Math is not easy. It's normal to forget even the basics when dealing with dozens of different principles and solution methods at the same time. This article will show you how to simplify fractions.
Method 1 of 4: Using the greatest common divisor
Step 1. List the numerator and denominator factors
Factors are numbers that, when multiplied, result in another value. For example, 3 and 4 are both factors of 12, because you can multiply them to get 12. To list the factors of a number, you simply have to list all the numbers that can be multiplied together to get to it.
- List the factors for that number from smallest to largest, not forgetting to include 1 or the number itself. For example, check out below how we could list the numerator and denominator factors of the 24/32 fraction:
- 24: 1, 2, 3, 4, 6, 8, 12, 24.
- 32: 1, 2, 4, 8, 16, 32.
Step 2. Find the greatest common divisor (CDM) for the numerator and denominator
The greatest common divisor is the highest value that can act as the divisor for two or more numbers. After listing all the factors of the numbers to be worked on, just find the highest value that repeats in both lists.
- 24: 1, 2, 3, 4, 6,
Step 8., 12, 24.
- 32: 1, 2, 4,
Step 8., 16, 32.
The MDC (maximum common divisor) for 24 and 32 is 8, as this is the highest value that can act as a divisor for both 24 and 32.
Step 3. Divide the numerator and denominator by the MDC
That way, you'll be able to simplify the fraction as much as possible. Note below:
- 24/8 = 3.
- 32/8 = 4.
- The simplified form of the fraction is 3/4.
Step 4. Check the result
Simply multiply the simplified fraction by the greatest common divisor to get the original fraction. Let's see the example below:
- 3 * 8 = 24.
- 4 * 8 = 32.
- In this way, it was possible to return to the original 24/32 fraction.
You can also check whether the fraction has been simplified to the maximum. Since 3 is a prime number, it can only be divided by 1 and by itself. 4 cannot be divided by 3. Therefore, the fraction cannot be simplified any further
Method 2 of 4: Using Continuous Division by a Small Number
Step 1. Choose a small number
When using this method, all you have to do is choose a small number such as 2, 3, 4, 5 or 7 to start. Pay attention to the fraction to check that each component of the fraction is divisible by the chosen number at least once. For example, when working with the fraction 24/108, avoid choosing the number 5, as none of the components of the fraction are divisible by it. On the other hand, 5 is a good choice if we are going to simplify the 25/60 fraction.
For the 24/32 fraction, the number 2 is a good choice. Since both components of the fraction are even numbers, they can be divided by 2
Step 2. Divide the numerator and denominator of the fraction by the chosen number
In this way, a new, simpler fraction can be obtained, with a smaller numerator and a smaller denominator. Note how this is done:
- 24/2 = 12.
- 32/2 = 16.
The simplified fraction results in 12/16.
Step 3. Repeat the process explained above
Since the numbers resulting from dividing by 2 remain even, they can continue to be divided by 2. If, in the process, the numerator or denominator becomes an odd number, you can try to divide both by another number. Let's see how to proceed to the fraction we arrived at in the step above, 12/16:
- 12/2 = 6.
- 16/2 = 8.
The result is the new 6/8 fraction.
Step 4. Continue dividing the numerator and denominator until you can no longer do this
In our example, as the resulting numbers are still even, they can still be divided by 2. Let's look at the solution below:
- 6/2 = 3.
- 8/2 = 4.
Now we have the new 3/4 fraction.
Step 5. Check if the fraction is already simplified as much as possible
In our ¾ example, 3 is a prime number. So its factors are just the 1 and itself. 4 cannot be divided by 3. Conclusion: the fraction has already been simplified to the maximum.
Now let's look at the 10/40 fraction and divide both the numerator and denominator by the number 5. The result is 2/8. Here, we can't continue dividing both numbers by 5, but we can choose another number: the 2. That way we'll get the final 1/4 result
Step 6. Check the result
Reverse the process by multiplying 3/4 by 2/2 three times to get the original 24/32 fraction. Note the calculation below:
- 3/4 * 2/2 = 6/8.
- 6/8 * 2/2 = 12/16.
- 12/16 * 2/2 = 24/32.
- Notice that you divided 24/32 by 2 * 2 * 2, which is the same thing as dividing it by 8, the MDC (maximum common divisor) of 24 and 32.
Method 3 of 4: Making the Factor List
Step 1. Know how to work the fraction
Leave plenty of space on the right side of the paper - it will be needed to write down all the factors.
Step 2. Make a list of factors for the numerator and one for the denominator
It's easier if one list is above the other. Start with number 1 as the first factor.
For example, let's look at how to work the 24/60 fraction. Let's start with 24.
Let's write the list of factors like this: 24 -- 1, 2, 3, 4, 6, 8, 12, 24
Now, let's proceed with 60.
Let's write: 60 -- 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Step 3. Find the greatest common divisor and divide the numerator and denominator by it
In our example, the greatest common divisor for both 24 and 60 is 12. So let's divide 24 by 12 and 60 by 12. That way we'll get the simplified result 2/5.
Method 4 of 4: Using Prime Factor Trees
Step 1. Find the prime factors of the numerator and denominator
A prime number is one that, to yield an integer, can only be divided by 1 and by itself. Examples of prime numbers include 2, 3, 5, 7, and 11.
Start with the numerator. Starting from 24, branch to 2 and 12. Since 2 is already a prime number, the tree here is done! Now, break 12 into two other numbers, 2 and 6. 2 is already a prime number. Then divide the 6 by two numbers: the 2 and the 3. See? Now we have 2, 2, 2 and 3 as their prime numbers.
Proceed with the denominator. Starting from 60, make two branches, one for 2 and one for 30. Continuing the branch, 30 will split into numbers 2 and 15. Now, 15 will split into numbers 3 and 5, both primes. As a result, we will get 2, 2, 3 and 5 as prime numbers.
Step 2. Decompose into prime factors for each number
Make a list of the prime numbers you have for each value to multiply them in the next step.
- So, for 24, we have 2 x 2 x 2 x 3 = 24.
- For the 60, we will have 2 x 2 x 3 x 5 = 60.
Step 3. Eliminate common factors
Any value that you perceive to be part of both the numerator and denominator can be dropped. In our case, the numbers that are repeated in both components of the fraction are 2 (twice) and 3. Time to say goodbye!
Now, what's left is the 2 and the 5 - or rather, 2/5! The same answer we got with the method above.
- If the numerator and denominator are equal, divide both by two. Keep doing this until they become numbers too small to divide.