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Chapter 11: Problem 48

Express each percent as a fraction in simplest form. $$25 \%$$

Short Answer

Expert verified
25% as a fraction is \( \frac{1}{4} \).

Step by step solution

01

Write the Percent as a Fraction

Start by expressing the percent as a fraction over 100. Since you are given \( 25\% \), write it as \( \frac{25}{100} \). This is equivalent to the decimal 0.25, but we are focusing on the fraction form for this exercise.
02

Simplify the Fraction

To simplify \( \frac{25}{100} \), find the greatest common divisor (GCD) of 25 and 100. The GCD is 25. Divide both the numerator and the denominator by 25 to simplify the fraction: \( \frac{25 \div 25}{100 \div 25} = \frac{1}{4} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Fractions
Simplifying fractions means reducing a fraction to its simplest form. This happens when the numerator and the denominator have no common divisors other than 1.
For example, in the fraction \(\frac{25}{100}\), both numbers can be divided evenly by 25, which simplifies the fraction.
Here鈥檚 how you go about simplifying fractions:
  • Identify the greatest common divisor (GCD) of both numbers.
  • Divide both the numerator and the denominator by that GCD.
Simplification is important because it makes computations easier and the resulting numbers simpler to understand. In our original exercise, after finding that the GCD of 25 and 100 is 25, dividing both by 25 gives us the simplified fraction \(\frac{1}{4}\). This form is much easier to work with in further calculations or comparisons with other fractions.
Greatest Common Divisor
The greatest common divisor (GCD) is the largest integer that divides two numbers without leaving a remainder.
Finding the GCD is crucial in simplifying fractions. It tells us how much we can "shrink" the fraction.
Here's a simple way to find the GCD:
  • List the factors of each number.
  • Identify the largest factor that appears in both lists.
In our exercise example for \(\frac{25}{100}\), we find the factors of 25 (1, 5, 25) and 100 (1, 2, 4, 5, 10, 20, 25, 50, 100).
The largest common factor is 25, making it the GCD. By using the GCD, we ensure the fraction is reduced completely.
Mathematics Education
Mathematics education is about understanding concepts, not just memorizing procedures.
When you learn concepts like simplifying fractions and using the GCD, you develop critical problem-solving and analytical skills.
This educational process involves:
  • Engaging with practical exercises, like converting percentages to fractions.
  • Developing a deep understanding of why and how mathematical processes work.
  • Gaining skills that can be applied to real-world scenarios.
Through consistent practice, students become more confident in tackling math problems.
This helps in building a solid foundation for future learning in mathematics and other analytical subjects.
In our exercise, understanding how to simplify \(\frac{25}{100}\) to \(\frac{1}{4}\) isn't just about the math but about learning a skill that enhances logical thinking.

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