Simplifying Fractions (Lowest Terms): Fifth Grade Mathematics

Simplifying fractions into lowest terms expresses a fractional amount using the smallest possible whole numbers for the numerator and denominator.

What Are Lowest Terms?

A fraction is in simplest form (or lowest terms) when the Greatest Common Factor (GCF) of the numerator and the denominator is 1. This means the numerator and denominator share no common factors other than 1.

+--------------------------------------------------------------------------+
|                  TWO METHODS FOR SIMPLIFYING FRACTIONS                   |
+---------------------+----------------------------------------------------+
| Method              | Procedure                                          |
+---------------------+----------------------------------------------------+
| Single-Step (GCF)   | Find the GCF of numerator and denominator; divide  |
|                     | both terms by the GCF in one single step.          |
| Multi-Step Repeated | Divide numerator and denominator repeatedly by any |
| Division            | shared common factor until no more exist.          |
+---------------------+----------------------------------------------------+

Method 1: The One-Step GCF Technique

Simplifying 24/36 in one step:
- Factors of 24: {1, 2, 3, 4, 6, 8, 12, 24}

- Factors of 36: {1, 2, 3, 4, 6, 9, 12, 18, 36}

- GCF = 12.

- Divide numerator and denominator by 12: (24 / 12) / (36 / 12) = 2/3.

- Because GCF(2, 3) = 1, 2/3 is in simplest form.

Method 2: Repeated Division

If finding the GCF immediately is difficult, use divisibility rules to divide by smaller common factors repeatedly:
- 24/36: Both numbers are even, divide by 2 -> 12/18.

- 12/18: Both numbers are even, divide by 2 -> 6/9.

- 6/9: Both numbers divisible by 3, divide by 3 -> 2/3.

- Both methods yield the identical simplified fraction.

Recognizing Fractions Already in Simplest Form

You can quickly tell a fraction is already in simplest form if:
- The numerator is 1 (any unit fraction like 1/5 is in simplest form).

- The numerator and denominator are consecutive whole numbers (like 7/8 or 14/15; consecutive numbers share no common factors other than 1).

- Both the numerator and denominator are prime numbers (like 3/7).

Exercises and Solutions

Exercise 1: Single-Step GCF Simplification. Simplify the fraction 28/42 to lowest terms using the Greatest Common Factor method. State the GCF and show the division.

Solution 1: Find the factors of 28: {1, 2, 4, 7, 14, 28}. Find the factors of 42: {1, 2, 3, 6, 7, 14, 21, 42}. The common factors are 1, 2, 7, and 14. The Greatest Common Factor is 14. Divide both the numerator and denominator by 14: 28 / 14 = 2, and 42 / 14 = 3. The simplified fraction in lowest terms is 2/3.

Exercise 2: Simplification Analysis. A student simplifies 30/45 by dividing both numbers by 5, obtaining 6/9, and stops there. Is the fraction in simplest form? If not, complete the simplification.

Solution 2: The fraction 6/9 is not in simplest form because the numerator 6 and the denominator 9 still share a common factor of 3 (GCF is not 1). To complete the simplification, divide both 6 and 9 by 3: 6 / 3 = 2, and 9 / 3 = 3. The true simplest form is 2/3. The student could have reached 2/3 in one step by dividing by the GCF of 30 and 45, which is 15.