Mathematics is not an easy subject to tackle. When they are not applied frequently it is very easy to forget the concepts and methods to be used, especially when they are really many as in this case. This article shows several useful methods for simplifying a fraction.
Steps
Method 1 of 4: Use the Greatest Common Divider
Step 1. List the numerator and denominator factors
Factors are all those values which, when multiplied appropriately, give the initial number as a result. For example, the numbers 3 and 4 are both factors of the number 12, since multiplying them together makes the product equal to 12. To create a number's factor list, you simply list all of its divisors.
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Write the list of all the factors of the numerator and denominator in ascending order, not forgetting to include the number 1 and the starting values. For example, analyzing the fraction 24/32 below you will find the set of factors of the numerator and denominator:
- 24: 1, 2, 3, 4, 6, 8, 12, 24
- 32: 1, 2, 4, 8, 16, 32
Reduce Fractions Step 2 Step 2. Identify the greatest common divisor existing between the numerator and denominator of the fraction in question
This value represents the largest number that two or more numbers can be divided by. After creating the list of all the factors of the numerator and those of the denominator, you just have to find the largest number that is common to both.
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24: 1, 2, 3, 4, 6,
Step 8., 12, 24
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32: 1, 2, 4,
Step 8., 16, 32
- In this example, the greatest common divisor of the numbers 24 and 32 is 8, since 8 is the largest number that can fully divide the values 24 and 32.
Reduce Fractions Step 3 Step 3. Divide the numerator and denominator of the fraction by the greatest common factor you have found
Do this to minimize the fraction under consideration. Continuing with the previous example you will get:
- 24/8 = 3
- 32/8 = 4
- The simplified and equivalent fraction to the initial one is 3/4.
Reduce Fractions Step 4 Step 4. Verify that your work is correct
To figure out if you've simplified the fraction correctly, simply multiply the numerator and denominator of the new fraction by the greatest common factor you used to reduce it to its lowest terms. If the calculations are correct, you should get the original fraction as a result. Continuing with the previous example you will get:
- 3 * 8 = 24
- 4 * 8 = 32
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As you can see, you got the starting fraction 24/32, so the calculations are correct.
Also carefully check the fraction you simplified to make sure it cannot be reduced further. In this case the number 3 is present in the numerator, which is a prime number and therefore can only be divided by itself or by 1, so the fraction you have obtained cannot be simplified further
Method 2 of 4: Performing Multiple Divisions Using Small Numbers
Reduce Fractions Step 5 Step 1. Pick a small number
In order to practice this method, you just have to choose a small number, such as 2, 3, 4, 5 or 7, to use as a divisor. Look at the fraction to simplify to make sure that the chosen number can be used as a divisor for both the numerator and the denominator. For example, if you need to simplify the fraction 24/108, you cannot choose the number 5 as a divisor because it does not fully divide either the numerator or the denominator. Conversely, if you have to work on the fraction 25/60, the number 5 is perfect as a divisor.
Continuing with the previous example, 24/32, the number 2 is a great choice. Since both the numerator and denominator are even numbers they can be divided by 2
Reduce Fractions Step 6 Step 2. Divide the numerator and denominator of the fraction under consideration by the divisor you have chosen
The new fraction you will get will be composed of the result of dividing the original numerator and denominator by the selected number, ie 2. By performing the calculations you will get:
- 24/2 = 12
- 32/2 = 16
- The new fraction is therefore 12/16.
Reduce Fractions Step 7 Step 3. Repeat the previous step
Since the numerator and denominator of the new fraction are still even numbers, you can continue dividing them by 2. In case the numerator, denominator, or both are an odd number, you will need to try to find a new common divisor. Continuing with the example fraction, 12/16, you will get:
- 12/2 = 6
- 16/2 = 8
- The new simplified fraction is 6/8.
Reduce Fractions Step 8 Step 4. Continue the simplification process until you are able to perform the split
Again, both the numerator and the denominator of the new fraction are still even numbers, so you can further divide them by 2. By doing the calculations you will get:
- 6/2 = 3
- 8/2 = 4
- The new simplified fraction is 3/4.
Reduce Fractions Step 9 Step 5. Make sure the final fraction cannot be reduced any further
The new fraction 3/4 presents the numerator with the value 3, which represents a prime number divisible only by itself or by 1, while the denominator contains the value 4 which is not divisible by 3. For this reason you can say that the fraction initial was reduced to a minimum. If the numerator or denominator of the new fraction is no longer divisible by the chosen number, you may still be able to simplify it by using a new divisor.
For example, by looking at the fraction 10/40 and dividing the numerator and denominator by 5, you get the fraction 2/8. In this case you can't divide the numerator and denominator by 5 again, but you can simplify the fraction further by dividing both by 2 to get the final result 1/4
Reduce Fractions Step 10 Step 6. Check your work is correct
Reverse the process by multiplying the fraction 3/4 by 2/2 three times consecutively, resulting in the starting fraction, 24/32. This way you can be sure that your calculations are correct.
- 3/4 * 2/2 = 6/8
- 6/8 * 2/2 = 12/16
- 12/16 * 2/2 = 24/32.
- Note that you have divided the example fraction (24/32) by 2, three consecutive times, which is equivalent to using the number 8 as a divisor (2 * 2 * 2 = 8), which represents the greatest common divisor of 24 and 32.
Method 3 of 4: List the Factors
Reduce Fractions Step 11 Step 1. Make a note of the fraction to be simplified
Leave a large blank space on the right of the sheet in which to report all the factors of the fraction.
Reduce Fractions Step 12 Step 2. Write a list of all the factors of the numerator and denominator
Record them in two separate lists, each lined up next to the number they refer to. Start from number 1 and fill in the lists in ascending order.
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For example, if you need to simplify the fraction 24/60, you start by creating the list of factors in the numerator, i.e. 24.
You will get the following list: 24 - 1, 2, 3, 4, 6, 8, 12, 24
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At this point, create the list of denominator factors, i.e. 60.
You will get the following list: 60 - 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Reduce Fractions Step 13 Step 3. Now find the largest number common to both lists
The value you choose represents the greatest common divisor of the fraction under consideration. Ask yourself what is the largest number that is a divisor of both the numerator and the denominator of the fraction. Once located, use it to perform the calculations.
Continuing with the previous example, the greatest common divisor of the fraction under consideration is 12. Since 24 and 60 are divisible by 12, the final result of your work will be 2/5
Method 4 of 4: Use the Prime Factor Tree Diagram
Reduce Fractions Step 14 Step 1. Find all the prime factors of the numerator and denominator
A number is called "prime" when it is divisible only by 1 and itself. The numbers 2, 3, 5, 7 and 11 are examples of prime numbers.
- Start by analyzing the numerator. The number 24 can be factored into 2 and 12. Since the factor 2 is a prime number this part of the tree diagram is already complete. Analyze the number 12 and compose it into two other factors obtaining: 2 and 6. As in the previous case, 2 is a prime factor, so this branch of the diagram is also complete. Now look for two other factors of the number 6 which are: 2 and 3. The result of the decomposition highlighted the following prime factors: 2, 2, 2 and 3.
- Analyze the denominator. The number 60 can be broken down into 2 and 30. Two factors of the number 30 are represented by the values 2 and 15. The number 15 can be divided into 3 and 5 which are both prime numbers. In this case the prime factors of the denominator are 2, 2, 3 and 5.
Reduce Fractions Step 15 Step 2. Take note of the prime factors of the numerator and denominator
Create two lists of prime factors, one for the numerator and one for the denominator, in order to calculate the product. You will not have to perform the calculations, but you will need it to visualize the solution to be adopted in a simpler and faster way.
- For the numerator, 24, you get: 2 x 2 x 2 x 3 = 24
- For the denominator, 60, you get 2 x 2 x 3 x 5 = 60
Reduce Fractions Step 16 Step 3. Remove all prime factors that they have in common from the two lists
You will need to delete from the list all numbers that appear in both the denominator list and the numerator list. In this example, the common prime factors are the pairs of the numbers 2 and 3 that will need to be eliminated.
- The prime factors remaining after the cancellation are 2 and 5, which, arranged in the form of a fraction, become 2/5, exactly the final result of the reduction to the minimum terms of the fraction 24/60.
- If the numerator and denominator of the starting fraction are even numbers, start by dividing them in half and continue until you get prime numbers.
