Question

Составьте уравнение по условию задачи: премиальный фонд в 72000 рублей решено было распределить в конце года между сотрудниками отдела поровну. в течение года 6 человек ушли из отдела, поэтому каждый получил на 1000 рублей больше,чем предполагалось. сколько сотрудников было в отделе первоначально и сколько стало к концу года?

Answer 1

Х – количество сотрудников в начале года.
У - количество сотрудников в конце года
1) х – у = 6.
2) 72000/у – 72000/х=1000 → 72/у – 72/х = 1
72х – 72у = х у → 72х –72х+ 432 = х² - 6х → х² - 6х - 432 = 0
Х = 3 ± √(9 + 432) = 3 ± 21
По физическому смыслу подходит только первый корень х = 24 → у = 18.

Answer 2

Problem Analysis

To solve this problem, we need to find the initial number of employees in the department and the final number of employees at the end of the year. We are given that the total bonus fund is 72,000 rubles, and it is distributed equally among the employees. Additionally, we know that 6 employees left the department during the year, resulting in each remaining employee receiving 1,000 rubles more than originally planned.

Solution

Let's assume that the initial number of employees in the department is x. Since the bonus fund is distributed equally among the employees, each employee would receive 72,000 / x rubles initially.

However, due to the departure of 6 employees, the bonus fund is now distributed among (x - 6) employees. Each employee now receives 72,000 / (x - 6) rubles.

According to the given information, each employee received 1,000 rubles more than originally planned. Therefore, we can set up the following equation:

(72,000 / (x - 6)) - (72,000 / x) = 1,000

To solve this equation, we can multiply both sides by x(x - 6) to eliminate the denominators:

x(72,000) - (x - 6)(72,000) = 1,000x(x - 6)

Simplifying the equation:

72,000x - 72,000(x - 6) = 1,000x^2 - 6,000x

Expanding and rearranging the equation:

72,000x - 72,000x + 432,000 = 1,000x^2 - 6,000x

1,000x^2 - 6,000x - 432,000 = 0

Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula to find the values of x.

The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1,000, b = -6,000, and c = -432,000.

Plugging in the values into the quadratic formula:

x = (-(-6,000) ± √((-6,000)^2 - 4 * 1,000 * (-432,000))) / (2 * 1,000)

Simplifying the equation:

x = (6,000 ± √(36,000,000 + 1,728,000,000)) / 2,000

x = (6,000 ± √1,764,000,000) / 2,000

x = (6,000 ± 41,992.15) / 2,000

Now we have two possible values for x:

x1 = (6,000 + 41,992.15) / 2,000

x2 = (6,000 - 41,992.15) / 2,000

Calculating the values:

x1 = 47,992.15 / 2,000 = 23.9961

x2 = -35,992.15 / 2,000 = -17.9961

Since the number of employees cannot be negative, we discard the negative value. Therefore, the initial number of employees in the department was approximately 24.

To find the final number of employees at the end of the year, we subtract the number of employees who left during the year (6) from the initial number of employees:

Final number of employees = Initial number of employees - Number of employees who left

Final number of employees = 24 - 6 = 18

Therefore, the initial number of employees in the department was 24, and the final number of employees at the end of the year was 18.

Answer

The initial number of employees in the department was 24, and the final number of employees at the end of the year was 18.