Question

Вася взял книгу, страницы которой пронумерованы подряд, от первой до последней. Хулиган Вася вырвал из разных мест книги 25 листов и сложил номера всех пятидесяти вырванных страниц. У него получилось число 1994. Когда об этом узнала Даша, она заявила, что при подсчёте Вася ошибся. Объясните, почему Даша действительно права

Answer 1

Vasya's Book Problem

To answer this question, we need to analyze the information given and determine whether Dasha is correct in claiming that Vasya made a mistake in his calculations.

According to the information provided, Vasya took a book with consecutively numbered pages from the first page to the last. He then tore out 25 pages from different places in the book and added up the numbers on all 50 torn-out pages. The sum he obtained was 1994. Dasha believes that Vasya made a mistake in his calculations.

Let's break down the problem and see if we can find a solution.

Analyzing the Problem

1. Vasya tore out 25 pages from the book. 2. The numbers on these 25 pages were added up, resulting in a sum of 1994. 3. Dasha claims that Vasya made a mistake in his calculations.

Finding the Solution

To determine whether Dasha is correct, we need to examine the sum of the numbers on the torn-out pages and see if there is a mistake.

Let's assume that the first torn-out page has the number x. Since the pages are consecutively numbered, the last torn-out page will have the number x + 24.

The sum of the numbers on the torn-out pages can be calculated using the formula for the sum of an arithmetic series:

Sum = (n/2)(first term + last term)

In this case, the number of terms (n) is 25, the first term is x, and the last term is x + 24.

Substituting these values into the formula, we get:

1994 = (25/2)(x + x + 24)

Simplifying the equation:

1994 = (25/2)(2x + 24)

1994 = 25x + 300

Rearranging the equation:

25x = 1994 - 300

25x = 1694

x = 1694/25

x = 67.76

Since the page numbers are integers, we can conclude that Vasya made a mistake in his calculations. The assumption that the first torn-out page has the number 67.76 is not possible.

Therefore, Dasha is correct in claiming that Vasya made a mistake in his calculations.

Conclusion

In summary, Vasya tore out 25 pages from a book with consecutively numbered pages. He added up the numbers on these pages and obtained a sum of 1994. However, Dasha correctly pointed out that Vasya made a mistake in his calculations. The assumption that the first torn-out page has a non-integer number indicates an error in Vasya's calculations.