Problem 21 - Prealgebra

Question

Solve each of the following problems by first restating it as one of the three basic percent problems of Section 7.2 . In each case, be sure to show the equation. Population In a city of \(32,000\) people, there are \(10,000\) people under 25 years of age. What percent of the population is under 25 years of age?

Step-by-Step Solution

Answer

31.25% of the population is under 25 years of age.

Step 1: Restate the Problem into a Percent Equation

We need to determine what percent 10,000 people are of the total population of 32,000 people. This can be represented by the equation: \( \frac{10,000}{32,000} \times 100\% = x \% \)

Step 2: Simplify the Fraction

Simplify the fraction \( \frac{10,000}{32,000} \). Divide both the numerator and the denominator by their greatest common divisor, which is 2,000. This simplifies to \( \frac{5}{16} \).

Step 3: Convert the Fraction to a Decimal

Convert \( \frac{5}{16} \) to a decimal by dividing 5 by 16. This results in approximately \( 0.3125 \).

Step 4: Convert Decimal to a Percent

Convert the decimal \( 0.3125 \) to a percent by multiplying by 100. Therefore, \( 0.3125 \times 100 = 31.25\% \).

Step 5: Conclusion

The calculation shows that 31.25% of the city's population is under 25 years of age.

Key Concepts

Percent Equation

To solve percent problems, we often use the percent equation. This equation clearly sets up a relationship between a part, the whole, and the percent value. The typical equation used in percent problems is: \[ \text{Part} = (\text{Percent} \times \text{Whole}) \]When transformed to find the percent, it becomes: \[ \text{Percent} = \left( \frac{\text{Part}}{\text{Whole}} \right) \times 100 \% \]When you have 10,000 people in a group of 32,000, the percent equation allows you to express this relationship as a clear percentage of the total population. It's essentially about asking, "What fraction of the whole is this part, and what is that fraction as a percentage?" By using the percent equation, we can easily solve many real-world problems by understanding how different parts relate to their wholes in percentage terms.

Fractions

Fractions are a foundational concept in mathematics that express parts of a whole. A fraction consists of a numerator and a denominator. The numerator is the top number and indicates the parts we are considering, while the denominator is the bottom number and indicates how many equal parts make up the whole. In the original problem, the fraction \( \frac{10,000}{32,000} \) represents the part (people under 25) over the whole (the total population). To simplify a fraction, we need to divide both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD is 2,000, simplifying our fraction to \( \frac{5}{16} \).Simplifying fractions makes them easier to interpret and work with, especially as you progress to converting them between formats or using them in further calculations.

Decimal Conversion

Transforming a fraction to a decimal is an essential skill, especially when dealing with percentages. This conversion is straightforward. You divide the numerator by the denominator. For the fraction \( \frac{5}{16} \), dividing 5 by 16 gives us the decimal 0.3125. Once you have a decimal, converting it to a percentage is simple: just multiply by 100. In this example, multiplying 0.3125 by 100 results in 31.25. Thus, the decimal 0.3125 represents a 31.25% value of the whole.Decimals and percentages are simpler for most people to understand than fractions, primarily because we regularly encounter percentages in day-to-day life, whether it's sales tax, tips, or understanding how much of a group or quantity you're dealing with.