## Task 4 - 140724

Fritz borrowed a book from his friend Max for 6 days. On the morning of the 6th day, he tells his friend Paul, who wants to borrow the book after him:

"On the first day, I read the 12th part of the book, on the following 4 days I read an eighth each, and today, if I want to finish the book, I have to read 20 pages less than I have read in the past days combined.

How many pages does the book have in total?"

Investigate the possibilities for Paul to answer this question in such a way that all of Fritz's statements are true!
Assuming the statements are true if the book has $x$ pages. Of these, Fritz read $\frac{x}{12}$ pages on the first day, and over the following 4 days, he read $4 \cdot \frac{x}{8}$ pages, which together is $\frac{7}{12} x$ pages. Therefore, for the last day, $\frac{5}{12} x$ pages remained. According to the problem, this was 20 pages less than $\frac{7}{12} x$ pages. This leads to $\frac{2}{12} x = 20$ and thus $x = 6 \cdot 20 = 120$.

Therefore, the statement that the book has 120 pages can only be true if Fritz's statements are correct. Indeed, they are correct for 120 pages; because if the book has 120 pages, Fritz read 10 pages on the first day, 15 pages each on the following 4 days, totaling 70 pages by then; and if he reads another (20 pages less, i.e.,) 50 pages, the total number of pages in the book, 120 pages, is exactly reached.

Solutions of the 2nd Round 1974 adopted from [5]

### 4.16.3 3rd Round 1974, Class 7