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question:John is holding a poker night with his friends and is getting the decks of cards ready before his friends show up. After losing so many cards from various decks, John thinks he needs to count exactly how many cards he has to make sure he has enough full decks for the night. Each deck of cards should hold 52 cards. He finds 3 half-full decks of cards and x full decks of cards. As he is arranging them, he realizes that the quality of a lot of the cards is quite poor and decides to throw 34 of the cards in the trash. How many cards does John now have? If we know the answer to the above question is 200, what is the value of unknown variable x?
answer:John has 3 half-full decks of cards, which means he has 3 * (52/2) = 78 cards from those decks. He also has x full decks of cards, which means he has x * 52 cards from those decks. In total, before throwing any cards away, John has 78 + x * 52 cards. After throwing 34 cards away, John now has 78 + x * 52 - 34 cards. We are given that the total number of cards John has is 200, so we can write: 78 + x * 52 - 34 = 200. Simplifying the equation, we get: 52x + 44 = 200. Subtracting 44 from both sides, we get: 52x = 156. Dividing both sides by 52, we get: x = 3. The value of x is 3. 3 The answer is: 3
question:f there were initially 160 tissues inside a tissue box and Tucker bought 3 boxes, but used 210 tissues while he was sick with the flu, how many tissues would he have left?
answer:Tucker bought 3 boxes, so he would have a total of 3 * 160 = 480 tissues. He used 210 tissues while he was sick, so he would have 480 - 210 = 270 tissues left. 270 The answer is: 270
question:The matrices [begin{pmatrix} a & 2 X & 4 end{pmatrix} quad text{and} quad begin{pmatrix} -frac{2}{7} & frac{1}{7} b & frac{3}{14} end{pmatrix}]are inverses. Enter the ordered pair (a,b). The answer is 14. What is the value of unknown variable X?
answer:To solve this problem, we need to determine the values of a and b that make the given matrices inverses of each other. Two matrices are inverses if their product is the identity matrix. The identity matrix for a 2x2 matrix is: [begin{pmatrix} 1 & 0 0 & 1 end{pmatrix}] So, we can set up the equation as follows: [begin{pmatrix} a & 2 x & 4 end{pmatrix} cdot begin{pmatrix} -frac{2}{7} & frac{1}{7} b & frac{3}{14} end{pmatrix} = begin{pmatrix} 1 & 0 0 & 1 end{pmatrix}] Let's simplify the matrix multiplication: [begin{pmatrix} a(-frac{2}{7}) + 2b & a(frac{1}{7}) + frac{3}{7} x(-frac{2}{7}) + 4b & x(frac{1}{7}) + frac{12}{14} end{pmatrix} = begin{pmatrix} 1 & 0 0 & 1 end{pmatrix}] Now, we can equate the corresponding entries of the matrices: a(-frac{2}{7}) + 2b = 1 a(frac{1}{7}) + frac{3}{7} = 0 x(-frac{2}{7}) + 4b = 0 x(frac{1}{7}) + frac{12}{14} = 1 Let's solve these equations: From the first equation, we have: -frac{2}{7}a + 2b = 1 Multiplying by 7, we get: -2a + 14b = 7 From the second equation, we have: frac{1}{7}a + frac{3}{7} = 0 Multiplying by 7, we get: a + 3 = 0 a = -3 Substituting a = -3 in the first equation, we have: -2(-3) + 14b = 7 6 + 14b = 7 14b = 7 - 6 14b = 1 b = frac{1}{14} From the third equation, we have: -frac{2}{7}x + 4b = 0 Multiplying by 7, we get: -2x + 28b = 0 Substituting b = frac{1}{14}, we have: -2x + 28(frac{1}{14}) = 0 -2x + 2 = 0 -2x = -2 x = 1 From the fourth equation, we have: frac{1}{7}x + frac{12}{14} = 1 Multiplying by 14, we get: 2x + 12 = 14 2x = 14 - 12 2x = 2 x = 1 So, the ordered pair (a, b) is (-3, frac{1}{14}). The value of x is 1. The answer is: 1
question:abella has three times as many green houses as yellow houses. She also has 40 fewer yellow houses than red houses. If she has 90 green houses, how many of her houses are not yellow?
answer:If Isabella has 90 green houses, and she has three times as many green houses as yellow houses, then she has 90/3 = 30 yellow houses. If Isabella has 40 fewer yellow houses than red houses, then she has 30+40 = 70 red houses. The total number of houses Isabella has is 90 green + 30 yellow + 70 red = 190 houses. The number of houses that are not yellow is 190 - 30 = 160 houses. 160 The answer is: 160