Each fraction has a variety of names that we may need to use in different scenarios. For example, the number 1 has a lot of distinct names. Remember that the result of dividing any number (except 0) by itself is 1. Other names for 1 include 2/2, 6/6, 93/93, and so on. Another property of the number 1 is that when any integer is multiplied by 1, the original number remains unchanged. These two features of the number 1 can be used to rename fractions.
By renaming fractions, it can be easier to then convert fractions to decimals, as well as tackle questions where you have to add, subtract, divide, or multiply fractions.
How to rename fractions
Fractions are frequently renamed by students. When a student renames a fraction, it transforms into a mixed number, which is the fraction’s proper form. This makes it easier to convert fractions into decimals. To convert a mixed number into a fraction, you can rename it as an improper fraction. If you need to put the fraction in its simplest form, you must also simplify it.
Converting to an Improper Fraction
Let us take the fraction 5 4/10. In this example 5 is the whole number and 4/10 is the proper fraction. To rename this as an improper fraction, multiple the whole number (5) by the denominator (10):
5 x 10 = 50
Then, using this number, add the numerator. This will give you the new numerator of the improper fraction.
50 + 4 = 54
Therefore 5 4/10 renamed is 54/10.
Simplifying
Sometimes you may find that a question will ask the fraction to be renamed in its simplest form. This means that once you have converted the mixed number into an improper fraction, you should simplify it.
Here is an example:
Rename 2 4/6 in its simplest form. Firstly, we must convert this to an improper fraction as we did in the previous section:
2 x 6 = 12
12 + 4 = 16
Therefore, 2 4/6 as an improper fraction is 16/6.
Now we must rename this fraction in a simpler form. To do so we need to find the greatest common factor (GCF) of the numerator and denominator and divide. A factor is a number that is generated when two numbers are evenly split. In this context, a factor is also known as a divisor. The greatest common factor of two or more numbers is the highest number shared by all the factors.
To find the GCF, you have to list the factors of 16 and 6:
- 16: 1, 2, 4, 8, 16
- 6: 1, 2, 3, 6
Notice that both share 2 as a factor, as this is the highest common factor in the list so it’s the GCF. Using this, we now divide the numerator and denominator by the Greatest Common Factor:
- Numerator: 16 / 2 = 8
- Denominator: 6 / 2 = 3
We can therefore rename the improper fraction 16/6 as 8/3. It is that simple!
Simplifying with prime numbers
Let us look at this example:
Rename 1 3/4 as an improper fraction in its simplest form.
To start, you should find the new numerator:
1 x 4 = 4
4 + 3 = 7
Therefore 1 3/4 as an improper fraction is 7/4.
The question also asks us to rename the fraction in its simplest form. As such, let us list the factors of 7 and 4:
- 4: 1, 2, 4
- 7 is a prime number, meaning its only factors are 1 and 7
Since these numbers do not share a common factor, they cannot be simplified. 7/4 is therefore the simplest form of 1 3/4!
How do you rename a fraction as a mixed number?
Any number can be renamed as a mixed number so long as the numerator is larger than the denominator. 2/3, for example, cannot be expressed as a mixed number!
Here is an example:
Rename the improper fraction: 19/7. To form the whole number, we need to find out how many 7’s go into 19. To figure this out, you should list the multiples of 7:
- 7, 14, 21, 28
As we can see, 7 goes into 19 twice with a remainder of 5 (7×2 = 14, +5 = 19)
The number you multiply the denominator by, which is 2 in this case, becomes the whole number, and the remainder becomes the new numerator: 19/7 becomes 2 5/7.
Simplifying
In the same way that you can simplify improper fractions, you can also simplify mixed numbers – in two different ways.
Method 1:
Let us look at the example 28/8. 8 goes into 28 three times with a remainder of 4, you can express this as such:
- Whole number: 8 x 3 = 24
- Remainder: 28 – 24 = 4
- Mixed Number: 3 4/8
You may notice that the fraction 4/8 is half, meaning you can simplify this as 1/2. Remember that as these are equivalent fractions, 4/8 and 1/2 are exactly the same.
So, 28/8 can be renamed as 3 ½.
Method 2:
You can actually simplify the improper fraction before you calculate it as a mixed number. This is especially useful if you have a massive fraction with a numerator in the hundreds.
Let us look again at 28/8. You should start by listing the factors of the numerator and denominator:
- 28: 1, 2, 4, 7, 14, 28
- 8: 1, 2, 4, 8
4 is the greatest common factor of these two numbers, so let us divide both by 4:
- 28 / 4 = 7
- 8 / 4 = 2
Now the improper fraction is 7/2 which is a lot simpler to work with. The answer, of course, will be 3 1/2.
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