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question:Two integers are relatively prime if they have no common factors other than 1 or -1. What is the probability that a positive integer less than or equal to X is relatively prime to 30? Express your answer as a common fraction. If we know the answer to the above question is frac{4}{15}, what is the value of unknown variable X?
answer:We want to find the value of X in the given situation. To find the probability that a positive integer less than or equal to X is relatively prime to 30, we need to determine the value of X. Since we are looking for the probability, X should be a positive integer. The probability of two numbers being relatively prime can be calculated using Euler's totient function, which gives the count of positive integers less than or equal to a given number that are relatively prime to it. The value of Euler's totient function for 30 is 8. This means that there are 8 positive integers less than or equal to 30 that are relatively prime to 30. To find the probability, we can divide the count of relatively prime numbers by the total count of positive integers less than or equal to X. The total count of positive integers less than or equal to X is X. So, the probability can be calculated as frac{8}{X}. Given that the probability is frac{4}{15}, we can write: frac{4}{15} = frac{8}{X} To solve for X, we can cross-multiply and solve the equation: 4X = 8 * 15 4X = 120 Dividing both sides of the equation by 4, we find: X = 30 The value of X is 30. The answer is: 30
question:Susan wants to throw a party for her mom. She is planning on having 30 guests. For dinner she is making a recipe that makes 2 servings each. Each batch of the recipe calls for 4 potatoes and 1 teaspoon of salt. A potato costs .10 and a container of salt costs 2 at the supermarket. If each container of salt has 5 teaspoons, how much money will Susan spend on food?
answer:Susan is making 2 servings per batch of the recipe, so she needs 30/2 = 15 batches of the recipe. Each batch of the recipe calls for 4 potatoes, so Susan needs 15 * 4 = 60 potatoes. Each batch of the recipe calls for 1 teaspoon of salt, and Susan needs 15 * 1 = 15 teaspoons of salt. Each container of salt has 5 teaspoons, so Susan needs 15/5 = 3 containers of salt. A potato costs 0.10, so Susan will spend 60 * 0.10 = 6 on potatoes. A container of salt costs 2 and Susan needs 3 containers, so she will spend 3 * 2 = 6 on salt. In total, Susan will spend 6 + 6 = 12 on food. 12 The answer is: 12
question:After purchasing six dozen eggs from the grocery store, Addilynn used half of them after two weeks. While moving the eggs to clean the shelves, she accidentally broke 15 of the remaining eggs. How many eggs are left on the shelf?
answer:Six dozen eggs is equal to 6 x 12 = 72 eggs. After two weeks, Addilynn used half of the eggs, so there are 72 / 2 = 36 eggs left. After accidentally breaking 15 eggs, there are 36 - 15 = 21 eggs left on the shelf. 21 The answer is: 21
question:Lindsey saved 50 in September, she saved 37 in October, and 11 in November. Lindsey's mom said that since Lindsey had saved more than 75, she would give Lindsey 25. Then Lindsey spent 87 on a video game. How much money did Lindsey have left?
answer:Lindsey saved a total of 50 + 37 + 11 = 98. Since Lindsey saved more than 75, her mom gave her an additional 25. So Lindsey had a total of 98 + 25 = 123. After spending 87 on a video game, Lindsey had 123 - 87 = 36 left. 36 The answer is: 36