Question
The population of a small town decreases by 50 people each year due to migration. If the current population is
2500, how many years will it take for the population to reach 2000? Set up an inequality to solve this problem
and explain your reasoning
2500, how many years will it take for the population to reach 2000? Set up an inequality to solve this problem
and explain your reasoning
Answers
GPT-4o mini
To determine how many years it will take for the population of the town to decrease from 2500 to 2000 due to migration, we can set up an equation based on the annual decrease of 50 people.
Let \( y \) represent the number of years that will pass. Each year, the population decreases by 50, so after \( y \) years, the population will be:
\[
\text{Population after } y \text{ years} = \text{Current Population} - (\text{Decrease per year} \times \text{Years})
\]
Substituting the known values into the equation:
\[
\text{Population after } y \text{ years} = 2500 - 50y
\]
We want to find out when this population will be less than or equal to 2000:
\[
2500 - 50y \leq 2000
\]
Now, we will solve this inequality step by step.
1. **Subtract 2500 from both sides:**
\[
-50y \leq 2000 - 2500
\]
\[
-50y \leq -500
\]
2. **Divide both sides by -50:** (Note that when dividing or multiplying by a negative number, the inequality sign flips.)
\[
y \geq \frac{-500}{-50}
\]
\[
y \geq 10
\]
This means that it will take at least 10 years for the population to decrease to 2000 or less. Specifically, after 10 years, the population will be exactly 2000:
\[
\text{Population after 10 years} = 2500 - 50 \times 10 = 2500 - 500 = 2000.
\]
In conclusion, the inequality \( 2500 - 50y \leq 2000 \) helps us find the minimum number of years, \( y \), required for the population to decrease from 2500 to 2000, which is at least 10 years.
Let \( y \) represent the number of years that will pass. Each year, the population decreases by 50, so after \( y \) years, the population will be:
\[
\text{Population after } y \text{ years} = \text{Current Population} - (\text{Decrease per year} \times \text{Years})
\]
Substituting the known values into the equation:
\[
\text{Population after } y \text{ years} = 2500 - 50y
\]
We want to find out when this population will be less than or equal to 2000:
\[
2500 - 50y \leq 2000
\]
Now, we will solve this inequality step by step.
1. **Subtract 2500 from both sides:**
\[
-50y \leq 2000 - 2500
\]
\[
-50y \leq -500
\]
2. **Divide both sides by -50:** (Note that when dividing or multiplying by a negative number, the inequality sign flips.)
\[
y \geq \frac{-500}{-50}
\]
\[
y \geq 10
\]
This means that it will take at least 10 years for the population to decrease to 2000 or less. Specifically, after 10 years, the population will be exactly 2000:
\[
\text{Population after 10 years} = 2500 - 50 \times 10 = 2500 - 500 = 2000.
\]
In conclusion, the inequality \( 2500 - 50y \leq 2000 \) helps us find the minimum number of years, \( y \), required for the population to decrease from 2500 to 2000, which is at least 10 years.