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question:Ralph is going to practice playing tennis with a tennis ball machine that shoots out tennis balls for Ralph to hit. He loads up the machine with 175 tennis balls to start with. Out of the first 100 balls, he manages to hit 2/5 of them. Of the next 75 tennis balls, he manages to hit 1/3 of them. Out of all the tennis balls, how many did Ralph not hit?
answer:Out of the first 100 balls, Ralph hits 2/5 of them, so he hits 2/5 * 100 = 40 balls. Out of the next 75 balls, Ralph hits 1/3 of them, so he hits 1/3 * 75 = 25 balls. In total, Ralph hits 40 + 25 = 65 balls. Since Ralph started with 175 balls, he did not hit 175 - 65 = 110 balls. 110 The answer is: 110
question:Find the fifth term of the geometric sequence with first term 2 and second term frac{1}{4}.
answer:The common ratio of the sequence is frac{frac{1}{4}}{2} = frac{1}{8}. To find the fifth term, we multiply the first term by the common ratio four times. So, the fifth term is 2 cdot left(frac{1}{8}right)^4 = boxed{frac{1}{2048}}. The answer is: frac{1}{2048}
question:f f is a function defined on the positive integers such that f(xy) = f(x) + f(y) for all positive integers x and y, and f(10) = 14 and f(40) = 20, what is the value of f(500)?
answer:We have f(10) = f(2 cdot 5) = f(2) + f(5) and f(40) = f(2^3 cdot 5) = 3f(2) + f(5). Subtracting the first equation from the second equation, we get f(40) - f(10) = 2f(2), so 20 - 14 = 2f(2) and f(2) = 3. Then, f(500) = f(2^2 cdot 5^3) = 2f(2) + 3f(5) = 2(3) + 3f(5) = 6 + 3f(5). We also have f(10) = f(2 cdot 5) = f(2) + f(5) = 3 + f(5), so f(5) = f(10) - 3 = 14 - 3 = 11. Therefore, f(500) = 6 + 3f(5) = 6 + 3(11) = 6 + 33 = boxed{39}.The answer is: 39
question:f William goes to the library 2 times per week, how many times does Jason go to the library in 4 weeks if he goes 4 times more often than William?
answer:If William goes to the library 2 times per week, then Jason goes 4 times more often, which means he goes 2 * 4 = 8 times per week. If there are 4 weeks in total, then Jason goes to the library 8 * 4 = 32 times in 4 weeks. 32 The answer is: 32