A county planner prepares the following table showing population trends in three local towns. The first column gives the name of the town. The second column gives the population as of the last census. The third column gives the estimated increase or decrease in population since that census, expressed as a percentage of the previous population.

Question

Town: Pleasantville, Census Population: x, Percentage Change: –2%
Town: Amity Center, Census Population: y, Percentage Change: +5%
Town: Harmony Corners, Census Population: z. Percentage Change: +7%

a. Choose one of these towns. Use the second and third columns to write two expressions showing the current estimated population of the town in terms of x, y, or z.

b. Add a fourth row to the table showing the population information for the town of Concord. Concord’s population as of the last census was p people; for its percentage change, choose a negative integer not divisible by two. Write two expressions showing the current estimated population of Concord in terms of p.

c. Choose a number greater than 5,000 and less than 10,000 for the value of p. Based on the percentage change you chose in part b, calculate the estimated current population of Concord, rounding to the nearest whole number if needed. Explain why you used the expression you did to make your calculation.

Answer 1

a. Let's choose Pleasantville for this problem.

The Census Population is \( x \) and the Percentage Change is \(-2%\). The current estimated population can be calculated by applying the percentage change to the census population. Here are the two expressions:

1. The current population after a decrease of 2%: \[ \text{Current Population} = x + \left(-0.02 \times x\right) = x(1 - 0.02) = 0.98x \]

2. Another way to express the current population is by calculating the decrease explicitly: \[ \text{Current Population} = x - 0.02x = 0.98x \]

Both expressions represent the same current population in terms of \( x \).

b. Let's add a fourth row for the town of Concord.

Concord’s population as of the last census is \( p \), and let's choose a negative integer for the percentage change that is not divisible by two; we can use \(-3%\).

Here’s the updated table including Concord:

| Town | Census Population | Percentage Change | |------------------|------------------|-------------------| | Pleasantville | \( x \) | -2% | | Amity Center | \( y \) | +5% | | Harmony Corners | \( z \) | +7% | | Concord | \( p \) | -3% |

Now, the expressions for the current estimated population of Concord:

1. The current population after a decrease of 3%: \[ \text{Current Population} = p + \left(-0.03 \times p\right) = p(1 - 0.03) = 0.97p \]

2. Another way to express the current population is: \[ \text{Current Population} = p - 0.03p = 0.97p \]

Both expressions also represent the same current population in terms of \( p \).

c. Let's choose a number for \( p \) between 5,000 and 10,000. I'll choose \( p = 8,000 \).

Now, we will calculate the estimated current population of Concord using the expression we derived:

\[ \text{Current Population} = 0.97p \] Substituting \( p = 8,000 \): \[ \text{Current Population} = 0.97 \times 8,000 = 7,760 \]

Thus, the estimated current population of Concord is 7,760.

I chose this expression \( 0.97p \) because it accurately factors in the -3% decrease from the original population, yielding the updated estimate post-census. This expression reflects the population after the initial census figure has been adjusted for the specified percentage change.