Simplifying fractions involves reducing a fraction to its simplest form. A fraction in simplest form has a numerator and denominator that don't share any factors except 1.
For example, . All of these are equivalent fractions, but only is in simplest form. We can simplify all of the other equivalent fractions and find that the simplest form of those fractions will be .
There are two main methods for simplifying fractions. One uses prime numbers, while the other uses the greatest common factor (GCF) of the numerator and denominator.
How to simplify fractions using prime numbers
This method may be slightly simpler to perform than the GCF method, but it can be much less efficient.
To use this method,
- Divide each number in the fraction by a shared prime number that divides the numerator and denominator exactly (result should be a whole number). Start from low to high
- Repeat step 1 until there are no more shared prime numbers that the fraction can be divided by
The reason we use prime numbers is because since we start dividing from low to high prime numbers, we don't need to check multiples of prime numbers such as 2. 2 × 2 = 4. 4 × 2 = 8, and so on. If we follow the steps above and divide by 2 until we can't anymore, we don't need to test if we can divide by 4 because there will never be a case where we've fully divided by 2, and could still divide by 4, or 8, etc. This is true for all non-prime numbers.
Since even numbers are divisible by 2, we try using 2 as our first prime number:
Because 29 is a prime number, it cannot share another factor with 8, so even though we could still divide 8 by 2 twice more, is as much as we can simplify this fraction. Lets look at another example.
2. Simplify :
In Example 2 above, we could have immediately divided the numerator and denominator by 18, since 2 × 3 × 3 = 18. 18 in this example is the greatest common factor.
How to simplify fractions using their GCF
Simplifying fractions using their GCF is more efficient than dividing the numerator and denominator of the fraction by prime numbers until their only shared factor is 1.
To use this method of simplification, use the following steps:
- Find the GCF between the numerator and denominator of the fraction being simplified
- Divide the numerator and denominator of the fraction by the GCF
The GCF is 24, so:
Refer to the GCF page for details on how to find the GCF. Briefly, the prime factorizations of 48 and 72 are:
48 = 2 × 2 × 2 × 2 × 3
72 = 2 × 2 × 2 × 3 × 3
So, 48 and 72 share 4 prime factors: 2, 2, 2, and 3. To find the GCF, we multiply these 4 prime factors,
2 × 2 × 2 × 3 = 24 = GCF
and then simplify the fraction by dividing the GCF.