Problem 72 - Algebra for College Students

Question

Between 2000 and \(2016,\) the estimated population of metro Chicago, Illinois, grew from 9,098,316 to \(9,512,999 .\) What was the percent increase to the nearest tenth?

Step-by-Step Solution

Answer

The percent increase in population is approximately 4.6%.

Step 1 - Identify Initial and Final Population

The initial population in 2000 is 9,098,316. The final population in 2016 is 9,512,999.

Step 2 - Calculate the Change in Population

Subtract the initial population from the final population: \[ \text{Change in Population} = 9,512,999 - 9,098,316 = 414,683 \]

Step 3 - Determine the Relative Increase

Divide the change in population by the initial population to get the relative increase: \[ \text{Relative Increase} = \frac{414,683}{9,098,316} \approx 0.0456 \]

Step 4 - Convert to Percent

Multiply the relative increase by 100 to get the percent increase: \[ \text{Percent Increase} = 0.0456 \times 100 = 4.56 \% \]

Step 5 - Round to the Nearest Tenth

Round 4.56% to the nearest tenth: \[ \text{Rounded Percent Increase} \approx 4.6 \% \]

Key Concepts

population change

Population change refers to the variation in the number of individuals in a population over a specific period. It's a key metric for understanding demographics and planning for resources. To calculate the population change, we subtract the initial population from the final population.

For example, in the exercise, the population of metro Chicago increased from 9,098,316 in 2000 to 9,512,999 in 2016. This means the change in population is:

\[ \text{Change in Population} = 9,512,999 - 9,098,316 = 414,683 \baş\]

This value reflects the growth in the population over the 16-year period.

relative increase

Relative increase gives an idea of how significant the change in population is in relation to the initial population. It offers a better perspective than the absolute change alone, as it accounts for the size of the initial population.

To calculate the relative increase, we divide the change in population by the initial population:

\[ \text{Relative Increase} = \frac{414,683}{9,098,316} \baş\]

This equals approximately 0.0456. The result tells us that the population increased by 4.56% relative to its size in the year 2000.

percent increase

Percent increase translates the relative increase into a percentage, making it easier to understand and communicate.

The formula for calculating the percent increase is: \[ \text{Percent Increase} = \text{Relative Increase} \times 100 \baş\]

In the exercise, we found the relative increase to be 0.0456. To find the percent increase, we multiply this by 100:

\[ \text{Percent Increase} = 0.0456 \times 100 = 4.56 \baş\]

For more clarity, we often round the percent increase to the nearest tenth, giving us 4.56% rounded to approximately 4.6%.