Problem 53 - Prealgebra

Question

If 12 of the 20 students in a class are boys, what percent are boys?

Step-by-Step Solution

Answer

60% of the students are boys.

Step 1: Understand the Question

We are given the number of boys in a class (12) and the total number of students (20). We need to calculate the percentage of students who are boys, which is the fraction of boys relative to the total, converted into a percentage.

Step 2: Set Up the Fraction

The fraction of students who are boys is the number of boys divided by the total number of students. This can be expressed as \( \frac{12}{20} \).

Step 3: Simplify the Fraction

To make calculations easier, simplify \( \frac{12}{20} \) by finding the greatest common divisor of 12 and 20, which is 4. Divide both the numerator and the denominator by 4 to get \( \frac{3}{5} \).

Step 4: Convert the Fraction to a Percentage

To convert \( \frac{3}{5} \) into a percentage, multiply by 100: \[ \frac{3}{5} \times 100 = 60 \% \]. Thus, 60% of the students are boys.

Key Concepts

Understanding Fractions

Fractions are a way to represent a part of a whole. They consist of two parts: the numerator, which is the top number, and the denominator, which is the bottom number.
The numerator shows how many parts we are considering, while the denominator shows the total number of equal parts the whole is divided into. For example, in the fraction \( \frac{12}{20} \), the numerator is 12, indicating the 12 boys we're looking at, and the denominator is 20, representing all the students in the class. Fractions help us understand proportions, making it easier to compare different quantities and solve real-world problems.

• Numerator: The top number indicating parts of interest.
• Denominator: The bottom number indicating total parts.
• Proportion: Represents comparative size relative to a whole.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This involves finding the greatest common divisor (GCD) and dividing both the numerator and the denominator by it. This step makes fractions easier to work with and understand. In the example \( \frac{12}{20} \), the GCD is 4. By dividing both 12 and 20 by 4, we simplify the fraction to \( \frac{3}{5} \). Simplified fractions maintain the same value as the original, but are easier to interpret and use.

• Greatest Common Divisor (GCD): The largest number that divides both the numerator and the denominator without leaving a remainder.
• Maintained Value: Simplified fractions still represent the same quantity or proportion.
• Ease in Use: Simplified fractions make further calculations straightforward.

Converting Fractions to Percentages

To convert a fraction to a percentage, multiply the fraction by 100. This shows the part of the whole as a percentage, an important tool for understanding proportions in a clearer manner. Using our simplified fraction \( \frac{3}{5} \), we multiply by 100 to find the percentage: \[\frac{3}{5} \times 100 = 60\%\]Therefore, 60% of the students are boys. Converting fractions to percentages is useful for comparisons across different contexts, making data analysis more intuitive and accessible to audiences familiar with percentage representation.

• Multiplication by 100: Converts a fraction into a percentage.
• Percentage Representation: Provides a clearer picture of proportions and comparisons.
• Broad Application: Useful in various fields like statistics, finance, and daily life calculations.