Question 33·Hard·One-Variable Data Distributions; Measures of Center and Spread
A company has 40 employees whose current mean annual salary is $52,000.
For this year, the company announces a compensation change consisting of two parts:
- Every employee will receive the same flat raise of dollars.
- In addition, each of the 5 executives will receive a one-time bonus of $10,000 on top of the flat raise.
After the raise and the one-time bonuses are applied this year, the mean total pay across all 40 employees is $58,000.
What is the value of , the flat raise each employee receives, in dollars?
(Express the answer as an integer)
For problems about mean salary (or any mean), first convert the means into totals by multiplying the mean by the number of data points (here, employees). Compare the total before and after the change to find the total increase, then carefully break that increase into its components (uniform raises plus any special bonuses or adjustments). Write an equation representing “sum of all individual changes = total change,” solve for the unknown, and double-check that you used the correct number of people in each term so you don’t miscount bonuses or divides.
Hints
Relate mean to total
Mean (average) salary is total salary divided by the number of employees. How can you find the total salary before and after the change using the means and 40 employees?
Find the total change in pay
Once you know the total salary before and after the change, subtract them. What does that difference represent?
Account for both raises and bonuses
The total increase in money comes from the flat raises to all 40 employees and the 5 bonuses of . Can you write an equation that adds those two parts and sets them equal to the total increase?
Solve for r
Your equation should have the form . Solve this equation for .
Desmos Guide
Compute total salaries before and after the change
In Desmos, type 40*52000 to see the original total salary and 40*58000 to see the new total salary. Then subtract them with 40*58000 - 40*52000 to confirm the total increase in pay for all employees.
Set up the expression for the flat raise
To incorporate the bonuses, type the expression (40*58000 - 40*52000 - 5*10000)/40. This corresponds to taking the total increase, subtracting the total bonuses, and then dividing by 40 employees.
Read off the flat raise value
Look at the numeric output of the expression from step 2. That value is the flat raise each employee receives.
Step-by-step Explanation
Convert the mean salaries to total salaries
The mean (average) salary is the total salary divided by the number of employees.
- Original mean salary: for 40 employees
- Total original salary:
- New mean total pay: for 40 employees
- Total new salary:
Find the total increase in salary for all employees
The total increase in money paid out by the company is the difference between the new total and the original total:
So the company is paying out more in total this year.
Express the total increase in terms of the raises and bonuses
This total increase of comes from two parts:
- Every employee gets the flat raise dollars. There are 40 employees, so the total from flat raises is .
- Each of the 5 executives gets a one-time bonus, so total bonus money is:
So we can write the equation for the total increase:
Solve the equation for the flat raise r
Now solve the equation
Subtract from both sides:
Divide both sides by 40:
So the flat raise each employee receives is dollars.