Question

Заведующий отдела получил премию, в размере 40 процентов своего оклада, а его заместитель 30 процентов своего оклада. Премия заведующего отдела оказалос на 4500 тенге больше премии заместителя. Каков оклад заведующего отдела, если он на5000 тенге больше оклада заместителя.

Answer 1

Calculation of the Department Head's Salary

To calculate the salary of the department head, let's assume the salary of the deputy is represented by x tenge.

According to the given information: - The department head received a bonus of 40% of their salary. - The deputy received a bonus of 30% of their salary. - The department head's bonus was 4500 tenge more than the deputy's bonus. - The department head's salary is 5000 tenge more than the deputy's salary.

Let's break down the information into equations:

1. The department head's bonus is 4500 tenge more than the deputy's bonus: - Department Head's Bonus = Deputy's Bonus + 4500

2. The department head's salary is 5000 tenge more than the deputy's salary: - Department Head's Salary = Deputy's Salary + 5000

3. The department head's bonus is 40% of their salary: - Department Head's Bonus = 0.4 * Department Head's Salary

4. The deputy's bonus is 30% of their salary: - Deputy's Bonus = 0.3 * Deputy's Salary

Now, let's solve these equations to find the salary of the department head.

Using equation 1, we can substitute the value of the department head's bonus from equation 3: Department Head's Bonus = Deputy's Bonus + 4500 0.4 * Department Head's Salary = 0.3 * Deputy's Salary + 4500

Using equation 2, we can substitute the value of the department head's salary from equation 4: 0.4 * (Deputy's Salary + 5000) = 0.3 * Deputy's Salary + 4500

Simplifying the equation: 0.4 * Deputy's Salary + 2000 = 0.3 * Deputy's Salary + 4500

Subtracting 0.3 * Deputy's Salary from both sides: 0.1 * Deputy's Salary + 2000 = 4500

Subtracting 2000 from both sides: 0.1 * Deputy's Salary = 2500

Dividing both sides by 0.1: Deputy's Salary = 2500 / 0.1 Deputy's Salary = 25000#### Calculation of Salaries and Bonuses

To calculate the salary of the department head and their deputy, we can use the given information about their bonuses and the difference in their salaries.

Let's assume the salary of the deputy is x tenge. According to the information provided, the department head received a bonus of 40% of their salary, while the deputy received a bonus of 30% of their salary. Additionally, the department head's bonus was 4500 tenge more than the deputy's bonus.

Based on this information, we can set up the following equations:

Equation 1: Department Head's Bonus = 40% of Department Head's Salary Equation 2: Deputy's Bonus = 30% of Deputy's Salary Equation 3: Department Head's Bonus - Deputy's Bonus = 4500 tenge Equation 4: Department Head's Salary - Deputy's Salary = 5000 tenge

Let's solve these equations to find the salaries of the department head and their deputy.

Solution

Let's start by assigning variables to the unknowns: - Department Head's Salary = y tenge - Deputy's Salary = x tenge

Using Equation 1, we can write: Department Head's Bonus = 0.4 * y

Using Equation 2, we can write: Deputy's Bonus = 0.3 * x

Using Equation 3, we can write: 0.4 * y - 0.3 * x = 4500

Using Equation 4, we can write: y - x = 5000

Now, we have a system of two equations with two unknowns. We can solve this system of equations to find the values of x and y.

To solve the system of equations, we can use the substitution method or the elimination method. Let's use the elimination method:

Multiplying Equation 3 by 10, we get: 4y - 3x = 45000

Adding this equation to Equation 4, we get: 4y - 3x + y - x = 45000 + 5000 5y - 4x = 50000

Now, we have a new equation: 5y - 4x = 50000 ---(Equation 5)

We can solve Equations 5 and 4 simultaneously to find the values of x and y.

Let's subtract Equation 4 from Equation 5: 5y - 4x - (y - x) = 50000 - 5000 4y - 3x = 45000

We have obtained the same equation as Equation 3. This means that the system of equations is dependent, and we can solve it by substituting one variable in terms of the other.

From Equation 4, we can write: y = x + 5000

Substituting this value of y in Equation 3, we get: 0.4 * (x + 5000) - 0.3 * x = 4500

Simplifying the equation, we get: 0.4x + 2000 - 0.3x = 4500 0.1x = 2500 x = 2500 / 0.1 x = 25000

Now that we have found the value of x, we can substitute it back into Equation 4 to find the value of y: y = x + 5000 y = 25000 + 5000 y = 30000

Answer

Therefore, the salary of the department head is 30000 tenge, and the salary of their deputy is 25000 tenge.

Please note that the calculations are based on the given information and assumptions made during the process.