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
Let's use the variables A for assistant caregivers and S for senior caregivers.
We know that currently there are 4 assistant caregivers, so A = 4.
We also know that currently there are 12 senior caregivers, so S = 12.
In order to have an equal number of assistant and senior caregivers, we want A = S.
The owner is going to be hiring 3 assistant caregivers per month, so the number of assistant caregivers will increase by 3 each month: A = 4 + 3t, where t represents the number of months.
Similarly, the owner is going to be hiring 1 senior caregiver per month, so the number of senior caregivers will increase by 1 each month: S = 12 + t.
We want A = S, so we can set up the equation: 4 + 3t = 12 + t.
Simplifying the equation, we get 2t = 8.
Dividing both sides of the equation by 2, we get t = 4.
This means it will take 4 months to have an equal number of assistant caregivers and senior caregivers.
Plugging in this value of t into the equations for A and S, we find:
A = 4 + 3(4) = 4 + 12 = 16 assistant caregivers.
S = 12 + 4 = 16 senior caregivers.
So after 4 months, there will be 16 assistant caregivers and 16 senior caregivers in total.