Simplifying Fractions

Simplifying fractions is an essential skill in mathematics that makes calculations easier and helps you better understand the relationship between numbers. But what does it actually mean to simplify a fraction, and why is it so important? Let’s break it down.

What Does It Mean to Simplify a Fraction?

This process continues until the numerator and denominator share no common divisors other than 1.

At this point, the fraction is considered reduced to its simplest form.

A Practical Example

Let’s simplify the fraction \( \frac{18}{24} \):

1. Identify the common divisors of 18 and 24. Both are divisible by 2.
2. Divide the numerator and denominator by 2:  $$ \frac{18}{24} = \frac{18 \div 2}{24 \div 2} = \frac{9}{12} $$

Notice that \( \frac{3}{4} \) is an equivalent fraction to \( \frac{9}{12} \), since we applied the property of invariance. Both fractions have the same decimal value (0.75).

$$ \frac{3}{4} = \frac{9}{12} = 0.75 $$

However, \( \frac{3}{4} \) is made up of smaller numbers, making it much easier to work with. This is called the simplified form.

The fraction \( \frac{9}{12} \) can still be simplified further because both 9 and 12 are divisible by 3:

$$ \frac{9}{12} = \frac{9 \div 3}{12 \div 3} = \frac{3}{4} $$

Now, \( \frac{3}{4} \) is fully simplified and reduced to its lowest terms, as 3 and 4 have no common divisors other than 1.

Why Simplify Fractions?

Simplifying fractions makes calculations easier and more intuitive, especially when performing operations like addition, subtraction, multiplication, and division.

For instance, working with smaller numbers is more straightforward. The fraction \( \frac{3}{4} \) is far easier to handle than \( \frac{18}{24} \), even though they represent the same decimal value (0.75).

$$ \frac{3}{4} = \frac{18}{24} = 0.75 $$

A simplified fraction is also clearer and more readable, as it provides a direct understanding of the relationship between the numerator and denominator.

Additionally, working with large numbers increases the likelihood of errors during calculations.

While unsimplified fractions can sometimes be confusing, there are cases where it’s necessary to temporarily work with them. For example, when adding \( \frac{1}{2} \) and \( \frac{5}{6} \): $$
\frac{1}{2} + \frac{5}{6} $$ You need to convert both fractions to a common denominator before adding. In this case, multiply the numerator and denominator of \( \frac{1}{2} \) by 3 to match the denominator of \( \frac{5}{6} \): $$ \frac{1 \cdot 3}{2 \cdot 3} + \frac{5}{6} = \frac{3}{6} + \frac{5}{6} = \frac{3+5}{6} = \frac{8}{6} $$ After performing the addition, simplify the result by reducing it to its lowest terms: $$ \frac{8}{6} = \frac{8 \div 2}{6 \div 2} = \frac{4}{3} $$ This example highlights how unsimplified fractions can be useful during intermediate steps, but simplifying the final result is essential for clarity and understanding.

Try It Yourself

Simplify the fraction \( \frac{36}{48} \):

Start by finding the common divisors of 36 and 48, and determine their greatest common divisor.

In this case, the GCD of 36 and 48 is 12:

$$ GCD(36,48)=12 $$

Next, apply the property of invariance and divide both the numerator and denominator by 12:

$$ \frac{36}{48} = \frac{36 \div 12}{48 \div 12} = \frac{3}{4} $$

Now the fraction is fully simplified!

The fractions \( \frac{36}{48} \) and \( \frac{3}{4} \) are equivalent, sharing the same decimal value (0.75):

$$ \frac{36}{48} = \frac{3}{4} = 0.75 $$

However, it’s clear that \( \frac{3}{4} \) is much easier to work with. This simplified form not only makes calculations quicker but also immediately shows the relationship between the two numbers: 3 parts out of 4.

In conclusion, simplifying fractions is a key skill that makes math simpler and more efficient.

Next time you encounter a fraction, remember: there’s always a way to simplify and let clarity take center stage!