Fractions were originally used in 1600 BC in Ancient Egypt, making the concept fairly old. There are various forms of fractions. A proper fraction is when the numerator is less than the denominator, for example 1/2. However, when the denominator is less than or equal to the numerator, the fraction is improper, 14/9 is an example of this. An improper fraction can be converted into a mixed fraction.
Fractions with numerators equal to or larger than the denominator are known as improper fractions. They always have a value of one or greater. Improper fractions are frequently stated in a simplified mixed number form, as mixed fractions are easier to understand.
What are improper fractions?
If you come across a fraction where the numerator is larger than the denominator, such as 5/2 or 8/5, for example, this is considered to be an improper fraction.
There are two parts to every fraction: the numerator and the denominator. Proper fractions and improper fractions are the two primary forms of fractions in mathematics based on the numerator and denominator values.
What are mixed numbers?
A mixed fraction or mixed number is a fraction that has both a whole number and a fractional component. 3 1/2, for example, is a mixed fraction.
What does the whole number mean? Well, in the example above (3 1/2), if we were to express the whole number as a fraction, it could be expressed as 3/1, but a better representation of 3 is 6/2 as this is equivalent to the proper fraction (1/2) attached to it. Understanding what whole numbers mean and what they equate to is useful when converting mixed numbers to improper fractions as the whole number is key.
Converting improper fractions into mixed numbers
Let’s quickly review the definitions of mixed numbers and improper fractions before learning how to convert an improper fraction into a mixed number. A mixed fraction has a whole number portion and a proper fraction, and its value is always greater than 1.
3 2/5, for example, is a mixed number. The numerator of an improper fraction is always higher than or equal to the original denominator. Examples of improper fractions include 17/2, 197/3, 9/5, and so on.
For improper fraction conversion, we must divide the numerator by the denominator to convert an improper fraction to a mixed number. The mixed number is generated after division in such a way that the acquired quotient becomes the whole number, the remainder becomes the new numerator, and the denominator remains the same.
- Example of how to convert an improper fraction into a mixed number
Let us look at the fraction 19/6 and convert this into a mixed number. To begin with, we have to know how much of the denominator goes into the numerator. To do this we divide the numerator by the denominator. The answer will tell us the whole number, and the remainder will give us the new numerator.
6 goes into 19 three times (6 x 3 = 18) and there is a remainder of 1. Therefore, the whole number is 3 and the new numerator is 1. If we construct our mixed number, 19/6 becomes 3 1/6. It is that simple!
Here is another example:
Let us look at the fraction 42/13 and convert this into a mixed number. This is a bit tricky, but do not worry. The first thing you should do is list the 13 times table:
13, 26, 39, 52, 65…
As we can see, 13 goes into 42 three times with a remainder of 3. Therefore, 42/13 as a mixed number is 3 3/13!
Simplification
You may come across a complex fraction such as 64/12. This looks insanely difficult, so what we can do is try to simplify the fraction first and then convert it. To simplify the numerator and denominator we need to find the greatest common factor between the two numbers, let us list the factors of each number to begin with:
- 64: 1, 2, 4, 8, 16, 32, 64
- 12: 1, 2, 3, 4, 6, 12
Looking at the factors, we can see that both share the factors of 1, 2, and 4. Of these, 4 is the greatest/highest. If we then divide the numerator and denominator by 4, it will give us a simplified version of this fraction.
64 / 4 = 16
12 / 4 = 3
As such, 64/12 can be simplified to 16/3. It will now be a lot easier to convert this into a mixed number:
3 goes into 16 five times with a remainder of 1. This means that 16/3 as a mixed number and simplified fraction is 5 1/3.
Converting mixed fractions into improper fractions
Converting a mixed number into an improper fraction is essentially the previous method, but in reverse. However, you now have to multiply the whole number by the denominator and then add the numerator. The result of this is the new, improper numerator.
Here is an example:
Convert 4 7/10 into an improper fraction. First, let us multiply the denominator (10) by the whole number (4):
4 x 10 = 40
Then, add the numerator (7) to this answer:
40 + 7 = 47
This is the new numerator. As such, if we were to express 4 7/10 as an improper fraction, it would be 47/10.
Simplifying
What happens if you get a nasty fraction like 5 84/24? Not many of us will easily be able to multiply with 24 and then add 84! When simplifying at this stage, we can ignore the whole number and just focus on the numerator and denominator. Let’s list their factors:
- The denominator (84): 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
- The numerator (24): 1, 2, 3, 4, 6, 8, 12, 24.
The numerator and denominator share the factors 1, 2, 3, 4, 6, and 12. Of these, 12 is the highest so let us divide the numerator and denominator by this:
84 / 12 = 7
24 / 12 = 2
As such, 5 84/24 can be simplified to 5 7/2. Let us now convert this much easier mixed fraction:
2 x 5 = 10
10 + 7 = 17
Therefore 5 84/24 as an improper fraction is 17/2
Why is it important to know how to convert fractions?
Once you understand how to convert different types of fractions, you will be able to tackle more difficult questions that involve both mixed numbers and improper fractions. Here is an example:
3 1/2 + 27/6 = ?
To answer such a question, you have to decide whether to answer the question as an improper fraction or mixed number. Whichever you decide, the fractions must both be in that form. For example, if we wanted to take this question as a mixed number:
3 1/2 is already a mixed number!
So, 27/6 must be converted.
6 goes into 27 four times with a remainder of 3. Therefore, 27/6 can be expressed as 4 3/6 which can be simplified to 4 1/2.
The question then becomes:
3 1/2 + 4 1/2 = ?
This is much easier to answer as you can simply add them together to get the answer.
3 + 4 = 7
1/2 + 1/2 = 1
Therefore 3 1/2 + 4 1/2 = 8
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