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Fractions are numbers that have a numerator and denominator like this, \(\frac{1}{2}\), where 1 is the numerator and 2 is the denominator. Fractions can also be considered as a ratio of two numbers. For example, if 4 mangoes are distributed among 2 students, in fractions the share of each student would be \(\frac{4}{2}\) = 2.
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Key Takeaways: Types of fractions, Proper fractions, improper fractions, Mixed fractions, Like fractions, Unlike fractions, Equivalent fractions, Unit fractions, Addition of fractions
Parts of a Fraction
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Numerator: The number of parts of the whole which you have. Here you have 3 parts of a whole that is 4. It is always put in the top half of the fraction.
Denominator: The number representing the whole of the fraction and is always put in the bottom half of the fraction.
Line: The line separating 3 and 4 in the above fraction
\(\frac{3}{4}\)
Fraction
Types of Fractions
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Based on the numerator and the denominator, we mainly classify fractions into a total of 7 types. We would be explaining each type with suitable examples below:
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Proper Fractions: These are the fractions where the numerator is always less than the denominator. To put it simply, Numerator < Denominator. For example,
\(\frac{3}{6}\)
Proper Fraction
Here, Numerator 3 is less than Denominator 6. So, such kinds of fractions are called Proper fractions.
Learning: When you simplify a proper fraction, the final value would always be less than 1.
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Improper Fractions: It is simply the opposite of proper fractions. Here, the numerator is always greater than the denominator. To put it simply, Numerator > Denominator. For example,
\(\frac{8}{4}\)
Improper Fraction
Here, numerator 8 is greater than denominator 4. So, such kinds of fractions are called improper fractions.
Learning:
- When you simplify an Improper fraction, the final value would always be greater than or equal to 1 but never less than 1.
- Since in such a fraction, the denominator is always 1, any natural number can be represented as an improper fraction.
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Mixed Fractions: Any fraction which is a combination of both a fraction and any natural number. For example,
\(\Large3\frac{4}{5}\)
Mixed Fraction
Here, we have ‘3’ as a natural number and 4/5 is a fraction. So, such kinds of fractions are called Mixed fractions.
Learning:
- A mixed fraction can always be converted to an Improper fraction
- An Improper fraction similarly can always be converted into a mixed fraction
- When you simplify a mixed fraction, the final value would always be greater than 1.
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Like Fractions: These fractions have the same denominators. For example,
\(\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}\)
Like Fractions
Here, all the four denominators are the same, that is 5. Such kinds of fractions are called Like Fractions.
Learning: Simplification that includes addition, subtraction etc. of such fractions becomes very easy due to common denominator. For example, if we need to add the above fractions, we just need to add the numerators (1+2+3+4=10) and the denominator would remain the same. In this case, the fraction would be 10/5.
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Unlike Fractions: These fractions do not have common denominators. For example,
\(\frac{1}{5}, \frac{2}{4}, \frac{3}{2}, \frac{4}{3}\)
Unlike Fractions
Here, as we see, we do not have any common denominator. Such kinds of fractions are called, Unlike Fractions.
Learning: Simplification of Unlike Fractions is not as direct as Like fractions. It is a difficult and tedious task. For example, if we need to perform addition for the above fractions, we need to follow these steps:
- First, calculate the LCM of denominators 5 and 4
- The LCM of the denominators, in this case, would be 20.
- Now, you need to multiply 1/5 with 4 and 2/4 with 5 both in the numerator and the denominator.
- The resultant fraction upon performing the above simplification is 4/20 and 10/20
- Now, you just need to add the resultant fraction and the final result would be 14/20
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Equivalent Fractions: These fractions upon simplification give the same value. For example,
\(\frac{1}{2}, \frac{2}{4}\)
Equivalent Fractions
Here, if we further simplify both the fractions, the value would be the same that is 0.5 for both the fractions. Such kinds of fractions are called equivalent fractions.
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Unit Fractions: These fractions have 1 as numerator always and the denominator consists of any positive integer. For example,
\(\frac{1}{6}, \frac{2}{4}\)
Unit Fractions
Here, if we simplify both the fractions, the numerator would always be 1 and the denominator would be some positive integer. Such kinds of fractions are called Unit Fractions.
Things to Remember
- Fractions are also called the ratio of two numbers
- Every unique fraction would have one numerator and denominator
- When numerator > denominator- Improper fraction, whereas numerator < denominator is Proper fraction. The value of the Improper fraction would always be greater than or equal to 1 but never less than 1 whereas, in a Proper fraction, the value would be always less than 1.
- The calculation of Like fractions is easy as it has common denominators but the calculation of Unlike fractions becomes very tedious as it involves taking out LCM of denominators for addition and subtraction tasks.
- A mixed fraction is a combination of a natural number and a fraction. It can easily be converted to Improper fraction and vice versa.
Sample Questions
Ques. What is the difference between Proper fractions and Improper fractions? (2 marks)
Ans: A fraction having a numerator smaller than the denominator is called a proper fraction whereas, in the case of Improper fraction, the numerator is greater than the denominator. So, the difference is mainly in terms of whether a fraction has a larger numerator or a denominator. Also, in the case of proper fraction, the final value of fraction upon simplifying is always less than 1 whereas, in improper fractions, the final value always comes out to greater than or equal to 1.
Ques. Solve: (3 marks)
1) \(\frac{2}{3}+\frac{4}{3}\) 2) \(\frac{3}{7}-\frac{2}{7}\)
Ans:
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Since we have common denominator 3, in the fraction \(\frac{2}{3}\) + \(\frac{4}{3}\), we just need to add the numerators 2 and 4 for addition. So, the final result would be \(\frac{6}{3}\) = 2
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In this fraction, we have a common denominator 7. So, we just subtract the numerators. Here, 3 - 2 = 1. So final result is \(\frac{1}{7}\)
Ques. Solve: (4 marks)
1) \(\frac{1}{3}+\frac{2}{4}\) 2) \(\frac{1}{2}-\frac{1}{4}\)
Ans:
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In this fraction, we have an unequal denominator. So, in addition, we first take out LCM of denominators 3 and 4.
The LCM of 3 and 4 is 12. Now multiply \(\frac{1}{3}\) by 4 and \(\frac{2}{4}\) by 3 in both numerator and denominator.
The resultant fraction is \(\frac{4}{12}\) and \(\frac{6}{12}\)
Just add the numerator now since we have a common denominator of 12. Adding 4 and 6 = 10
So final result is \(\frac{10}{12}\)= \(\frac{5}{6}\)
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We have an unequal denominator in this fraction as well. We will repeat the previous steps.
Take out the LCM of denominators 2 and 4 that is 4
Multiply \(\frac{1}{2}\) by 2 and \(\frac{1}{4}\) by 1
The resultant fraction is \(\frac{2}{4}\) and \(\frac{1}{4}\)
Simply subtract the numerator since we have a common denominator now.
So final result is \(\frac{1}{4}\)
Ques. Solve: (4 marks)
1) \(\frac{2}{3}+\frac{4}{3}+\frac{5}{3}\) 2) \(\frac{12}{2}-\frac{6}{2}-\frac{4}{2}\)
Ans:
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As we can see in the given fractions, denominator is common which is 3
So, keeping denominator in common we would just add 2 + 4 + 5 = 11
Hence, final fraction will be \(\frac{11}{3}\)
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In the given fraction, common denominator is 2
Subtracting the numerator 12 - 6 - 4 = 2
Hence, final fraction is \(\frac{2}{2}\)
On further solving we get 1 as result
Ques. Solve: (4 marks)
1) \(\frac{4}{3}+ \frac{5}{4}+ \frac{6}{5}\) 2) \(\frac{15}{2} - \frac{8}{3} - \frac{9}{5} \)
Ans:
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In this fraction we don’t have a common denominator. So we need to first find out the LCM of 3,4 and 5
LCM of 3,4 and 5 is 60
Now we need to multiply \(\frac{4}{3}\) by 20, \(\frac{5}{4}\) by 15 and \(\frac{6}{5}\) by 12 in both numerator and denominator so that we have a common denominator 60
By solving we get \(\frac{80}{60}\), \(\frac{75}{60}\), \(\frac{72}{60}\)
Just adding numerator now 80 + 75 + 72 = 227
so final fraction is \(\frac{227}{60}\)
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Take out the LCM of 2,3 and 5. Upon solving we get 30
Now multiply \(\frac{15}{2}\) by 15, \(\frac{8}{3}\) by 10 and \(\frac{9}{5}\) by 6 in both numerator and denominator so that we have common denominator 30
Upon solving we get \(\frac{225}{30}\), \(\frac{80}{30}\) and \(\frac{54}{30}\)
Now just do subtraction in numerator 225 - 80 - 54 = 91
Hence, final fraction is \(\frac{91}{30}\)
Ques. Solve: (4 marks)
1) \(\frac{3}{2} \times \frac{4}{2} \times \frac{5}{2}\) 2) \(\frac{7}{2} \times \frac{5}{3} \times \frac{4}{2}\)
Ans:
a. For multiplication, we multiply numerator with numerator and denominator with denominator. So, we multiply numerator 3 × 4 × 5 = 60, and denominator 2 × 2 × 2 = 8
so, fraction is \(\frac{60}{8}\) = \(\frac{15}{2}\)
b. For multiplication with different denominators too, we just multiply numerator with numerator and denominator with denominator. So, multiplying numerator 7 × 5 × 4 = 140 and denominator 2 × 3 × 2 = 12
so, fraction is \(\frac{140}{12}\) = \(\frac{35}{3}\)
Ques. Define Unit fraction with a suitable example. (1 mark)
Ans: Any fraction which has a numerator as 1 and denominator is any positive integer is termed as Unit fraction. For example, \(\frac{1}{4}\) wherein the numerator is 1 and the denominator 4 is a positive integer.
Ques. Define and give some examples of Equivalent fraction. (1 mark)
Ans: Those fractions whose value comes out to be the same after simplification is termed as Equivalent fraction. Example – \(\frac{2}{4}\), \(\frac{3}{6}\), \(\frac{4}{8}\).
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