In​ 2020, there were 15 comma 800 students at college​ A, with a projected enrollment increase of 500 students per year. In the same​ year, there were 32 comma 300 students at college​ B, with a projected enrollment decline of 1000 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?
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Part 1
In the year

enter your response here​, the enrollment at both colleges will be the same.

Let the number of years after 2020 be denoted as \( t \).

- College A starts with 15,800 students and gains 500 students per year.
  The number of students at College A after \( t \) years is:
  \[
  \text{Enrollment at College A} = 15800 + 500t
  \]

- College B starts with 32,300 students and loses 1000 students per year.
  The number of students at College B after \( t \) years is:
  \[
  \text{Enrollment at College B} = 32300 - 1000t
  \]

We want to find the value of \( t \) when both colleges have the same enrollment. Set the two equations equal to each other:
\[
15800 + 500t = 32300 - 1000t
\]

### Solve for \( t \):
1. Add \( 1000t \) to both sides:
   \[
   15800 + 1500t = 32300
   \]
2. Subtract 15,800 from both sides:
   \[
   1500t = 16500
   \]
3. Divide by 1500:
   \[
   t = \frac{16500}{1500} = 11
   \]

Thus, the colleges will have the same enrollment **11 years after 2020**, which is in the year **2031**.

### Part 2: Find the enrollment in that year:
Substitute \( t = 11 \) into either equation to find the enrollment.

Using College A's equation:
\[
\text{Enrollment at College A} = 15800 + 500(11) = 15800 + 5500 = 21300
\]

Thus, the enrollment at both colleges will be **21,300** in the year **2031**.

Would you like further details or explanations?

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Here are 5 related questions:
1. How do changes in enrollment growth rates affect future projections?
2. Can we model enrollment trends for other years based on these equations?
3. What happens if College A’s growth rate changes over time?
4. How would the solution change if College B’s decline rate is reduced to 500 students per year?
5. How can we predict enrollment in 2035 for both colleges?

**Tip:** When working with word problems, translating the information into mathematical equations is key to solving them effectively!

In 2020, there were 15,800 students at college A, with a projected enrollment increase of 500 students per year. In the same year, there were 32,300 students at college B, with a projected enrollment decline of 1,000 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?

Learn how to calculate when two colleges will have the same enrollment using linear equations. This problem involves projecting changes in enrollment over time and solving for the point of equality. Suitable for Grades 8-10, with a detailed step-by-step solution.

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