Simplifying Fractions

When it comes to simplifying fractions, the main goal is to express a fraction in its lowest terms. This means that the numerator (the top number) and the denominator (the bottom number) have no common factors other than 1. Simplifying makes fractions easier to understand and work with, especially in more complex mathematical problems.

To simplify a fraction, follow these simple steps:

Steps to Simplify Fractions

  1. Identify the Numerator and Denominator: A fraction is expressed as \(\frac{a}{b}\) where \(a\) is the numerator and \(b\) is the denominator.

  2. Find the Greatest Common Divisor (GCD): The GCD of two numbers is the largest number that divides both without leaving a remainder. To find the GCD, you can use the prime factorization method or the Euclidean algorithm.

  3. Divide Both the Numerator and Denominator by the GCD: Once you have the GCD, divide both numbers by it to arrive at the simplified fraction.

  4. Check for Further Simplification: Lastly, ensure that the new numerator and denominator do not have any common factors other than 1.

Let’s look into these steps a bit deeper with some examples.

Example 1: Simplifying the Fraction \(\frac{12}{16}\)

  1. Identify the Numbers: Here, the numerator is 12, and the denominator is 16.

  2. Find the GCD:

    • The factors of 12 are: 1, 2, 3, 4, 6, 12
    • The factors of 16 are: 1, 2, 4, 8, 16

    The largest common factor is 4. Thus, GCD(12, 16) = 4.

  3. Divide Both by the GCD:

    • \(\frac{12 \div 4}{16 \div 4} = \frac{3}{4}\)

  4. Check for Further Simplification: The numbers 3 and 4 have no common factors other than 1, so \(\frac{3}{4}\) is in its simplest form.

Example 2: Simplifying the Fraction \(\frac{18}{24}\)

  1. Identify the Numbers: Here, the numerator is 18, and the denominator is 24.

  2. Find the GCD:

    • The factors of 18 are: 1, 2, 3, 6, 9, 18
    • The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24

    The GCD is 6.

  3. Divide Both by the GCD:

    • \(\frac{18 \div 6}{24 \div 6} = \frac{3}{4}\)

  4. Check for Further Simplification: Again, 3 and 4 have no common factors other than 1, confirming that \(\frac{3}{4}\) is in its simplest form.

Example 3: Simplifying the Fraction \(\frac{45}{60}\)

  1. Identify the Numbers: Numerator: 45, Denominator: 60.

  2. Find the GCD:

    • The factors of 45 are: 1, 3, 5, 9, 15, 45
    • The factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

    The GCD is 15.

  3. Divide Both by the GCD:

    • \(\frac{45 \div 15}{60 \div 15} = \frac{3}{4}\)

  4. Check for Further Simplification: The simplified fraction is once again \(\frac{3}{4}\).

Example 4: Simplifying the Fraction \(\frac{25}{30}\)

  1. Identify the Numbers: Numerator: 25, Denominator: 30.

  2. Find the GCD:

    • The factors of 25 are: 1, 5, 25
    • The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30

    The GCD is 5.

  3. Divide Both by the GCD:

    • \(\frac{25 \div 5}{30 \div 5} = \frac{5}{6}\)

  4. Check for Further Simplification: With 5 and 6 having no common factors other than 1, \(\frac{5}{6}\) is in its simplest form.

Example 5: Simplifying the Fraction \(\frac{50}{100}\)

  1. Identify the Numbers: Numerator: 50, Denominator: 100.

  2. Find the GCD:

    • The factors of 50 are: 1, 2, 5, 10, 25, 50
    • The factors of 100 are: 1, 2, 4, 5, 10, 20, 25, 50, 100

    The GCD is 50.

  3. Divide Both by the GCD:

    • \(\frac{50 \div 50}{100 \div 50} = \frac{1}{2}\)

  4. Check for Further Simplification: Here, 1 and 2 have no common factors other than 1, thus \(\frac{1}{2}\) is in its simplest form.

Quick Method: Using Prime Factorization

Sometimes using prime factorization can streamline the process of finding the GCD. Here’s how it works:

  1. Prime Factorization of Both Numbers: Write both the numerator and the denominator as products of their prime factors.

  2. Identify Common Prime Factors: Identify the common prime factors and multiply them to find the GCD.

  3. Divide by the GCD as Before: Follow the earlier steps to simplify.

Example: Simplifying \(\frac{72}{90}\) Using Prime Factorization

  1. Prime Factorization:

    • 72 = \(2^3 \times 3^2\)
    • 90 = \(2^1 \times 3^2 \times 5^1\)

  2. Common Prime Factors: The common primes are \(2^1\) and \(3^2\). Thus, GCD = \(2 \times 9 = 18\).

  3. Divide:

    • \(\frac{72 \div 18}{90 \div 18} = \frac{4}{5}\)

  4. Check Simplicity: The numbers 4 and 5 have no common factors other than 1.

Exercises to Practice Simplifying Fractions

  1. Simplify \(\frac{36}{48}\)
  2. Simplify \(\frac{84}{126}\)
  3. Simplify \(\frac{14}{35}\)
  4. Simplify \(\frac{32}{96}\)
  5. Simplify \(\frac{60}{90}\)

Answers to Exercises

  1. \(\frac{3}{4}\)
  2. \(\frac{2}{3}\)
  3. \(\frac{2}{5}\)
  4. \(\frac{1}{3}\)
  5. \(\frac{2}{3}\)

Conclusion

Simplifying fractions is a fundamental skill in mathematics that applies to many areas of math and real-life situations. By mastering the steps of finding the GCD and dividing both the numerator and denominator, you can easily express fractions in their simplest form. Practicing with exercises will further strengthen your understanding and proficiency in working with fractions. Happy simplifying!