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question:Given any two positive real numbers x and y, then x , Diamond , y is a positive real number defined in terms of x and y by some fixed rule. Suppose the operation x , Diamond , y satisfies the equations (xy) , Diamond , y=x(y , Diamond , y) and (x , Diamond , 1) , Diamond , x = x , Diamond , 1 for all x,y>0. Given that 1 , Diamond , X, find 19 , Diamond , 98. If we know the answer to the above question is 19, what is the value of unknown variable X?
answer:We are given that 1 , Diamond , X = 19, and we need to find the value of X. Using the second equation, we have (x , Diamond , 1) , Diamond , x = x , Diamond , 1. Substituting 1 for x, we have (1 , Diamond , 1) , Diamond , 1 = 1 , Diamond , 1. Simplifying, we have 1 , Diamond , 1 = 1 , Diamond , 1. Using the first equation, we have (xy) , Diamond , y = x(y , Diamond , y). Substituting 1 for x and 1 for y, we have (1 cdot 1) , Diamond , 1 = 1(1 , Diamond , 1). Simplifying, we have 1 , Diamond , 1 = 1 , Diamond , 1. Since 1 , Diamond , X = 19 and 1 , Diamond , 1 = 1 , Diamond , 1, we can conclude that X = 1. Therefore, the value of X is 1. The answer is: 1
question:What is the units digit of the product of the first three positive composite numbers?
answer:The first three positive composite numbers are 4, 6, and 8. The units digit of the product 4 times 6 times 8 is 4 times 6 times 8 equiv 4 times 6 times 8 equiv 24 times 8 equiv 4 times 8 equiv boxed{2}. The answer is: 2
question:John's weight bench can support x pounds. He wants to make sure to stay 20% under that weight for safety. He can put 550 pounds on the bar if he weighs 250 pounds. What is the value of unknown variable x?
answer:To solve this problem, we need to determine the value of x, which represents the maximum weight the bench can support in pounds. We know that John wants to stay 20% under that weight for safety. Therefore, he can lift 80% of the maximum weight. We also know that John can put 550 pounds on the bar if he weighs 250 pounds. We can set up the equation as follows: 80% of the maximum weight = 550 pounds + John's weight 0.8x = 550 + 250 Let's simplify and solve for x: 0.8x = 800 To isolate x, we divide both sides of the equation by 0.8: x = 800 / 0.8 x = 1000 The value of x is 1000. 1000 The answer is: 1000
question:The average age of 33 fifth-graders is X. The average age of 55 of their parents is 33. The average age of all of these parents and fifth-graders is 24.75. What is the value of unknown variable X?
answer:To solve this problem, we need to determine the value of x, which represents the average age of the fifth-graders. Let's break down the information given: Number of fifth-graders: 33 Average age of fifth-graders: X Number of parents: 55 Average age of parents: 33 Average age of all fifth-graders and parents: 24.75 We can set up the equation as follows: (Number of fifth-graders * Average age of fifth-graders) + (Number of parents * Average age of parents) = Total age of all fifth-graders and parents (33 * X) + (55 * 33) = (33 + 55) * 24.75 Let's simplify and solve for x: 33X + 1815 = 88 * 24.75 33X + 1815 = 2178 To isolate x, we subtract 1815 from both sides of the equation: 33X + 1815 - 1815 = 2178 - 1815 33X = 363 To solve for x, we divide both sides of the equation by 33: X = 363 / 33 X = 11 The value of x is 11. The answer is: 11