Problem 23 - Calculate With Confidence

Question

In a class of 30 students, 6 students did not pass an exam. What percentage of the students did not pass the exam? _______

Step-by-Step Solution

Answer

20% of the students did not pass the exam.

Step 1: Identify the Total and Unsuccessful Students

The total number of students in the class is 30, and the number of students who did not pass is 6.

Step 2: Use the Percentage Formula

The percentage of students who did not pass can be found by dividing the number of students who did not pass by the total number of students, then multiplying by 100 to convert it to a percentage. The formula is: \[ \text{Percentage} = \left( \frac{\text{Number of students who didn't pass}}{\text{Total number of students}} \right) \times 100 \]

Step 3: Substitute the Values into the Formula

Substitute 6 for the number of students who didn't pass and 30 for the total number in the formula: \[ \text{Percentage} = \left( \frac{6}{30} \right) \times 100 \]

Step 4: Calculate the Fraction

First, calculate \( \frac{6}{30} \):\[ \frac{6}{30} = 0.2 \]

Step 5: Convert the Fraction to a Percentage

Multiply the fraction by 100 to find the percentage: \[ 0.2 \times 100 = 20 \]

Step 6: Conclusion

Therefore, 20% of the students did not pass the exam.

Key Concepts

Understanding Mathematics Education

Mathematics education is essential as it builds the foundation for logical reasoning and problem-solving skills that students use in everyday life. Learning math involves understanding various methods for calculations and solutions to real-world problems. This particular exercise focuses on finding percentages, which is a versatile skill useful in areas such as finance, science, and general decision-making. By practicing these calculations, students develop a better ability to analyze and interpret data.

Mastering mathematics also fosters critical thinking and nurtures a student's ability to approach problems methodically and creatively. Therefore, even a simple exercise such as converting fractions to percentages plays a critical role in the broader educational journey.

Changing Fractions to Percentages

Fractions and percentages are both ways to express parts of a whole. Converting a fraction to a percentage involves interpreting the fraction into a more relatable format for many applications. The percentage provides a clear understanding of the proportion with respect to 100, making comparisons easier.

To convert a fraction such as \( \frac{6}{30} \), follow these steps:

• First, divide the numerator (top number) by the denominator (bottom number): \( \frac{6}{30} = 0.2 \).
• Then, convert the result to a percentage by multiplying by 100: \( 0.2 \times 100 = 20 \).

Thus, 20% of the students did not pass, showing how fractions translate into percentages to provide clarity in situations.

Importance of Basic Arithmetic

Basic arithmetic is the foundation of all mathematics, including addition, subtraction, multiplication, and division. These operations are key in this exercise which involves dividing and multiplying to reach a final percentage.

In arithmetic, understanding how to divide fractions properly is essential. When dividing 6 by 30, we aim to simplify or reduce the fraction to its lowest form before converting it into a decimal. This step is crucial in ensuring the accuracy of the conversion to a percentage.

Having a solid grasp of arithmetic skills allows students to tackle more complex problems with confidence. It also enhances their ability to interpret everyday numerical information efficiently.

Steps to Solve the Problem

Breaking down a problem into clear steps is a vital problem-solving skill. Whether it's basic math or complex algorithms, organizing the approach helps in understanding and solving effectively. Here’s how we solve the given exercise systematically:

• Identify the essential numbers. Here, the total students and the ones who did not pass.
• Set up the formula. Utilize the percentage formula \( \left( \frac{\text{Number of students who didn't pass}}{\text{Total number of students}} \right) \times 100 \).
• Insert the given values into the formula. This translates the situation into mathematical terms.
• Perform the arithmetic operations. First, calculate the fraction \( \frac{6}{30} \) and then multiply by 100.
• Conclude with a clear statement. The final step provides the result that 20% of students didn’t pass.

Structuring problems in steps not only simplifies the process but also highlights areas of improvement where students can double-check their work.