Question

Решите задачу: На вопрос о том, сколько у него учеников, древнегреческий учёный Пифагор Пифагор отвечал так: "Половина моих учеников изучает математику, четвертая часть изучает природу, седьмая часть проводит время в молчаливом размышлении, остальную часть составляют 3 девы". Сколько учеников было у Пифагора?

Answer 1

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Answer 2

#### Problem Analysis The ancient Greek scientist Pythagoras responded to the question of how many students he had with the following statement: "Half of my students study mathematics, a quarter study nature, a seventh spend time in silent contemplation, and the remaining three are maidens." We need to determine the total number of students Pythagoras had based on this information. #### Solution Let's break down the information provided by Pythagoras and solve the problem step by step: - **Half of the students study mathematics**: This means that half of the total number of students are studying mathematics. - **A quarter study nature**: This means that a quarter of the total number of students are studying nature. - **A seventh spend time in silent contemplation**: This means that a seventh of the total number of students spend time in silent contemplation. - **The remaining three are maidens**: This means that the remaining students, after accounting for those studying mathematics, nature, and silent contemplation, are three maidens. To find the total number of students, we can use the following equation: Total number of students = Number of students studying mathematics + Number of students studying nature + Number of students in silent contemplation + Number of maidens Let's calculate each component: - Number of students studying mathematics: Half of the total number of students. - Number of students studying nature: A quarter of the total number of students. - Number of students in silent contemplation: A seventh of the total number of students. - Number of maidens: Three. Now, let's substitute these values into the equation to find the total number of students. #### Calculation Let's denote the total number of students as 'x'. - Number of students studying mathematics: **x/2** (half of the total number of students) - Number of students studying nature: **x/4** (a quarter of the total number of students) - Number of students in silent contemplation: **x/7** (a seventh of the total number of students) - Number of maidens: **3** Using the equation mentioned earlier, we can write: x = x/2 + x/4 + x/7 + 3 To solve this equation, we can find a common denominator and simplify: Multiplying the equation by 28 (the least common multiple of 2, 4, and 7) to eliminate the denominators, we get: 28x = 14x + 7x + 4x + 84 Combining like terms, we have: 28x = 25x + 84 Subtracting 25x from both sides, we get: 3x = 84 Dividing both sides by 3, we find: x = 28 #### Answer Therefore, Pythagoras had **28 students**. #### Explanation According to Pythagoras, half of his students studied mathematics, which would be 14 students (28/2). A quarter of his students studied nature, which would be 7 students (28/4). A seventh of his students spent time in silent contemplation, which would be 4 students (28/7). The remaining three students were maidens. Adding all these numbers together, we get a total of 28 students.