Problem 118 - Elementary Algebra

Question

In 10 years, the population of Detroit fell from 950,000 to about 712,500 . Find the percent decrease.

Step-by-Step Solution

Answer

The percent decrease in population is 25%.

Step 1: Identify initial and final population

The initial population of Detroit was 950,000. The final population after 10 years is 712,500.

Step 2: Calculate the decrease in population

Subtract the final population from the initial population to find the decrease: \[ 950,000 - 712,500 = 237,500 \]

Step 3: Find the percent decrease

Use the formula for percent decrease, which is: \[ \text{Percent Decrease} = \frac{\text{Decrease in Population}}{\text{Initial Population}} \times 100 \]Substitute the values: \[ \frac{237,500}{950,000} \times 100 = 0.25 \times 100 = 25\text{%} \]

Key Concepts

Population Change

Population change is a key concept in understanding how the number of inhabitants in a location varies over a period. It occurs due to factors like birth rates, death rates, and migration patterns. In our example, Detroit's population changed over a span of 10 years.
To clearly see this change, we calculate the difference between the initial population and the final population. Knowing the magnitude of change helps us understand the dynamics and severity of the shifts happening in the population.

Initial and Final Values

In any calculation involving changes, it is crucial to identify the initial and final values.
Here, the initial population of Detroit was 950,000. This is the starting value before the demographic changes took place.
The final population, which is 712,500, represents the population after the 10-year period.
Determining these values accurately ensures that our calculations reflect the true extent of change. Initial and final values provide a benchmark that lets us measure how much change has occurred, making them indispensable in percentage change calculations.

Percent Decrease Formula

The percent decrease formula is a tool used to quantify the reduction of an initial value in percentage terms.
To find percent decrease, you use the formula:
\[ \text{Percent Decrease} = \frac{\text{Decrease in Population}}{\text{Initial Population}} \times 100 \]
This equation helps us convert the actual decrease in numbers to a percentage, making it easier to comprehend the scale of the reduction.
For Detroit, we found the decrease to be 237,500 people (950,000 - 712,500).
Substituting into the formula, it becomes:
\[ \frac{237,500}{950,000} \times 100 = 25\% \]
This means Detroit's population decreased by 25% over 10 years. Understanding this formula is useful for analyzing reductions in various fields, not just population studies.