💡First, I need to calculate the least common multiple (LCM) for each pair of numbers involving 18 and the given numbers: 3, 6, 9, 12, 15, and 18.1. **LCM of 18 and 3**: - 18 is already a multiple of 3, so the LCM is 18.2. **LCM of 18 and 6**: - 18 is already a multiple of 6, so the LCM is 18.3. **LCM of 18 and 9**: - 18 is already a multiple of 9, so the LCM is 18.4. **LCM of 18 and 12**: - Factorize both numbers: - 18 = 2 × 3² - 12 = 2² × 3 - Take the highest powers of all prime factors: - 2² × 3² = 4 × 9 = 36 - So, the LCM is 36.5. **LCM of 18 and 15**: - Factorize both numbers: - 18 = 2 × 3² - 15 = 3 × 5 - Take the highest powers of all prime factors: - 2 × 3² × 5 = 2 × 9 × 5 = 90 - So, the LCM is 90.6. **LCM of 18 and 18**: - The LCM of any number with itself is the number itself, so the LCM is 18.After calculating all the LCMs, the values are 18, 18, 18, 36, 90, and 18. The largest value among these is 90.

💡Okay, so I have this equation to solve: ( x - frac{5}{x - 4} = 2 - frac{5}{x - 4} ). Hmm, let me see. I need to find the roots of this equation, and then choose the correct option from the given choices. The options are about how many roots there are and whether they are integral or not. First, I should probably simplify the equation. Let me write it down again: ( x - frac{5}{x - 4} = 2 - frac{5}{x - 4} ). I notice that both sides of the equation have the term ( -frac{5}{x - 4} ). Maybe I can eliminate that term by adding ( frac{5}{x - 4} ) to both sides. Let me try that.Adding ( frac{5}{x - 4} ) to both sides gives me:( x = 2 ).Wait, that seems too straightforward. So, does that mean the solution is just ( x = 2 )? Let me check if that's correct by substituting ( x = 2 ) back into the original equation.Plugging ( x = 2 ) into the left side: ( 2 - frac{5}{2 - 4} = 2 - frac{5}{-2} = 2 + frac{5}{2} = 2 + 2.5 = 4.5 ).Now, plugging ( x = 2 ) into the right side: ( 2 - frac{5}{2 - 4} = 2 - frac{5}{-2} = 2 + frac{5}{2} = 2 + 2.5 = 4.5 ).Okay, so both sides equal 4.5 when ( x = 2 ), which means ( x = 2 ) is indeed a solution. But wait, I should also make sure that ( x = 2 ) doesn't make any denominators zero in the original equation. The denominator here is ( x - 4 ). Plugging ( x = 2 ) into ( x - 4 ) gives ( 2 - 4 = -2 ), which is not zero. So, there's no division by zero issue here. That means ( x = 2 ) is a valid solution.Now, looking back at the options provided:A) infinitely many integral rootsB) no rootC) one integral rootD) two equal integral rootsE) two equal non-integral rootsSince we found that ( x = 2 ) is the only solution, and it's an integer, the correct answer should be C) one integral root.But just to be thorough, let me consider if there could be any other solutions. The equation simplified to ( x = 2 ), which suggests there's only one root. There's no quadratic term or anything that would indicate multiple roots. So, I don't think there are any other solutions.Also, since ( x = 2 ) is an integer, it fits the description of option C. There's no indication of multiple roots or non-integer roots here. So, I'm confident that C is the correct choice.**Final Answer**boxed{C}

💡First, determine Cameron's work rate by calculating the fraction of the task he completes in one day.Next, calculate the amount of work Cameron completes in the initial 9 days working alone.Subtract the work done by Cameron from the total task to find the remaining work after 9 days.Let the combined work rate of Cameron and Sandra be represented by a variable. Set up an equation based on the time they take to complete the remaining task together.Solve the equation to find the combined work rate.Finally, determine the total time it would take for both Cameron and Sandra to complete the entire task together using their combined work rate.

💡Okay, so I have this problem here: John buys 20 bars of soap, each weighing 1.5 pounds, and the cost is 0.5 per pound. I need to figure out how much money John spent on soap. Alright, let's break this down step by step.First, I need to understand what exactly is being asked. The problem is about calculating the total cost John incurred when purchasing these soap bars. To find the total cost, I probably need to know the total weight of all the soap bars and then multiply that by the cost per pound.So, let's see. John bought 20 bars of soap. Each bar weighs 1.5 pounds. To find the total weight, I should multiply the number of bars by the weight per bar. That makes sense because if each bar is 1.5 pounds, then 20 bars would be 20 times that weight.Let me write that down: Total weight = Number of bars × Weight per bar. Plugging in the numbers, that would be 20 bars × 1.5 pounds per bar. Hmm, 20 times 1.5... Let me calculate that. 20 times 1 is 20, and 20 times 0.5 is 10. So, adding those together, 20 + 10 equals 30. So, the total weight is 30 pounds.Alright, now that I have the total weight, I need to find out the total cost. The cost is given as 0.5 per pound. So, to find the total cost, I should multiply the total weight by the cost per pound. That would be Total cost = Total weight × Cost per pound.Plugging in the numbers, that's 30 pounds × 0.5 per pound. Let me do that multiplication. 30 times 0.5 is 15. So, the total cost is 15.Wait a minute, let me double-check that to make sure I didn't make a mistake. So, 20 bars at 1.5 pounds each is indeed 30 pounds. And 30 pounds at 0.5 per pound is 15. Yeah, that seems right.Just to think about it another way, if one pound costs 0.5, then two pounds would cost 1.0, right? So, for 30 pounds, it's like 15 times two pounds, which would be 15 times 1.0, which is 15. Yep, that checks out.I guess another way to look at it is by breaking down the cost per bar first. If each bar is 1.5 pounds and costs 0.5 per pound, then the cost per bar would be 1.5 × 0.5, which is 0.75. Then, for 20 bars, it would be 20 × 0.75, which is also 15. So, both methods give me the same answer, which makes me more confident that 15 is the correct total cost.I think I've got it. John spent 15 on soap.

💡Okay, so I have this trigonometry problem here: If sinθ cosθ > 0, then θ is in which quadrants? The options are A: first and second, B: first and third, C: first and fourth, D: second and fourth. Hmm, let me think about this step by step.First, I remember that in the unit circle, the signs of sine and cosine depend on the quadrant where the angle θ is located. Sine corresponds to the y-coordinate, and cosine corresponds to the x-coordinate. So, in the first quadrant, both sine and cosine are positive. In the second quadrant, sine is positive but cosine is negative. In the third quadrant, both sine and cosine are negative. And in the fourth quadrant, sine is negative while cosine is positive.The problem says that sinθ cosθ is greater than zero. That means the product of sine and cosine is positive. For a product of two numbers to be positive, both numbers must be positive or both must be negative. So, either both sinθ and cosθ are positive, or both are negative.Let me consider the quadrants where this can happen. In the first quadrant, as I mentioned, both sine and cosine are positive. So their product will definitely be positive. That's one possibility.Now, in the third quadrant, both sine and cosine are negative. When you multiply two negative numbers, the result is positive. So, sinθ cosθ will also be positive in the third quadrant. That's another possibility.What about the other quadrants? In the second quadrant, sine is positive and cosine is negative. Multiplying a positive and a negative gives a negative result, which doesn't satisfy sinθ cosθ > 0. Similarly, in the fourth quadrant, sine is negative and cosine is positive, so their product is also negative. So, neither the second nor the fourth quadrants work.Therefore, the only quadrants where sinθ cosθ is positive are the first and third quadrants. Looking back at the options, that's option B.Wait, let me double-check. If θ is in the first quadrant, sinθ and cosθ are both positive, so their product is positive. If θ is in the third quadrant, both are negative, so their product is positive. In the second and fourth quadrants, one is positive and the other is negative, so their product is negative. Yeah, that makes sense.I think I'm confident now. The answer should be B: The first and third quadrants.

💡Okay, so I need to figure out how much I should be willing to pay for this piece of clothing that's marked at 100 yuan. The problem says that individual clothing sales can make a profit if they sell it for 20% more than what they paid for it. But bosses usually mark up the price by 50% to 100%. Hmm, that means the actual cost price is lower than the marked price, and the marked price is already higher than what the boss paid.First, let me understand what a markup means. If something is marked up by 50%, that means the selling price is 50% higher than the cost price. Similarly, a 100% markup means the selling price is double the cost price. So, if the marked price is 100 yuan, and the boss marked it up by 50%, then the cost price would be lower than 100 yuan.Let me denote the cost price as 'C'. If the boss marked it up by 50%, then the selling price (which is the marked price) would be C plus 50% of C, so that's 1.5C. Similarly, if the boss marked it up by 100%, then the selling price would be C plus 100% of C, which is 2C.Given that the marked price is 100 yuan, I can set up two equations:1. For a 50% markup: 1.5C = 1002. For a 100% markup: 2C = 100Let me solve these equations to find the possible cost prices.Starting with the 50% markup:1.5C = 100To find C, I'll divide both sides by 1.5:C = 100 / 1.5C = 66.666... yuanSo, if the boss marked it up by 50%, the cost price would be approximately 66.67 yuan.Now, for the 100% markup:2C = 100To find C, I'll divide both sides by 2:C = 100 / 2C = 50 yuanSo, if the boss marked it up by 100%, the cost price would be 50 yuan.This means the cost price could be anywhere between 50 yuan and approximately 66.67 yuan, depending on how much the boss marked it up.Now, the problem also mentions that individual clothing sales can profit as long as the selling price is 20% higher than the cost price. So, even if I'm buying it from an individual seller, they need to make at least a 20% profit on their cost price.Let me calculate the minimum selling price that would allow the seller to make a 20% profit.First, for the lower end of the cost price, which is 50 yuan:Profit = 20% of 50 yuan = 0.2 * 50 = 10 yuanSo, the selling price should be 50 + 10 = 60 yuanFor the higher end of the cost price, which is approximately 66.67 yuan:Profit = 20% of 66.67 yuan = 0.2 * 66.67 ≈ 13.33 yuanSo, the selling price should be 66.67 + 13.33 ≈ 80 yuanTherefore, the seller would need to sell the clothing for at least 60 yuan to make a 20% profit if the cost price is 50 yuan, and at least 80 yuan if the cost price is approximately 66.67 yuan.Since the marked price is 100 yuan, and the seller's cost price is somewhere between 50 yuan and 66.67 yuan, I should aim to bargain within the range that allows the seller to make their 20% profit. That means I should be willing to pay between 60 yuan and 80 yuan.But wait, let me double-check my calculations to make sure I didn't make any mistakes.For the 50% markup:1.5C = 100C = 100 / 1.5 ≈ 66.67 yuan20% of 66.67 ≈ 13.33 yuanSelling price ≈ 66.67 + 13.33 ≈ 80 yuanFor the 100% markup:2C = 100C = 50 yuan20% of 50 = 10 yuanSelling price = 50 + 10 = 60 yuanYes, that seems correct. So, the range I should consider for bargaining is between 60 yuan and 80 yuan.I think that makes sense because if the boss marked it up by 50%, the seller's cost is higher, so they need a higher selling price to make their 20% profit. Conversely, if the boss marked it up by 100%, the seller's cost is lower, so they can afford to sell it for less and still make a profit.Therefore, when bargaining, I should aim for a price between 60 yuan and 80 yuan to ensure that I'm paying a fair price that allows the seller to make a reasonable profit, while also getting a good deal for myself.