In 2005 , there were 19,700 students at college A, with a projected enrollment increase of 1250 students per year. In the same year, there were 35,450 students at college B, with a projected enrollment decline of 500 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 enrollment at each college will be \( \square \) students.

To solve this problem, we need to set up an equation that represents the enrollment at each college over time and then find the year when the enrollments will be equal. Let's denote the number of years after 2005 as \( n \). The enrollment at College A can be represented as: \[ A(n) = 19,700 + 1,250n \] The enrollment at College B can be represented as: \[ B(n) = 35,450 - 500n \] We want to find the value of \( n \) for which \( A(n) = B(n) \). So, we set the two equations equal to each other: \[ 19,700 + 1,250n = 35,450 - 500n \] Now, we solve for \( n \): \[ 1,250n + 500n = 35,450 - 19,700 \] \[ 1,750n = 15,750 \] \[ n = \frac{15,750}{1,750} \] \[ n = 9 \] So, it will take 9 years after 2005 for the enrollments to be the same. To find the year, we add 9 to 2005: \[ 2005 + 9 = 2014 \] In the year 2014, the enrollment at both colleges will be the same. Now, let's find the enrollment in each college at that time. We can use either of the two equations, but let's use the one for College A: \[ A(9) = 19,700 + 1,250 \times 9 \] \[ A(9) = 19,700 + 11,250 \] \[ A(9) = 30,950 \] So, in the year 2014, the enrollment at each college will be 30,950 students.