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.
Town Census Population Percentage Change
Pleasantville x –2%
Amity Center y +5%
Harmony Corners z +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.
Let's choose Pleasantville.
To find the current estimated population of Pleasantville, we need to subtract 2% of x from x.
So the expression for the current estimated population of Pleasantville is: x - 0.02x = x(1 - 0.02) = x(0.98)
Therefore, the current estimated population of Pleasantville is 0.98x.
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.
Town | Census Population | Percentage Change
------------------------------------------
Pleasantville | x | -2%
Amity Center | y | +5%
Harmony Corners | z | +7%
Concord | p | -9%
To find the current estimated population of Concord, we need to subtract 9% of p from p.
So the expression for the current estimated population of Concord is: p - 0.09p = p(1 - 0.09) = p(0.91)
Therefore, the current estimated population of Concord is 0.91p.
Another expression can be written as follows: p * (1 - 9/100) = p * (1 - 0.09) = p * 0.91
Therefore, the current estimated population of Concord is also 0.91p.
. 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.
Let's choose p = 7000.
Using the expression 0.91p, we can calculate the current estimated population of Concord:
0.91 * 7000 = 6370
So, based on the chosen percentage change of -9% and the value of p = 7000, the estimated current population of Concord is 6370 (rounded to the nearest whole number).
We used the expression 0.91p because the percentage change given for Concord is -9%. To calculate the estimated current population, we need to subtract 9% of the previous population. Multiplying by 0.91 (1 - 0.09) is the same as subtracting 9%.
To write expressions showing the current estimated population of a town in terms of x, y, or z, we need to consider the initial population and the percentage change.
Let's take the town Pleasantville as an example.
The second column gives the population as of the last census, which is represented by x.
The third column gives the estimated decrease in population since that census, expressed as a percentage of the previous population, which is -2%.
To calculate the current estimated population, we need to apply the percentage change to the previous population.
First, we calculate the percentage decrease by multiplying x by -2% or -0.02:
Decrease = -0.02x
Next, we subtract the decrease from the initial population to get the current estimated population:
Current estimated population of Pleasantville = x + (-0.02x)
Simplifying this expression gives us:
Current estimated population of Pleasantville = (1 - 0.02)x
Therefore, the expression for the current estimated population of Pleasantville in terms of x is: (1 - 0.02)x, where x represents the population at the last census.
Similarly, you can write expressions for the other towns, Amity Center and Harmony Corners, using the given data for y and z respectively.