A mean tracks total pay per person, so compare the old and new means to find the average increase. If everyone gets a raise but only a few get bonuses, spread the bonuses across the whole group before finding the raise. Then a one-variable Desmos regression can solve the pay-change equation. Treating the entire average increase as the raise ignores the bonuses.
Hints
- Hint 1
The mean is total pay divided by the number of employees. Subtract the old mean from the new one to find the average increase. Does all of that increase come from the flat raise?
- Hint 2
Only the executives get bonuses, but the new mean covers all 40 employees. Multiply the number of bonuses by the bonus amount, then divide by 40 to find how much the bonuses add to the mean.
- Hint 3
The flat raise goes to everyone, so it increases the mean by exactly . Add that to the bonuses' effect on the mean. What should their sum equal?
Step-by-step
Separate the raise from the bonuses
Step 1Find the increase in mean pay
Subtract the old mean from the new mean to find the average increase per employee: . The $6,000 isn't automatically the flat raise: the bonuses also raised the mean.
- Step 2
Find the bonuses' effect on the mean
There are bonuses of $10,000, but the mean includes all 40 employees. Type in Desmos; it prints . So the bonuses add to the company-wide mean. Dividing by instead would give the bonus per executive, not its effect on that mean.
- Step 3
Write the pay-change equation
The flat raise reaches every employee, so it adds exactly to the mean. The bonuses add , and together those changes make the increase: . Here means the flat raise in dollars.
- Step 4
Solve for the flat raise
Below the earlier line, type in Desmos. Use for and instead of ; this regression finds the value that makes both sides match. Under PARAMETERS, Desmos shows . So each employee's flat raise is $4,750. Grid in 4750.