 1 cup, he would need 3 scoops because 1/3 cup times 3 equals 1 cup. So, for 3 cups, he would need 3 times that amount. That means 3 scoops per cup times 3 cups equals 9 scoops. Okay, so he needs 9 scoops of flour.Now, onto the sugar. He needs 2 cups of sugar, and again, he's using a 1/3 cup scoop. Using the same logic as before, one cup requires 3 scoops. So, for 2 cups, he would need 3 scoops per cup times 2 cups, which equals 6 scoops. Got it, so he needs 6 scoops of sugar.To find the total number of scoops he needs, I just add the scoops for flour and sugar together. That's 9 scoops for flour plus 6 scoops for sugar, which equals 15 scoops in total.Wait a second, let me double-check that. If 1/3 cup times 9 equals 3 cups, that's correct because 9 times 1/3 is 3. And 1/3 cup times 6 equals 2 cups, since 6 times 1/3 is 2. So, adding those together, 3 cups plus 2 cups equals 5 cups total, but in terms of scoops, it's 15 scoops. Yeah, that makes sense.I think I've got it right. So, Mitch needs to use his 1/3 cup scoop 15 times in total to measure out all the flour and sugar he needs for his cake.

answer：First, I need to find the value of ( f(4) ) using the given piecewise function.Since ( 4 > 1 ), I will use the second part of the function:[f(4) = -4 + 3 = -1]Next, I need to find ( f(f(4)) ), which is ( f(-1) ).Since ( -1 leq 1 ), I will use the first part of the function:[f(-1) = -1 + 1 = 0]Therefore, the final answer is ( 0 ).