Questions: In 2020, there were 12,300 students at college A, with a projected enrollment increase of 500 students per year. In the same year, there were 21,050 students at college B, with a projected enrollment decline of 750 students per year. According to these projections, when will the colleges have the same enrollment? What will be the enrollment in each college at that time? In the year . the enrollment at both colleges will be the same. The total enrollment at each college will be students.

In 2020, there were 12,300 students at college A, with a projected enrollment increase of 500 students per year. In the same year, there were 21,050 students at college B, with a projected enrollment decline of 750 students per year. According to these projections, when will the colleges have the same enrollment? What will be the enrollment in each college at that time?

In the year . the enrollment at both colleges will be the same.

The total enrollment at each college will be students.

Transcript text: In 2020, there were 12,300 students at college $A$, with a projected enrollment increase of 500 students per year. In the same year, there were 21,050 students at college $B$, with a projected enrollment decline of 750 students per year. According to these projections, when will the colleges have the same enrollment? What will be the enrollment in each college at that time? In the year $\square$ . the enrollment at both colleges will be the same. The total enroliment at each colloge will be $\square$ students.

Solution

Solution Steps

Step 1: Define the Variables

Let $ t $ be the number of years after 2020. We need to find the year when the enrollments at both colleges are equal.

Step 2: Write the Equations

For college $ A $, the enrollment after $ t $ years is given by: \[ E_A(t) = 12300 + 500t \]

For college $ B $, the enrollment after $ t $ years is given by: \[ E_B(t) = 21050 - 750t \]

Step 3: Set the Equations Equal

To find when the enrollments are the same, set $ E_A(t) = E_B(t) $: \[ 12300 + 500t = 21050 - 750t \]

Step 4: Solve for $ t $

Combine like terms: \[ 12300 + 500t + 750t = 21050 \] \[ 12300 + 1250t = 21050 \]

Subtract 12300 from both sides: \[ 1250t = 8750 \]

Divide by 1250: \[ t = \frac{8750}{1250} = 7 \]

Step 5: Calculate the Year

The year when the enrollments are equal is: \[ 2020 + 7 = 2027 \]

Step 6: Calculate the Enrollment

Substitute $ t = 7 $ back into either equation to find the enrollment: \[ E_A(7) = 12300 + 500 \times 7 = 12300 + 3500 = 15800 \]

Final Answer

In the year $\boxed{2027}$, the enrollment at both colleges will be the same. The total enrollment at each college will be $\boxed{15800}$ students.

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