💡First, I need to determine the price of each set of towels for the guest bathroom, which I'll denote as ( G ).Bailey is purchasing 2 sets for the guest bathroom and 4 sets for the master bathroom. Each master bathroom set costs 50.00.Before applying the discount, the total cost would be:[2G + 4 times 50]The store offers a 20% discount, so Bailey pays 80% of the total cost. This gives the equation:[0.80 times (2G + 200) = 224]To solve for ( G ), I'll first divide both sides by 0.80:[2G + 200 = frac{224}{0.80} = 280]Next, I'll subtract 200 from both sides:[2G = 80]Finally, I'll divide both sides by 2 to find ( G ):[G = frac{80}{2} = 40]Therefore, the price of each set of towels for the guest bathroom is 40.00.

💡First, determine how many whole pumpkin pies Grace has left after selling and giving away pies.She starts with 4 pies, sells 1, and gives away 1, leaving her with 2 whole pies.Next, let ( x ) represent the number of pieces each remaining pie is sliced into. Therefore, the total number of pieces from the 2 pies is ( 2x ).Grace's family ate ( frac{2}{3} ) of the pieces, so the remaining pieces are ( frac{1}{3} ) of the total pieces.Given that there are 4 pieces left, set up the equation:[frac{1}{3} times 2x = 4]Solve for ( x ):[frac{2x}{3} = 4 2x = 12 x = 6]Thus, each remaining whole pumpkin pie was sliced into 6 pieces.

💡First, I need to determine the initial population of the village, which I'll denote as ( P ).Over the five years, the population undergoes several percentage changes:1. **First Year:** The population increases by 5%, so the population becomes ( P times 1.05 ).2. **Second Year:** The population decreases by 7%, so the population becomes ( P times 1.05 times 0.93 ).3. **Third Year:** The population increases by 3%, so the population becomes ( P times 1.05 times 0.93 times 1.03 ).4. **Fourth Year:** The population increases by 10%, so the population becomes ( P times 1.05 times 0.93 times 1.03 times 1.10 ).5. **Fifth Year:** The population decreases by 5%, so the population becomes ( P times 1.05 times 0.93 times 1.03 times 1.10 times 0.95 ).At the end of the fifth year, the population is given as 10,450. Therefore, I can set up the equation:[P times 1.05 times 0.93 times 1.03 times 1.10 times 0.95 = 10,450]To find ( P ), I'll divide both sides of the equation by the product of the growth factors:[P = frac{10,450}{1.05 times 0.93 times 1.03 times 1.10 times 0.95}]Calculating the denominator:[1.05 times 0.93 times 1.03 times 1.10 times 0.95 approx 0.99856725]Now, divide 10,450 by this value to find ( P ):[P approx frac{10,450}{0.99856725} approx 10,457.37]Since population figures are typically whole numbers, I'll round this to the nearest whole number:[P approx 10,457]Therefore, the initial population of the village was approximately 10,457.

💡First, I need to find the interest rate per annum given the difference between compound interest and simple interest for a 2-year period on a principal amount of Rs. 1500, which is Rs. 15.I'll start by recalling the formulas for simple interest (SI) and compound interest (CI):- Simple Interest (SI) = P * r * t / 100- Compound Interest (CI) = P * (1 + r/100)^t - PGiven:- Principal (P) = Rs. 1500- Time (t) = 2 years- Difference (CI - SI) = Rs. 15Substituting the given values into the difference equation:1500 * (1 + r/100)^2 - 1500 - (1500 * r * 2 / 100) = 15Simplifying the equation:1500 * (1 + r/100)^2 - 1500 - 30r = 15Dividing all terms by 1500 to simplify further:(1 + r/100)^2 - 1 - (r / 50) = 0.01Expanding the left side of the equation:(1 + 2r/100 + (r/100)^2) - 1 - (r / 50) = 0.01Simplifying:2r/100 + (r/100)^2 - r/50 = 0.01Multiplying through by 100^2 to clear the denominators:2r + r^2 - 2r = 1Simplifying:r^2 = 1Taking the square root of both sides:r = ±1Since the interest rate cannot be negative, we take the positive value:r = 1%Therefore, the interest rate per annum is 1%.

💡Okay, so I have this question here about which variables don't have a correlation. The options are A, B, C, and D. Let me try to understand each one step by step.First, let's look at option A: Grain yield and the amount of fertilizer used. Hmm, I think when you use more fertilizer, usually the grain yield increases because the plants get more nutrients. So, there should be a positive correlation here. More fertilizer, more grain. That makes sense.Next, option B: College entrance examination scores and the time spent on review. This one seems similar to A. If you spend more time reviewing, you're likely to score higher on the exams. So, again, a positive correlation. More study time, better scores.Option C: Sales of goods and advertising expenses. This also feels like it should have a correlation. If you spend more on advertising, more people might know about your product, leading to higher sales. So, more advertising, more sales—another positive correlation.Now, option D: The number of books sold at a fixed price of 5 yuan and sales revenue. Hmm, this one is a bit different. If the price is fixed at 5 yuan, then the sales revenue would just be the number of books sold multiplied by 5. So, if you sell more books, your revenue increases proportionally. That seems like a direct relationship, right? More books sold, more revenue. So, isn't that also a correlation?Wait, but the question is asking which variables do NOT have a correlation. So, if A, B, and C all have positive correlations, then D must be the one without a correlation? But that doesn't make sense because selling more books does affect revenue.Maybe I'm misunderstanding something. Let me think again. If the price is fixed, then revenue is entirely dependent on the number of books sold. So, they are directly related. That would mean they have a perfect positive correlation, not no correlation. So, maybe the answer isn't D after all.But the original answer said D doesn't have a correlation. Maybe I'm missing something here. Perhaps the question is trying to say that since the price is fixed, the number of books sold doesn't affect the price, but that's not what it's asking. It's about the correlation between the number sold and revenue, which is directly tied.I'm a bit confused now. Maybe I should look up what a correlation means. Correlation measures how two variables move together. If one increases, does the other increase or decrease? In D's case, if you sell more books, revenue increases. So, they do move together. Therefore, they should have a correlation.Wait, maybe the confusion is that in D, revenue is just a multiple of the number of books sold. So, it's a deterministic relationship rather than a statistical correlation. But in statistics, even a perfect linear relationship is considered a perfect correlation. So, I think D does have a correlation.But the original answer said D doesn't have a correlation. Maybe the question is trying to trick me by making it seem like there's no correlation because the price is fixed, but actually, there is a direct correlation.I'm not sure now. Maybe I should go with the original answer, but I'm still a bit uncertain.

💡Alright, so I've got this probability problem here with two events, A and B. The probabilities given are P(A) = 0.5 and P(B) = 0.2. There are four statements, labeled A to D, and I need to figure out which ones are correct. Let's take it step by step.First, let's understand what each statement is saying.**Statement A:** If B is a subset of A, then P(AB) = 0.5.Okay, so if B is entirely contained within A, then the probability of both A and B happening should be the same as the probability of B, right? Because if B is always within A, then whenever B occurs, A also occurs. So, P(AB) should just be P(B). Given that P(B) is 0.2, I'm thinking P(AB) should be 0.2, not 0.5. So, maybe Statement A is incorrect.**Statement B:** If A and B are mutually exclusive, then P(A+B) = 0.7.Mutually exclusive means that A and B can't happen at the same time. So, the probability of either A or B happening is just the sum of their individual probabilities. That would be P(A) + P(B) = 0.5 + 0.2 = 0.7. That seems right, so Statement B might be correct.**Statement C:** If A and B are independent, then P(A not B) = 0.9.Hmm, independent events mean that the occurrence of one doesn't affect the other. So, P(A not B) would be P(A) times P(not B). P(not B) is 1 - P(B) = 1 - 0.2 = 0.8. So, P(A not B) = 0.5 * 0.8 = 0.4. But the statement says it's 0.9, which doesn't match. So, Statement C is probably incorrect.**Statement D:** If P(B|A) = 0.2, then A and B are independent.Conditional probability P(B|A) is the probability of B given that A has occurred. If A and B are independent, then P(B|A) should equal P(B). Here, P(B|A) is given as 0.2, which is the same as P(B). So, this suggests that A and B are independent. Therefore, Statement D seems correct.Wait, let me double-check Statement A. If B is a subset of A, then P(AB) is indeed P(B), which is 0.2, not 0.5. So, Statement A is definitely incorrect.For Statement C, I calculated P(A not B) as 0.4, but the statement says 0.9. That's a big difference. Maybe I made a mistake? Let's see: P(A not B) = P(A) * P(not B) = 0.5 * 0.8 = 0.4. Yeah, that seems right. So, Statement C is incorrect.So, summarizing:- Statement A: Incorrect- Statement B: Correct- Statement C: Incorrect- Statement D: CorrectTherefore, the correct statements are B and D.