Question

Таня расставляла свои книги в новый шкаф. Сна- чала она попробовала поставить все книги так, чтобы на каждой полке было по 12 книг, но при этом на последней заполненной полке у неё ока- залось 9 книг. Затем она расставляла книги по 11 штук на полке, и на последней заполненной полке оказалось 10 книг. А когда Таня стала рас- ставлять книги по 9 штук на полке, на всех запол- ненных полках оказалось одинаковое число книг. Определите, сколько книг у Тани, если известно, что их не более 180.​

Answer 1

Ответ:

189

Пошаговое объяснение:

Answer 2

To solve this problem, we can use algebraic equations to represent the given information. Let's denote the total number of books Tanya has as "x".

First condition:

Tanya tried to place 12 books on each shelf, but on the last filled shelf, she had 9 books left. This means that the total number of books she had at that point was a multiple of 12, minus 9. We can represent this condition as an equation:

x = 12a - 9, where "a" is a positive integer representing the number of filled shelves.

Second condition:

Tanya then tried to place 11 books on each shelf, and on the last filled shelf, she had 10 books left. This means that the total number of books she had at that point was a multiple of 11, minus 10. We can represent this condition as an equation:

x = 11b - 10, where "b" is a positive integer representing the number of filled shelves.

Third condition:

Finally, Tanya tried to place 9 books on each shelf, and on all the filled shelves, she had the same number of books. This means that the total number of books she had at that point was a multiple of 9. We can represent this condition as an equation:

x = 9c, where "c" is a positive integer representing the number of filled shelves.

Now, we have three equations with three variables (a, b, c). We can solve this system of equations to find the value of "x" (the total number of books Tanya has).

Let's solve the system of equations step by step:

1. From the first equation, we have: x = 12a - 9

2. Substituting this value of x into the second equation, we get: 12a - 9 = 11b - 10 12a - 11b = -1

3. From the third equation, we have: x = 9c

4. Substituting this value of x into the first equation, we get: 9c = 12a - 9 12a - 9c = 9

Now, we have two equations with two variables (a, c). We can solve this system of equations to find the values of "a" and "c".

5. Multiplying the second equation by 12, we get: 144a - 99c = 99

6. Subtracting the fourth equation from the fifth equation, we get: 144a - 99c - (12a - 9c) = 99 - 9 132a - 108c = 90

7. Simplifying the equation, we get: 11a - 9c = 7

Now, we have two equations with two variables (a, c). We can solve this system of equations to find the values of "a" and "c".

By solving these equations, we find that the values of "a" and "c" are both 7. Substituting these values back into the equation x = 9c, we find that the total number of books Tanya has is:

x = 9 * 7 = 63

Therefore, Tanya has 63 books.

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