💡First, I need to determine the individual work rates of each person. Person A can complete the job in 4 hours, so their work rate is ( frac{1}{4} ) of the job per hour.Person B can complete the job in 6 hours, so their work rate is ( frac{1}{6} ) of the job per hour.Person C can complete the job in 8 hours, so their work rate is ( frac{1}{8} ) of the job per hour.Next, to find the lowest fraction of the job that can be completed in 1 hour by two people working together, I should consider the two slowest workers, which are Person B and Person C.Adding their work rates together:[frac{1}{6} + frac{1}{8} = frac{4}{24} + frac{3}{24} = frac{7}{24}]Therefore, the lowest fraction of the job that can be done in 1 hour by two people working together is ( frac{7}{24} ) of the job.

💡First, I need to determine the cost price of each vase based on the given profit and loss percentages.For the vase sold at a 25% profit:- Let the cost price be ( C_1 ).- Selling Price = Cost Price + 25% of Cost Price = ( 1.25 times C_1 ).- Given Selling Price = 1.50, so ( 1.25 times C_1 = 1.50 ).- Solving for ( C_1 ), we get ( C_1 = frac{1.50}{1.25} = 1.20 ).For the vase sold at a 25% loss:- Let the cost price be ( C_2 ).- Selling Price = Cost Price - 25% of Cost Price = ( 0.75 times C_2 ).- Given Selling Price = 1.50, so ( 0.75 times C_2 = 1.50 ).- Solving for ( C_2 ), we get ( C_2 = frac{1.50}{0.75} = 2.00 ).Next, I calculate the total cost price and total revenue:- Total Cost Price = ( C_1 + C_2 = 1.20 + 2.00 = 3.20 ).- Total Revenue = ( 1.50 + 1.50 = 3.00 ).Finally, I determine the net result:- Net Result = Total Revenue - Total Cost = ( 3.00 - 3.20 = -0.20 ).This indicates a loss of 20 cents. However, the closest option provided is a loss of 12 cents.

💡Okay, so I'm trying to figure out this problem about the movie "Thirty Thousand Miles in Chang'an." It was released on July 8, 2023, and made about 100 million yuan on the first day. Then, on the third day, it made approximately 240 million yuan. The question is asking about the growth rate of the box office revenue, which is denoted as x. They want to know which equation correctly represents this situation.First, I need to understand what they mean by "growth rate." I think it means that each day, the revenue increases by a certain percentage or factor, which is x. So, if the revenue on the first day is 100 million yuan, then on the second day, it would be 100 million times (1 + x), and on the third day, it would be the second day's revenue times (1 + x) again. That makes sense because if something grows at a constant rate, you multiply by that growth factor each day.So, let's break it down step by step.1. **First Day Revenue**: 100 million yuan. Let's denote this as R1 = 100 million.2. **Second Day Revenue**: This would be R1 multiplied by (1 + x). So, R2 = R1 * (1 + x) = 100 * (1 + x).3. **Third Day Revenue**: This would be R2 multiplied by (1 + x). So, R3 = R2 * (1 + x) = 100 * (1 + x) * (1 + x) = 100 * (1 + x)^2.But according to the problem, the third day's revenue is 240 million yuan. So, R3 = 240 million.Putting it all together, we have:100 * (1 + x)^2 = 240To make it simpler, I can divide both sides by 100 to get:(1 + x)^2 = 2.4So, the equation that represents this situation is (1 + x)^2 = 2.4.Looking at the options provided:A: 1 + x = 2.4B: (1 + x)^2 = 2.4C: 1 + (1 + x) = 2.4D: 1 + (1 + x) + (1 + x)^2 = 2.4From my calculation, the correct equation is (1 + x)^2 = 2.4, which is option B.But just to make sure I didn't make a mistake, let me double-check.If the growth rate is x, then each day's revenue is the previous day's revenue multiplied by (1 + x). So, starting from 100 million:- Day 1: 100- Day 2: 100*(1 + x)- Day 3: 100*(1 + x)^2Given that Day 3 is 240 million, so 100*(1 + x)^2 = 240Divide both sides by 100: (1 + x)^2 = 2.4Yes, that seems right. So, option B is correct.I think I got it. The key was understanding that the growth rate applies each day, so it's compounded daily, leading to the squared term on the third day.

💡First, I need to determine the total number of drawings Mark has. He has 12 notebooks, each with 36 sheets, and each sheet holds 8 drawings. So, the total number of drawings is 12 multiplied by 36 multiplied by 8, which equals 3456 drawings.Next, I need to figure out how many sheets are required if each sheet now holds 5 drawings. I'll divide the total number of drawings by 5. 3456 divided by 5 equals 691.2 sheets. Since you can't have a fraction of a sheet, Mark needs 692 sheets in total.Mark fills up 6 complete notebooks, each containing 36 sheets. So, the number of sheets used in these 6 notebooks is 6 multiplied by 36, which equals 216 sheets. Subtracting this from the total sheets needed, 691 minus 216 equals 475 sheets remaining.In the 7th notebook, there are 29 fully occupied sheets, each with 5 drawings. The number of drawings in these 29 sheets is 29 multiplied by 5, which equals 145 drawings. To find out how many drawings are on the last sheet, I'll subtract the total number of drawings in the fully occupied sheets from the overall total. 3456 minus 145 equals 3311 drawings remaining.Finally, to determine the number of drawings on the last sheet, I'll divide the remaining drawings by the number of drawings per sheet. 3311 divided by 5 equals 662.2 sheets. Since only one sheet is partially filled, there is 1 drawing on the last sheet.

💡Okay, so I have this problem about an equilateral triangle with area A. The process is to form a new triangle by joining the midpoints of the sides of the first one, and then repeating this process indefinitely. I need to find the sum of the areas of all these triangles. Hmm, let me try to break this down step by step.First, I know that an equilateral triangle has all sides equal and all angles equal to 60 degrees. The area of the original triangle is given as A. Now, when we join the midpoints of the sides, we're essentially creating a smaller equilateral triangle inside the original one. I think this new triangle is similar to the original one because all the angles remain the same, and the sides are proportionally smaller.I remember that when you connect midpoints of a triangle, the new triangle formed is called the medial triangle. For an equilateral triangle, this medial triangle should also be equilateral. But how much smaller is it? Well, since we're connecting midpoints, each side of the new triangle is half the length of the original triangle's sides. So, if the original triangle has a side length of, say, 's', the new triangle will have sides of length 's/2'.Now, the area of a triangle is proportional to the square of its side length. So if the side length is halved, the area should be (1/2)^2 = 1/4 of the original area. That means the area of the first new triangle is A/4.But wait, the problem says this process is repeated indefinitely. So, after the first new triangle, we'll form another one by joining the midpoints of the sides of the A/4 triangle. Following the same logic, the area of this next triangle should be (A/4)/4 = A/16. Continuing this pattern, each subsequent triangle has an area that is 1/4 of the previous one.So, the areas of the triangles form a geometric series: A + A/4 + A/16 + A/64 + ... and so on. I need to find the sum of this infinite series. I recall that the sum of an infinite geometric series can be found using the formula S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio, provided that |r| < 1.In this case, the first term 'a' is A, and the common ratio 'r' is 1/4. Plugging these into the formula, we get S = A / (1 - 1/4) = A / (3/4) = (4/3)A.Wait, but let me double-check. The first term is A, the second term is A/4, the third is A/16, etc. So the series is indeed A + A/4 + A/16 + ... with each term being 1/4 of the previous one. So the common ratio is 1/4, which is less than 1, so the formula applies.Therefore, the sum should be (4/3)A. Looking back at the options, that's option B: 4A/3.But just to make sure I didn't make a mistake, let me think again. The original area is A. The first new triangle is A/4, then A/16, and so on. So the total area is A + A/4 + A/16 + ... which is a geometric series with a = A and r = 1/4. Sum is A / (1 - 1/4) = A / (3/4) = 4A/3. Yeah, that seems correct.I think I might have initially thought the ratio was 1/2, but since area depends on the square of the side length, the ratio is actually (1/2)^2 = 1/4. So that's why the common ratio is 1/4, not 1/2. That was a crucial point.So, yeah, I'm confident now that the sum of the areas is 4A/3.

💡Okay, so I've got this sequence of numbers: 11, 23, 47, 83, 131, 191, 263, 347, 443, 551, 671. The question is asking for the tenth number in the sequence. Hmm, let's see. I need to figure out the pattern here to determine what comes next.First, I'll list out the numbers again to make sure I have them right: 11, 23, 47, 83, 131, 191, 263, 347, 443, 551, 671. Wait, that's actually 11 numbers listed, but the question is about the tenth number. Maybe the list includes the tenth number already? Let me count: 11 is the first, 23 is the second, 47 is the third, 83 is the fourth, 131 is the fifth, 191 is the sixth, 263 is the seventh, 347 is the eighth, 443 is the ninth, 551 is the tenth, and 671 is the eleventh. So, the tenth number is 551. But the question seems to imply that I need to find it, so maybe I'm misunderstanding.Alternatively, perhaps the sequence provided is up to the ninth number, and 551 and 671 are the tenth and eleventh numbers. Let me check the original problem again: "The sequence of numbers starts with 11, 23, 47, 83, 131, 191, 263, 347, 443, and the following numbers are 551 and 671. What is the tenth number in the sequence?" Okay, so the first nine numbers are given, and then it says the following numbers are 551 and 671. So, 551 is the tenth number, and 671 is the eleventh. But the question is asking for the tenth number, which would be 551. But that seems too straightforward. Maybe I'm missing something.Perhaps the user wants me to confirm the pattern and show how 551 is indeed the tenth number. Let's try that approach. To understand the pattern, I'll look at the differences between consecutive numbers.Starting with 11 to 23: 23 - 11 = 12.Next, 23 to 47: 47 - 23 = 24.Then, 47 to 83: 83 - 47 = 36.83 to 131: 131 - 83 = 48.131 to 191: 191 - 131 = 60.191 to 263: 263 - 191 = 72.263 to 347: 347 - 263 = 84.347 to 443: 443 - 347 = 96.443 to 551: 551 - 443 = 108.551 to 671: 671 - 551 = 120.Okay, so the differences are: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120.Looking at these differences, I notice that each difference increases by 12. For example, 24 - 12 = 12, 36 - 24 = 12, and so on. So, the pattern is that each time, the difference between consecutive numbers increases by 12.If I want to find the tenth number, which is 551, I can see that it's already given as the number following 443. But to confirm, I can calculate it by adding the next difference to 443. The last difference before 551 is 96 (from 347 to 443). Adding 12 to 96 gives 108. So, 443 + 108 = 551. That checks out.Similarly, to get from 551 to 671, the difference should be 120 (108 + 12). And indeed, 551 + 120 = 671.So, the pattern is consistent. Each time, the difference increases by 12. Therefore, the tenth number in the sequence is 551.Wait, but the initial problem statement says that the sequence starts with those nine numbers, and then the following numbers are 551 and 671. So, 551 is the tenth number, and 671 is the eleventh. Therefore, the answer should be 551.But just to be thorough, let me outline the steps clearly:1. List the given numbers: 11, 23, 47, 83, 131, 191, 263, 347, 443.2. Calculate the differences between consecutive numbers: - 23 - 11 = 12 - 47 - 23 = 24 - 83 - 47 = 36 - 131 - 83 = 48 - 191 - 131 = 60 - 263 - 191 = 72 - 347 - 263 = 84 - 443 - 347 = 963. Observe that each difference increases by 12.4. To find the tenth number, add the next difference to the ninth number (443). The last difference was 96, so the next difference is 96 + 12 = 108.5. Therefore, the tenth number is 443 + 108 = 551.6. Verify by checking the next number: 551 + 120 = 671, which matches the given sequence.So, everything adds up correctly. The tenth number in the sequence is indeed 551.