The manager, Kirk, Brian, and Matt's ages are 24, 28, 36, and 18 years, respectively, satisfying the given conditions. The total sum of their ages is 106 years.
X be the manager's age.
Kirk = 2(x - 10) (Kirk is twice ten years less than the manager's age).
Brian = 2x - 12 (Brian is 12 years younger than twice the manager's age).
Matt = (1/2)x + 6 (Matt is 6 years older than half the manager's age).
The total sum of their ages is 106 years.
Equation:
x + 2(x-10) + (2x-12) + ((1/2)x + 6) = 106
Solving:
x + 2x - 20 + 2x - 12 + (1/2)x + 6 = 106
Combine like terms:
5x + (1/2)x - 20 - 12 + 6 = 106
Combine constant terms:
5x + (1/2)x - 26 = 106
Combine the x-terms on one side:
(11/2)x - 26 = 106
Add 26 to both sides:
(11/2)x = 132
Multiply both sides by (2/11) to solve for x:
x = 24
Therefore, x = 24 (Manager's age).
Now, find the ages of the other employees:
Kirk: 2(24 - 10) = 28 years
Brian: 2(24) - 12 = 36 years
Matt: (1/2)(24) + 6 = 18 years
Complete question:
There is a competition at the local movie theater for free movie tickets. You must guess all four employees' ages given a few clues. The first clue is that when added together, their ages total 106 years. Kirk is twice ten years less than the manager's age, Brian is 12 years younger than twice the manager's age, and Matt is 6 years older than half the manager's age. What are all four of their ages? It may help to set up four let statements, one for each employee (including the manager).