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question:Let the distribution of the random variable X be P(X=i)=frac{i}{10} for i=1,2,3,4. If P(1≤X＜a)=frac{3}{5}, then the range of real number a is ______.
answer:To solve for the range of the real number a given the distribution of the random variable X and the condition P(1≤X＜a)=frac{3}{5}, we proceed as follows: 1. **Understanding the Distribution**: The distribution of X is given by P(X=i)=frac{i}{10} for i=1,2,3,4. This means the probabilities for X taking values 1, 2, 3, and 4 are frac{1}{10}, frac{2}{10}, frac{3}{10}, and frac{4}{10} respectively. 2. **Applying the Given Condition**: We are given that P(1≤X＜a)=frac{3}{5}. This means we are looking for the sum of probabilities from X=1 up to but not including X=a to equal frac{3}{5}. 3. **Calculating the Sum of Probabilities**: We calculate the sum of probabilities for X=1, 2, and 3 because we know a must be greater than 3 for the sum to reach frac{3}{5}: [ Pleft(1leqslant X lt aright) = Pleft(X=1right) + Pleft(X=2right) + Pleft(X=3right) = frac{1}{10} + frac{2}{10} + frac{3}{10} = frac{6}{10} = frac{3}{5}. ] 4. **Determining the Range of a**: Since the sum of probabilities for X=1, 2, and 3 equals frac{3}{5}, and we know P(X=4)=frac{4}{10} would exceed this sum, a must be greater than 3 but not exceed 4. Therefore, 3 < a leqslant 4. 5. **Conclusion**: The range of the real number a is (3,4]. Therefore, the final answer, encapsulated as required, is boxed{(3,4]}.
question:What is the smallest base10 integer that can be represented as 13_a in one base and 31_b in another base, where a and b are any bases larger than 3?
answer:Firstly, change the expressions 13_a and 31_b to base-10: [ 13_a = 1 cdot a + 3 = a + 3, ] [ 31_b = 3 cdot b + 1 = 3b + 1. ] Equating these since they represent the same number: [ a + 3 = 3b + 1 Rightarrow a = 3b - 2. ] To find the smallest possible values for a and b where both a > 3 and b > 3, let's try b = 4: [ a = 3 times 4 - 2 = 12 - 2 = 10. ] So a = 10 and b = 4. Let us calculate the base-10 integer: [ 13_{10} = 1 times 10 + 3 = 10 + 3 = 13, ] [ 31_4 = 3 times 4 + 1 = 12 + 1 = 13. ] We obtain boxed{13} in base-10.
question:David has a store where he sells cell phones. When he takes inventory at the end of the day, he notices that he has 10 Samsung cell phones but he started the day with 14 Samsung cell phones. Then he notices that he has 5 iPhones left, but started with 8. His staff then told him that in the afternoon they realized that 2 Samsung cell phones were damaged and 1 iPhone had a manufacturing defect in the screen, so they threw these out during the day. During the day, there are two types of discounts offered on each phone: a 10% discount on Samsung cell phones and a 15% discount on iPhones. The tax rates applied on the final price were different for Samsung and iPhones at 12% and 10%, respectively. Calculate the total revenue generated from the sales, considering discounts and taxes, if the retail price of a Samsung cell phone is 800, and an iPhone is 1,000.
answer:First, let's calculate the number of Samsung cell phones and iPhones sold, excluding the damaged ones. For Samsung cell phones: Started with: 14 Ended with: 10 Damaged: 2 Sold = Started - Ended - Damaged Sold (Samsung) = 14 - 10 - 2 = 2 For iPhones: Started with: 8 Ended with: 5 Damaged: 1 Sold = Started - Ended - Damaged Sold (iPhones) = 8 - 5 - 1 = 2 Now, let's calculate the revenue from each type of cell phone, considering the discounts and taxes. For Samsung cell phones: Retail price: 800 Discount: 10% Discounted price: 800 - (800 * 0.10) = 800 - 80 = 720 Tax: 12% Tax amount: 720 * 0.12 = 86.40 Final price after tax: 720 + 86.40 = 806.40 Total revenue from Samsung sales: 806.40 * 2 = 1612.80 For iPhones: Retail price: 1,000 Discount: 15% Discounted price: 1,000 - (1,000 * 0.15) = 1,000 - 150 = 850 Tax: 10% Tax amount: 850 * 0.10 = 85 Final price after tax: 850 + 85 = 935 Total revenue from iPhone sales: 935 * 2 = 1870 Total revenue generated from all sales: Total revenue = Revenue from Samsung + Revenue from iPhones Total revenue = 1612.80 + 1870 = 3482.80 David's store generated a total revenue of boxed{3482.80} from the sales of Samsung cell phones and iPhones, after considering discounts and taxes.
question:John spent 40 percent of his earnings last month on rent, he spent 30 percent less than what he spent on rent to purchase a new dishwasher, and he also spent 15 percent more on groceries than he spent on rent. What percent of last month's earnings did John have left over?
answer:Let's assume John's earnings last month were 100 (we can use any amount, but using 100 makes the percentages easy to calculate). John spent 40% of his earnings on rent: 100 * 40% = 40 on rent He spent 30% less on the dishwasher than he did on rent: 30% of 40 = 40 * 30% = 12 less So, he spent 40 - 12 = 28 on the dishwasher He spent 15% more on groceries than he spent on rent: 15% of 40 = 40 * 15% = 6 more So, he spent 40 + 6 = 46 on groceries Now, let's add up all his expenses: Rent: 40 Dishwasher: 28 Groceries: 46 Total expenses = 40 + 28 + 46 = 114 However, since we assumed his earnings were 100, and his total expenses cannot exceed his earnings, we need to adjust our assumption. Let's calculate the total percentage of his earnings that he spent and then subtract that from 100% to find out what percent he has left over. Total percent spent on rent, dishwasher, and groceries: Rent: 40% Dishwasher: 40% - 30% of 40% = 40% - 12% = 28% Groceries: 40% + 15% of 40% = 40% + 6% = 46% Total percent spent = 40% + 28% + 46% = 114% Since the total percent spent exceeds 100%, it's clear that our initial assumption of 100 earnings is not correct. However, we can still calculate the percentage of earnings he has left over by subtracting the total percent spent from 100%: Percent left over = 100% - (40% + 28% + 46%) = 100% - 114% = -14% This result indicates that John spent more than he earned. If we want to find the percent he has left over, we need to ensure that the total percent spent on rent, dishwasher, and groceries does not exceed boxed{100%} . Since the question implies that John did have some earnings left over, there might be a mistake in the interpretation of the question or the calculations. Please double-check the question and the calculations. If the question is correct as stated, then John would not have any earnings left over; instead, he would have spent more than he earned.