A daycare center in Oakland currently has 4 assistant caregivers and 12 senior caregivers. Since demand is high, the owner is going to be hiring 3 assistant caregivers per month and 1 senior caregiver per month. Her goal is to have a larger staff, including an equal number of assistant caregivers and senior caregivers. How long will that take? How many of each type will there be?
System of equations Use x and y
Let x represent the number of months and y represent the number of assistant caregivers and senior caregivers.
The initial number of assistant caregivers is 4 and the number of senior caregivers is 12, so we can write the following equation:
y = x + 4
The number of assistant caregivers increases by 3 per month and the number of senior caregivers increases by 1 per month, so we can write the following equation:
y + 3x = x + 12
Substituting the first equation into the second equation, we get:
x + 4 + 3x = x + 12
Combining like terms, we get:
4x + 4 = x + 12
Subtracting x from both sides, we get:
3x + 4 = 12
Subtracting 4 from both sides, we get:
3x = 8
Dividing both sides by 3, we get:
x = 8/3
Therefore, it will take approximately 2.67 months to achieve the owner's goal.
Substituting x = 8/3 into the first equation, we get:
y = 8/3 + 4
Simplifying, we get:
y = 8/3 + 12/3
Combining terms, we get:
y = 20/3
Therefore, there will be 2.67 assistant caregivers and 6.67 senior caregivers, which can be approximated to 3 assistant caregivers and 7 senior caregivers.