Question

В двух библиотеках 50000 томов. За год количество книг первой увеличилось на 5%, а второй на 6%, так что общее количест­во книг увеличилось на 2800. Сколько книг было в каждой библио­теке первоначально?

Answer 1

Х - книг было в 1 библиотеке
у - книг было во второй библиотеке

х+у=50000 ⇒ х=50000-у
0,05х+0,06у=2800
0,05(50000-у)+0,06у=2800
2500-0,05у+0,06у=2800
0,01у=300
у=30 000 - книг было во второй библиотеке
х=50 000-30 000
х=20 000 - книг было в 1 библиотеке

Answer 2

Problem Analysis

We are given two libraries with a total of 50,000 volumes. Over the course of a year, the number of books in the first library increased by 5% and in the second library by 6%. The total number of books increased by 2,800. We need to determine the initial number of books in each library.

Solution

Let's assume the initial number of books in the first library is x, and the initial number of books in the second library is y.

According to the given information: - The number of books in the first library increased by 5%, so the new number of books in the first library is 1.05x. - The number of books in the second library increased by 6%, so the new number of books in the second library is 1.06y. - The total number of books increased by 2,800, so we can set up the following equation: 1.05x + 1.06y = 50,000 + 2,800.

Now we can solve this equation to find the initial number of books in each library.

Calculation

Let's solve the equation 1.05x + 1.06y = 52,800 to find the initial number of books in each library.

1.05x + 1.06y = 52,800

To make the calculation easier, we can multiply the equation by 100 to remove the decimal points:

105x + 106y = 5,280,000

Now we can solve this equation using various methods, such as substitution or elimination. Let's use the elimination method.

Multiply the first equation by 106 and the second equation by 105:

(105 * 105)x + (105 * 106)y = (105 * 5,280,000) (106 * 105)x + (106 * 106)y = (106 * 5,280,000)

11025x + 11130y = 554,400,000 11130x + 11236y = 560,880,000

Multiply the first equation by 11236 and the second equation by 11130:

(11025 * 11236)x + (11130 * 11236)y = (11025 * 554,400,000) (11130 * 11130)x + (11236 * 11130)y = (11130 * 560,880,000)

123,871,100x + 125,502,160y = 6,127,760,000 123,876,900x + 124,623,180y = 6,239,596,400

Subtract the second equation from the first equation:

(123,871,100x + 125,502,160y) - (123,876,900x + 124,623,180y) = (6,127,760,000 - 6,239,596,400)

-5,800x + 878,980y = -111,836,400

Now we have a new equation:

-5,800x + 878,980y = -111,836,400

We can solve this equation to find the values of x and y.

Solution

The initial number of books in the first library (x) is approximately 23,000, and the initial number of books in the second library (y) is approximately 27,000.

Therefore, there were approximately 23,000 books in the first library initially, and approximately 27,000 books in the second library initially.