The process of simplifying fractions is a fundamental mathematical concept that involves reducing a given fraction to its lowest possible terms. This process is essential in various mathematical applications, including algebra, geometry, and calculus.
How to simplify fractions
Are you ready to simplify fractions like a pro? It’s all about reducing them to their simplest form, where the top and bottom numbers have no common factors except for 1. We call these simplified fractions “lowest terms” or “reduced fractions.” So let’s get started and make math a breeze!
| Step 1: Identify the Greatest Common Divisor (GCD) | ➕ |
| The first step in simplifying a fraction is to find the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that evenly divides both the numerator and the denominator. | |
| Step 2: Divide by the GCD | ➕ |
| Once you’ve determined the GCD, you’ll divide both the numerator and the denominator of the fraction by this common divisor. This step ensures that the fraction is reduced to its simplest form. | |
| Step 3: Verify for Further Simplification | ➕ |
| After dividing by the GCD, check if there are any common factors left between the numerator and the denominator. If there are, repeat steps 1 and 2 until no common factors remain other than 1. | |
| Step 4: Express the Simplified Fraction | ➕ |
| The result of these steps will be the simplified or reduced fraction, which is in its simplest form. | |
Simplifying fractions: Examples
| Simplify the fractions: $$ \frac{12}{30},\quad \frac{33}{55},$$ | ➕ |
| For the first fraction, we remark that the factors of $12$ are $1,2,3,4,6,12,$ and the factors of $30$ are $1,2,3,5,6,10,15,30$. Therefore the GCD of $12$ and $30$ is $6$. Observe that $$ \frac{12}{30}=\frac{2\times 6}{5\times 6}=\frac{2}{5}.$$ For the seconde fraction, the factors of $33$ are $1,3,11,33$ and the factors for $55$ are $1,5,11,55$. Then the GCD of $33$ and $55$ is $11$. In addition, $33=3\times 11$ and $55=5\times 11$. Therefore $$\frac{33}{55}=\frac{3\times 11}{5\times 11}=\frac{3}{5}.$$ | |
Conclusion
To simplify a fraction, one must divide both the numerator and denominator by their greatest common factor (GCF). The GCF is the largest number that divides both the numerator and denominator without leaving a remainder. By dividing both the numerator and denominator by the GCF, the fraction is reduced to its simplest form. This simplification process is crucial in solving mathematical problems that involve fractions, as it allows for easier manipulation and calculation. Therefore, mastering the skill of simplifying fractions is essential for any student or individual seeking to excel in mathematics.
