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question:Billy buys a pack of soda from the store. He has twice as many brothers as sisters, and he has 2 sisters. He can give 2 sodas to each of his siblings if he wants to give out the entire pack while giving each the same number. How many sodas are in the pack?
answer:Billy has 2 sisters. Since he has twice as many brothers as sisters, he has 2 * 2 = 4 brothers. The total number of siblings Billy has is 2 sisters + 4 brothers = 6 siblings. If he wants to give out the entire pack of soda by giving 2 sodas to each sibling, then the total number of sodas in the pack would be 6 siblings * 2 sodas per sibling = boxed{12} sodas.
question:A mobile phone's price was reduced to 1800 yuan, which is only 90% of its original price. How much was the price reduced by?
answer:To solve "How much was the price reduced by?", we first need to find out the original price. Then, we can calculate how much the price was reduced by. This problem is about the application of fractions, where we need to accurately identify the unit "1". If unknown, we can find it using division. Let's denote the original price as the unit "1". Since the reduced price, 1800 yuan, represents 90% (or frac{9}{10}) of the original price, we can find the original price by dividing 1800 by frac{9}{10}: text{Original Price} = 1800 div frac{9}{10} = 1800 times frac{10}{9} = 2000 text{ yuan} To find out how much the price was reduced by, we subtract the reduced price from the original price: text{Price Reduction} = text{Original Price} - text{Reduced Price} = 2000 - 1800 = 200 text{ yuan} Therefore, the price was reduced by boxed{200 text{ yuan}}.
question:The coefficients of the polynomial [ x^5 + bx^4 + cx^3 + dx^2 + ex + f = 0 ] are all integers. Let n be the exact number of integer roots of this polynomial, counting multiplicity. Determine all possible values of n.
answer:We analyze several simple scenarios to identify the valid values of n: 1. **x^5 + 1 = 0**: No integer roots. n = 0. 2. **x(x^4 + 2) = 0**: One integer root, x = 0. n = 1. 3. **x^2(x^3 + 1) = 0**: Two integer roots (0 with multiplicity of 2), n = 2. 4. **x^5 = 0**: One root, but with multiplicity 5. n = 5. 5. **Suppose there are three integer roots**: By Vieta’s formulas, the sum of the integer roots and the two additional roots (integer or non-integer) add up to -b. It is feasible to construct such a polynomial, e.g., (x^3(x + 1)(x - 1) = 0), giving integers 0, 1, -1. n = 3. 6. **Checking for four integer roots**: - Similar to the case with three roots, if there are four integer roots, adding any integer root will make n = 5. - Example: (x^4(x-1) = 0), roots are 0, 0, 0, 0, -1, thus n = 5 (not 4). Given the results and reasoning above, the possible values of n are boxed{0, 1, 2, 3, 5}.
question:Let A represent the number of apples Mrs. Hilt ate, and let H represent the number of hours it took her to eat them. If Mrs. Hilt ate A=12 apples in H=5 hours, how many apples per hour in average did she eat?
answer:To find the average number of apples Mrs. Hilt ate per hour, we divide the total number of apples she ate by the total number of hours it took her to eat them. So, if A represents the number of apples and H represents the number of hours, the average number of apples per hour is A/H. Given that A = 12 apples and H = 5 hours, we can calculate the average as follows: Average apples per hour = A / H Average apples per hour = 12 apples / 5 hours Average apples per hour = 2.4 apples per hour Therefore, Mrs. Hilt ate an average of boxed{2.4} apples per hour.