Every day we work with fractions but may not realize it. With that in mind, what is a fraction? In this article, we go over the fundamentals and see some examples of fraction calculations.
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What are fractions?
A fraction indicates the number of elements that make up a whole. The slash (/) that is printed between the two numbers distinguishes a fraction. A fraction is made up of a numerator at the top and a denominator at the bottom. For example, 1/2 is a fraction.
What exactly does this fraction imply? In the case of a pie, the bottom number indicates how many slices to cut (number of equal parts), and the top number indicates how many of those slices we can eat. So ½ indicates that we have cut our pie into two slices and can take one of them.
Types of fractions
Although there are different types of fractions such as equivalent fractions, the foundation of fractions in maths is based on these three types:
- Proper Fractions
- Improper Fractions
- Mixed Numbers
Proper Fractions
A proper fraction is defined solely by having a numerator that is smaller than the denominator. 3/12 and 2/5, for example, are valid fractions since 3 12 and 2 5 are the same number. Luke purchased a bar of chocolate and divided it into three equal portions. He kept one part and gave the other two to his friend Sara. Luke’s portion is 1/3 and Sara’s is 2/3. These two fractions are both proper fractions.
Improper Fractions
An improper fraction is essentially the opposite of a proper fraction as the numerator is always larger than the denominator. 8/3 and 18/7, for example, are incorrect fractions because 8 > 2 and 18 > 7.
Mixed Fractions
A mixed fraction, also known as a mixed number, always contains a whole number and a proper fraction. As an example:
- 1 ½
- 3 ¼
- 14 ¾
The whole number component is 1 in the first case, while the appropriate fraction is ½. The whole number component is 3 in the second case, and the appropriate fraction is ¼.
How do I convert fractions?
Improper Fractions into Mixed Numbers
We must divide the numerator by the denominator to convert improper fractions in mixed fractions. The result is then written as a whole number, the remainder as the numerator, and the divisor as the denominator in the mixed number form. For example, 15/4 is an incorrect fraction. Follow these steps to convert it to a mixed fraction:
We firstly must divide 15 by 4.
4 can go into 15 three times with a remainder of 3 (4 x 3 =12, +3 = 15). Therefore, the result is 3 and the remainder is 3.
The denominator remains the same, while the result becomes the whole number component and the residual 1 becomes the new numerator. This is what it looks like:
16/5 becomes 3 ¾
Mixed Numbers into Improper Fractions
What if a question gave you 10 ¾ and asked you to convert this into an improper fraction? A mixed fraction is when a full number and a correct fraction are combined. To make a mixed fraction into an improper fraction, multiply the denominator by the whole number part, then add the numerator to the product. The numerator will change but the denominator will stay the same.
With the question above, you should multiply the whole number (10) by the denominator (4) which gives you 40. Then you add the numerator (3) which gives you 43. As a result, 43 becomes the numerator of the improper fraction and the denominator stays the same.
What is the easiest way to calculate fractions?
Adding Fractions
To add fractions, you must find a common denominator.
Quiz Question: 2/7 + 3/9 =?
The easiest way to find a common denominator is to multiply the two denominators in the question.
7 times 9 = 63 so 63 is the common denominator. The next step is to replace the denominator of each fraction with the common denominator and convert the numerator.
For the first fraction:
2/7 becomes ?/63 by multiplying the denominator by 9. As such, we must also multiply the numerator by 9 so 2/7 becomes 18/63
Likewise with the second fraction. 3/9 becomes ?/63 if we multiply the denominator by 7. Therefore we also multiply the numerator by 7 which gives us 21/63. The question we have to tackle is now a lot easier because there is a common denominator:
18/63 + 21/63
Simply add the numerators together, 18 + 21 = 39, which gives us 39/63.
Always verify if the resulting fraction can be simplified further as a good practice.
We know that 39 is divisible by 3 in an even number of ways. The number 63 is also divisible by 3. The fraction will remain the same because the numerator and denominator are both divided by the same number. 39 divided by three equals 13, and 63 divided by three equals 21. The simplified answer is therefore 13/21.
Subtracting Fractions
Let’s begin with two basic fractions.
Quiz Question: 3/5 – 1/3 =?
We must find a common denominator, just like when adding fractions. As a result of multiplying our denominators, we get 3 times 5 = 15. After that, we must convert our numerators.
For the first fraction, we convert 5 into 15 by multiplying the denominator by 3. We must do the same with the numerator. 3 x 3 = 9. Therefore, the answer is 9/15.
For the second fraction, we convert the denominator 3 into 15 by multiplying it by 5. We must also multiply 1 by 5. This result is 5/15.
The quiz question now becomes a more manageable 9/15 – 5/15 =?
By subtracting 5 from 9 we get the answer of 4/15.
Multiplying Fractions
Quiz Question: Multiply 2/5 by 3/7.
The answer’s numerator will be the sum of the numerators of these fractions:
2 x 3 = 6
The answer’s denominator will be the product of these fractions’ denominators:
7 x 5 = 35
As a result, 2/5 multiplied by 3/7 equals 6/35.
Dividing Fractions
To divide fractions, we must multiply them.
Quiz Question: Divide 3/7 by 2/5.
Firstly, flip the second fraction so it becomes an improper fraction (this is the key to dividing fractions):
2/5 becomes 5/2
Then we multiply the two fractions:
3/7 x 5/2
Numerator: 3 x 5 = 15
Denominator: 7 x 2 = 14
The result is 15/14 or, 1 1/14
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