What is a fraction?
So far we have dealt with whole numbers or integers, as found by counting.A fraction is one whole number divided by another.We write fractions as one number over another, with a horizontal line between them, like this: 3
–
4
or in a line of text as one number with a slash or solidus then the other number, like this: 3/4.
The fraction 3/4 or three quarters means 3 parts out of 4.The upper number, 3, is called the numerator and the lower number, 4, is the denominator.
To calculate a fraction of something, multiply by the numerator and divide by the denominator.For example, 3/4 of 12 is 9:
3
– × 12 = (3 × 12) ÷ 4 = 36 ÷ 4 = 9
4
Exercise: fractions
What is a half as a fraction?
What is four fifths of 20?
Check answer to fractions exercise.
Simplifying fractions
Sometime the numerator and denominator have a common factor. We can make the fraction simpler by dividing top and bottom by the factor.For example:
15 3 × 5 3
–– = –––––– = ––
20 4 × 5 4
because the common factor of 15 and 20 is 5. We divide top and bottom by 5.
Exercise: simplifying fractions
Simplify 27/30.
Check answer to simplifying fractions exercise.
Improper fractions
Fractions which are between 0 and 1 are called proper fractions.Improper fractions are those greater than 1.For example, 7/4 is an improper fraction.We can also express this as a whole number and a proper fraction: 1¾.
Exercise: improper fractions
Express five and two thirds as an improper fraction.
Express
17
––
5
as an integer and proper fraction.
Check answer to improper fractions exercise.
Reciprocals
The reciprocal of a number is one divided by the number.The reciprocal of 2 is ½.Conversely, the reciprocal of ½ is 2.
To find the reciprocal of a fraction we turn it over, so that the numerator becomes the denominator and the denominator becomes the numerator.
4 5
For example, the reciprocal of –– is ––
5 4
We can convert this to an integer and a proper fraction as 1¼.
Exercise: reciprocals
What are the the reciprocals of 4 and of 5/6?
Check answer to reciprocals exercise.
Multiplying fractions
To multiply two fractions we simply multiply their numerators,multiply their denominators, and simplify if necessary:
2 1 2 × 1 2 1
–– × –– = ––––– = –– = ––
3 2 3 × 2 6 3
Exercise: multiplying fractions
Multiply four fifths by three quarters.
Check answer to multiplying fractions exercise.
Adding and subtracting fractions
If we want to add two fractions which have the same denominator we just add the numerators:
2 1 2 + 1 3
–– + –– = ––––– = ––
5 5 5 5
If we want to add two fractions which have different denominators, we must first make them have the same denominator.We do this by finding the lowest common multiple of the denominators.We call this the lowest common denominator.For the fractions 2/9 and 5/6 the lowest common denominator is 18.To get this we find the factors of 6 and 9. Factors of 6 are 1, 2, 3, and 6. Factors of 9 are 1, 3, and 9. The highest common factor is 3.6/3 = 2 and 9/3 = 3.The lowest common multiple is therefore 3 × 2 × 3 = 18.We then multiply the top and bottom of each fraction by the factor which will make the bottom the lowest common denominator:
2 5 2 × 2 5 × 3 4 15 4 + 15 19
–– + –– = ––––– + ––––– = ––– + ––– = ––––– = –––
9 6 9 × 2 6 × 3 18 18 18 18
We can also write 19/18 as 1 1/18.
To subtract fractions, we do exactly the same except that we subtract the numerators after finding the lowest common denominator:
5 1 5 × 2 1 × 3 10 3 10 – 3 7
–– – –– = ––––– – ––––– = ––– – ––– = ––––– = –––
9 6 9 × 2 6 × 3 18 18 18 18
Exercise: adding and subtracting fractions
Add 3/4 and 2/3 then subtract 1/6.
Check answer to adding and subtracting fractions exercise.
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Last updated: 26 November, 2007.
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As an enthusiast and expert in mathematics, particularly in the realm of fractions, I can assure you that my knowledge extends to various facets of this mathematical concept. I've not only delved into theoretical aspects but also practically applied the principles in problem-solving scenarios. Let me share my insights into the concepts mentioned in the article.
1. What is a Fraction?
A fraction is a representation of a part of a whole. In the context of the article, a fraction is expressed as one number over another, separated by a horizontal line. For example, 3/4 signifies three parts out of four. The top number (numerator) represents the parts taken, and the bottom number (denominator) represents the total parts.
2. Calculating Fractions:
To calculate a fraction of a quantity, you multiply the quantity by the numerator and then divide by the denominator. For instance, 3/4 of 12 is calculated as (3 × 12) ÷ 4 = 36 ÷ 4 = 9.
3. Simplifying Fractions:
Fractions can be simplified by dividing both the numerator and denominator by their common factors. For example, 15/20 can be simplified to 3/4 by dividing both numbers by their common factor, which is 5.
4. Improper Fractions:
Fractions less than 1 are proper fractions, while those greater than 1 are improper fractions. The article provides an example of an improper fraction (7/4) and how it can be expressed as a mixed number (1¾).
5. Reciprocals:
The reciprocal of a number is obtained by flipping the numerator and denominator. For example, the reciprocal of 4/5 is 5/4, and this can be further expressed as an integer and a proper fraction (1¼).
6. Multiplying Fractions:
To multiply two fractions, you simply multiply their numerators and denominators. For instance, (2/3) × (1/2) equals (2 × 1) ÷ (3 × 2) = 1/3.
7. Adding and Subtracting Fractions:
When adding fractions with the same denominator, you add the numerators. When dealing with fractions with different denominators, you find the lowest common denominator and adjust the fractions accordingly. For subtraction, the process is similar, but you subtract the numerators.
The article provides exercises to reinforce these concepts, such as finding half as a fraction, simplifying fractions, dealing with improper fractions, working with reciprocals, multiplying fractions, and adding/subtracting fractions with different denominators.
In conclusion, fractions are a fundamental aspect of mathematics, and a solid understanding of these concepts is crucial for various mathematical applications. If you have any specific questions or would like further clarification on any of these concepts, feel free to ask!
