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Chapter 9: Problem 82

Write each fraction in simplest form. $$ \frac{75}{100} $$

Short Answer

Expert verified
The simplest form of \(\frac{75}{100}\) is \(\frac{3}{4}\).

Step by step solution

01

Identify Greatest Common Factor

First, identify the greatest common factor (GCF) of both the numerator and the denominator. The numbers are 75 and 100.
02

Find the GCF of 75 and 100

List the factors of 75: (1, 3, 5, 15, 25, 75) and list the factors of 100: (1, 2, 4, 5, 10, 20, 25, 50, 100). The greatest common factor is 25.
03

Simplify the Fraction

Divide both the numerator and the denominator by their greatest common factor. For \(\frac{75}{100}\), divide by 25: \(\frac{75 \div 25}{100 \div 25} = \frac{3}{4}\).

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Key Concepts

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

Greatest Common Factor
The **Greatest Common Factor (GCF)** is crucial in simplifying fractions. It's the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCF makes fraction simplification straightforward.
To identify the GCF:
  • List all factors of each number.
  • Identify the greatest factor that appears in both lists.
For example, to simplify the fraction \( \frac{75}{100} \), list the factors:
  • Factors of 75: 1, 3, 5, 15, 25, 75
  • Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
The GCF is 25. The GCF helps to reduce fractions to their simplest form by dividing both numbers by it.
Numerator and Denominator
In any fraction, the **numerator** is the top number, and the **denominator** is the bottom number. The numerator tells you how many parts you have, and the denominator tells you how many parts make up a whole.
Consider the fraction \( \frac{75}{100} \):
  • Numerator: 75
  • Denominator: 100
These numbers help us understand what proportion of the whole we are dealing with. The goal in fraction simplification is often to reduce both numbers by finding their greatest common factor, allowing both the numerator and denominator to remain in smaller, more manageable terms while maintaining the fraction鈥檚 value.
Fraction Simplification
**Fraction Simplification** involves reducing a fraction to its simplest terms. This means making the numerator and denominator as small as possible while still retaining the same value.
Here's how it works:
  • Find the GCF of both the numerator and the denominator.
  • Divide both the numerator and the denominator by this GCF.
For \( \frac{75}{100} \), the GCF is 25, so we divide both numbers by 25:\[\frac{75 \div 25}{100 \div 25} = \frac{3}{4}\]This gives us a simplified fraction: \( \frac{3}{4} \). Simplifying fractions helps in making calculations easier and the results more interpretable.
Mathematics Education
**Mathematics Education** often involves guiding students through practical and understandable steps to grasp concepts such as simplifying fractions. Learning to simplify fractions is not just about following steps; it helps enhance mathematical literacy and problem-solving skills.
Education strategies include:
  • Using simple problems, gradually increasing in complexity.
  • Encouraging practice, as repetition aids comprehension.
  • Providing visual aids, like fraction bars, to demonstrate concepts.
Understanding processes like finding the GCF and simplifying fractions aids in building a strong foundation in mathematics. It encourages logical thinking and boosts students鈥� confidence in handling numbers.

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