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question:A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its twelfth move is m/n, where m and n are relatively prime positive integers, find m + n.
answer:Following the established relation P_n = frac{1}{2} (1 - P_{n-1}) and continuing from previously computed probabilities up to P_{10} = frac{171}{512}: - P_{11} = frac{1}{2}(1 - P_{10}) = frac{1}{2}(1 - frac{171}{512}) = frac{1}{2}(frac{341}{512}) = frac{341}{1024} - P_{12} = frac{1}{2}(1 - P_{11}) = frac{1}{2}(1 - frac{341}{1024}) = frac{1}{2}(frac{683}{1024}) = frac{683}{2048} Thus, m = 683 and n = 2048, so m + n = 683 + 2048 = boxed{2731}.
question:A country used to have a tax rate of 20%. They raised it to a new rate. In that same time frame, John went from making 1,000,000 a year to 1,500,000 a year. He pays 250,000 more in taxes now compared to then. What is the current tax rate?
answer:Let's denote the new tax rate as ( T ). When John was making ( 1,000,000 ) a year, he was paying ( 20% ) in taxes, which amounts to: [ 1,000,000 times 20% = 1,000,000 times 0.20 = 200,000 ] Now that he is making ( 1,500,000 ) a year, he is paying ( 250,000 ) more in taxes than before. So his current tax payment is: [ 200,000 + 250,000 = 450,000 ] This ( 450,000 ) is ( T% ) of his new income of ( 1,500,000 ). To find ( T ), we can set up the equation: [ 1,500,000 times T = 450,000 ] To solve for ( T ), we divide both sides by ( 1,500,000 ): [ T = frac{450,000}{1,500,000} ] [ T = 0.30 ] So the new tax rate ( T ) is ( boxed{30%} ).
question:Given that the universal set U = mathbb{R} and M = {x mid 3 leq x < 5}, find the complement of M with respect to U, denoted as U setminus M. A: {x mid x < 3 text{ or } x geq 5} B: {x mid x leq 3 text{ or } x > 5} C: {x mid x < 3 text{ and } x geq 5} D: {x mid x leq 3 text{ and } x > 5}
answer:Since the universal set U = mathbb{R} and M = {x mid -3 leq x < 5}, we need to find all the elements in U that are not in M. The set M contains all real numbers between -3 (inclusive) and 5 (exclusive). The complement of M, denoted U setminus M, will therefore consist of all real numbers less than -3, combined with all real numbers greater than or equal to 5, since these are the numbers not included in M. This means: U setminus M = {x mid x < -3} cup {x mid x geq 5} In interval notation, this is (-infty, -3) cup [5, infty). Comparing this with the given options, we see that option A matches our description: A: {x mid x < -3 text{ or } x geq 5} Therefore, the correct choice is: [boxed{A}]
question:Two numbers have a difference of 5 and their sum is increased by a factor of 2 to give 40. What is the larger of the two numbers?
answer:Let the numbers be x and y. According to the problem, the equations set up are: 1. x - y = 5 2. 2(x + y) = 40 which simplifies to x + y = 20. From x + y = 20, we can solve for y: [ y = 20 - x. ] Substituting y in the first equation: [ x - (20 - x) = 5 ] [ x - 20 + x = 5 ] [ 2x = 25 ] [ x = 12.5. ] Then, substitute x back into y = 20 - x: [ y = 20 - 12.5 ] [ y = 7.5. ] Hence, the larger number is boxed{12.5}.