NCERT Solutions Class 6 Maths Chapter 7 Fractions

NCERT Solutions for class 6 - Maths chapter 7 Fractions includes topics like mixed fractions, improper fractions, proper fractions, and how to represent a fraction on a number line. A fraction is a numerical value that denotes a portion of a whole. Even in real life, you need to execute this topic, so you must know about every concept of fractions.  

The NCERT Solutions for Class 6 Maths are based on the latest CBSE syllabus. Different types of questions will be based on every concept of class 6 Science Chapter 7, so you can easily grasp all principles and fundamentals. It also ensures that students can practice and reinforce their knowledge. 

1.0Download NCERT Solutions for Class 6 Maths Chapter 7 : Free PDF

We are providing NCERT Solutions prepared by the expert faculties of ALLEN in a downloadable PDF for Chapter 7 of Class 6 Maths. The solutions are detailed to make them clearer and easier to understand, helping students solve questions step by step. These solutions are especially important for revising and practising at home, making learning more convenient.

2.0Key Concepts Covered in Chapter 7 Fractions

This chapter introduces fractions as numbers that represent equal parts of a whole or a collection. Students learn how fractions are used to describe sharing, measuring, and comparing quantities in everyday situations.

• Understanding fractions as parts of a whole.
• Representing fractions using diagrams and models.
• Learning about fractional units like halves, thirds, and quarters.
• Placing and identifying fractions on a number line.
• Converting improper fractions and mixed fractions.
• Finding equivalent fractions.
• Comparing fractions and arranging them in order.
• Performing basic addition and subtraction of fractions.

3.0NCERT Solutions for Class 6 Maths Chapter 7 Fractions: All Exercises

In this section, you will find solutions to all the exercises from Chapter 7 of the Class 6 Maths book. It includes detailed answers for each question, helping students understand how to add, compare, subtract, and simplify fractions. These solutions will help students grasp the concept of fractions and apply them to real-world problems.

Exercise 7.1 – Fractional Units and Equal Shares

This exercise introduces fractions through everyday sharing situations such as dividing fruits, food, or quantities equally. Students learn how fractions represent equal parts of a whole and practise identifying fractional quantities when objects or amounts are shared among people.

Key Concepts Covered:

• Equal sharing concept

• Writing fractions from word problems

• Adding simple fractions (like 1/2 + 1/4)

• Converting word form to fraction form

• Ordering fractional quantities

Exercise 7.2 – Fractional Units as Parts of a Whole

In this exercise, students identify fractional parts of a whole using diagrams. Different pieces of a shape represent fractions of a whole object, helping students visualise how a whole can be divided into equal parts and represented as fractions.

Key Concepts Covered:

• Fraction as part of a whole

• Counting equal divisions

• Visual fraction representation

• Understanding denominator meaning

Exercise 7.3 – Measuring Using Fractional Units

This exercise focuses on understanding fractions through repeated addition and measurement. Students observe how fractional parts combine to form a whole and explore representations such as roti or paper strips to understand fractions like halves, quarters, thirds, and sixths.

Key Concepts Covered:

• Repeated addition of fractions

• Improper fractions from models

• Creating smaller fractions from larger ones

• Fraction multiplication through visuals

Exercise 7.4 – Marking Fractions on the Number Line

This exercise introduces how fractions can be placed on a number line. Students divide a unit into equal parts and mark fractions between 0 and 1, helping them understand fractional positions and the concept that many fractions exist between two numbers.

Key Concepts Covered:

• Number line representation

• Dividing unit interval equally

• Infinite fractions concept

• Improper fractions on number line

Exercise 7.5 – Mixed Fractions

This exercise explains mixed fractions and improper fractions. Students learn how to convert fractions greater than one into mixed numbers and understand how many whole units are contained in a fraction. It strengthens the connection between whole numbers and fractional values.

Key Concepts Covered:

• Improper to mixed conversion

• Mixed to improper conversion

• Identifying whole units

• Fractions greater than 1

Exercise 7.6 – Equivalent Fractions

This exercise introduces equivalent fractions and how different fractions can represent the same value. Students practise finding equivalent fractions, simplifying fractions, and understanding relationships between division, multiplication, and fractions.

Key Concepts Covered:

• Equivalent fractions rule

• Multiplying numerator & denominator

• Missing number problems

• LCM for common denominators

• Fractional representation of sharing

Exercise 7.7 – Comparing Fractions

In this exercise, students learn how to compare fractions using common denominators and multiples. They arrange fractions in ascending or descending order and determine which fraction is larger or smaller, improving their ability to analyse fractional values.

Key Concepts Covered:

• LCM method for comparison

• Common denominator technique

• Ascending and descending order

• Comparing unlike fractions

Exercise 7.8 – Addition and Subtraction of Fractions

This exercise focuses on performing operations with fractions. Students practise adding and subtracting fractions with same and different denominators, often using common multiples to simplify calculations and solve real-life problems involving fractional quantities.

Key Concepts Covered:

• LCM method for addition

• Adding unlike fractions

• Simplifying fractions

• Word problems involving fractions

4.0NCERT Questions with Solutions Class 6 Maths Chapter 7

7.1 Fractional Units and Equal Shares

Fill in the blanks with fractions.

• Three guavas together weight 1 kg . If they are roughly of the same size, each guava will roughly weigh kg. Sol. Since three guavas weight 1 kg ., So, one unit will be divided into three part to get weight of each guava. Each guava weight $\frac{1}{3}\mathrm{kg}$.
• A wholesale merchant packed 1 kg of rice in four packets of equal weight. The weight of each packet is kg. Sol. Since 1 kg of rice is packed in four packets of equal weight. Weight of each packet is $\frac{1}{4}\mathrm{kg}$.
• Four friends ordered 3 glasses of sugarcane juice and shared it equally among themselves. Each one drank glass of sugarcane juice. Sol. Since 3 glasses of sugarcane juice is to be shared equally among four friends, so each one drank $\frac{3}{4}$ glass of sugarcane juice.
• The big fish weighs $\frac{1}{2}\mathrm{kg}$. The small one weighs $\frac{1}{4}\mathrm{kg}$. Together they weigh kg. Sol. Total weight of both fish $=\frac{1}{2}+\frac{1}{4}$ $\frac{1\times 2}{2\times 2}=\frac{2}{4};\frac{1\times 1}{4\times 1}=\frac{1}{4}$ $\frac{1}{2}+\frac{1}{4}=\frac{2}{4}+\frac{1}{4}=\frac{2+1}{4}=\frac{3}{4}\mathrm{kg}$.
  
• Arrange these fraction words in order of size from the smallest to the biggest in the empty box below: One and a half, three quarters, one and a quarter, half, quarter, two and a half Sol. One and a half $=1+\frac{1}{2}=1\frac{1}{2}$, half $=\frac{1}{2}$, quarter $=\frac{1}{4}$ One and a quarter $=1+\frac{1}{4}=1\frac{1}{4}$, two and a half $=2+\frac{1}{2}=2\frac{1}{2}$

7.2 Fractional Units as Parts of A Whole

So, the required piece is $\frac{1}{8}$ chikki.

7.3 Measuring Using Fractional Units

Let us look at another example.

• Continue this table of $\frac{1}{2}$ for 2 more steps. Sol.
  
• Can you create a similar table for $\frac{1}{4}$ ? Sol. Here represents a whole roti.
  
• Making $\frac{1}{3}$ using a paper strip. Can you use this to also make $\frac{1}{6}$ ? Sol. Making $\frac{1}{3}$ using a paper strip.
  
  We can make $\frac{1}{6}$ by dividing the part $\frac{1}{3}$ into two equal parts.
  
• Draw a picture and write an addition statement as above to show: (a) 5 times $\frac{1}{4}$ of a roti (b) 9 times $\frac{1}{4}$ of a roti Sol. (a) 5 times $\frac{1}{4}$ of a roti.
  
  $\Rightarrow \frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{5}{4}$ $\Rightarrow$ Here, one full roti and $\frac{1}{4}$ part of roti. (b) 9 times $\frac{1}{4}$ of a roti.
  
  $\begin{array}{rl}& \underset{1}{\underbrace{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}}+\underset{1}{\underbrace{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}}}+\frac{1}{4}=\frac{9}{4}\\ & +\frac{1}{4}=2+\frac{1}{4}\end{array}$ $\Rightarrow$ Here, 2 full roti and $\frac{1}{4}$ part of roti.
• Match each fractional unit with the correct picture:
  
  Sol.
  

7.4 Marking Fraction Lengths on the Number Line

• On a number line, draw lines of lengths $\frac{1}{10},\frac{3}{10}$, and $\frac{4}{5}$. Sol. To represent $\frac{1}{10},\frac{3}{10},\frac{4}{5}\left(=\frac{8}{10}\right)$ on a number line, divide the distance between ' 0 ' and 1 to 10 equal parts.
  
  Point P represents $\frac{1}{10}$ Point $Q$ represents $\frac{3}{10}$ Point R represents $\frac{8}{10}$ or $\frac{4}{5}$
• Write five more fractions of your choice and mark them on the number line. Sol. Do it yourself.
• How many fractions lie between 0 and 1 ? Think, discuss with your classmates, and write your answer. Sol. There are infinite number of fractions lie between 0 and 1 . Example : $\frac{3}{5},\frac{4}{5},\frac{7}{9},\frac{8}{11},\frac{5}{6}$ etc.
• What is the length of the blue line and black line shown below? The distance between 0 and 1 is 1 unit long, and it is divided into two equal parts. The length of each part is $\frac{1}{2}$. So, the blue line is $\frac{1}{2}$ units long. Write the fraction that gives the length of the black line in the box.
  
  Sol. Length of blue line $=\frac{1}{2}$ Length of black line $=\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=\frac{3}{2}$
• Write the fraction that gives the lengths of the black lines in the respective boxes.
  
  Sol.
  

7.5 Mixed Fractions

• How many whole units are there in $\frac{7}{2}$ ? Sol. $\frac{7}{2}=3+\frac{1}{2}$ There are 3 whole units in $\frac{7}{2}$.
• How many whole units are there in $\frac{4}{3}$ and in $\frac{7}{3}$ ? Sol. $\frac{4}{3}=1+\frac{1}{3}$ and $\frac{7}{3}=2+\frac{1}{3}$ There is 1 whole unit in $\frac{4}{3}$ and 2 whole units in $\frac{7}{3}$
• Figure out the number of whole units in each of the following fractions: (a) $\frac{8}{3}$ (b) $\frac{11}{5}$ (c) $\frac{9}{4}$ Sol. (a) $\frac{8}{3}=2+\frac{2}{3}=2\frac{2}{3}$ (b) $\frac{11}{5}=2+\frac{1}{5}=2\frac{1}{5}$ (c) $\frac{9}{4}=2+\frac{1}{4}=2\frac{1}{4}$
• Can all fractions greater than 1 be written as such mixed numbers? Sol. Yes, all fractions greater than 1 can be written as mixed numbers/mixed fraction. Ex : $-\frac{12}{7}=1+\frac{5}{7}=1\frac{5}{7}$
• Write the following fractions as mixed fractions (e.g., $\frac{9}{2}=4\frac{1}{2}$ ): (a) $\frac{9}{2}$ (b) $\frac{9}{5}$ (c) $\frac{21}{19}$ (d) $\frac{47}{9}$ (e) $\frac{12}{11}$ (f) $\frac{19}{6}$ Sol. (a) $\frac{9}{2}=4+\frac{1}{2}=4\frac{1}{2}$ (b) $\frac{9}{5}=1+\frac{4}{5}=1\frac{4}{5}$ (c) $\frac{21}{19}=1+\frac{2}{19}=1\frac{2}{19}$ (d) $\frac{47}{9}=5+\frac{2}{9}=5\frac{2}{9}$ (e) $\frac{12}{11}=1\frac{1}{11}=1\frac{1}{11}$ (f) $\frac{19}{6}=3+\frac{1}{6}=3\frac{1}{6}$
• Write the following mixed numbers as fractions: (a) $3\frac{1}{4}$ (b) $7\frac{2}{3}$ (c) $9\frac{4}{9}$ (d) $3\frac{1}{6}$ (e) $2\frac{3}{11}$ (f) $3\frac{9}{10}$ Sol. (a) $3\frac{1}{4}=3+\frac{1}{4}=\frac{3\times 4+1}{4}=\frac{13}{4}$ (b) $7\frac{2}{3}=7+\frac{2}{3}=\frac{7\times 3+2}{3}=\frac{22}{3}$ (c) $9\frac{4}{9}=9+\frac{4}{9}=\frac{9\times 9+4}{9}=\frac{85}{9}$ (d) $3\frac{1}{6}=3+\frac{1}{6}=\frac{3\times 6+1}{6}=\frac{19}{6}$ (e) $2\frac{3}{11}=2+\frac{3}{11}=\frac{2\times 11+3}{11}=\frac{25}{11}$ (f) $3\frac{9}{10}=3+\frac{9}{10}=\frac{3\times 10+9}{10}=\frac{39}{10}$

7.6 Equivalent Fractions

• Are $\frac{3}{6},\frac{4}{8},\frac{5}{10}$ equivalent fractions? Why? Sol. Since, $\frac{3}{6}\Rightarrow \frac{3÷3}{6÷3}=\frac{1}{2}$ $\frac{4}{8}\Rightarrow \frac{4÷4}{8÷4}=\frac{1}{2}$ and $\frac{5}{10}\Rightarrow \frac{5÷5}{10÷5}=\frac{1}{2}$ Since three fractions have same value when simplified, so they are equivalent fractions.
• Write two equivalent fractions for $\frac{2}{3}$. Sol. $\frac{2}{6}\Rightarrow \frac{2\times 2}{6\times 2}=\frac{4}{12}$ $\Rightarrow \frac{2\times 3}{6\times 3}=\frac{6}{18}$
• $\frac{4}{6}=\square =\square =\square =$.................(write as many you can) $\frac{4}{6}=\frac{8}{12}=\frac{12}{18}=\frac{16}{24}=$ Sol. $\frac{4}{6}\Rightarrow \frac{4\times 2}{6\times 2}=\frac{8}{12};\frac{4\times 3}{6\times 3}=\frac{12}{18};\frac{4\times 4}{6\times 4}=\frac{16}{24}$; $\frac{4}{6}=\frac{8}{12}=\frac{12}{18}=\frac{16}{24}=$
• Three rotis are shared equally by four children. Show the division in the picture and write a fraction for how much each child gets. Also, write the corresponding division facts, addition facts, and, multiplication facts. Fraction of roti each child gets is . Division fact: Addition fact: Multiplication fact: Compare your picture and answers
  
  with your classmates! Sol.
  
  Division fact $\Rightarrow 3$ wholes divided in 4 parts $=3÷4=\frac{3}{4}$ Addition fact $\Rightarrow$ Four times $\frac{3}{4}$ added gives 3 wholes $=\frac{3}{4}+\frac{3}{4}+\frac{3}{4}+\frac{3}{4}=\frac{12}{4}=3$ Multiplication fact $\Rightarrow 4$ parts of $\frac{3}{4}$ make 3 wholes $=4\times \frac{3}{4}=3$
• Draw a picture to show how much each child gets when 2 rotis are shared equally by 4 children. Also, write the corresponding division facts, addition facts, and multiplication facts. Sol. As 2 rotis have to be shared equally by 4 children we divide each roti in 4 parts and give
  
  (a) 1 part of each roti to each-child as shown below :
  
  (b) 2 parts to each child as shown below :
  
  Division fact $\Rightarrow 2$ whole divides in 4 parts $2÷4$ or $\frac{2}{4}=\frac{1}{2}$ Addition fact $\Rightarrow \frac{2}{4}+\frac{2}{4}+\frac{2}{4}+\frac{2}{4}=\frac{8}{4}=2$ Multiplication fact $=4\times \frac{2}{4}=2$
• Anil was in a group where 2 cakes were divided equally among 5 children. How much cake would Anil get? Sol.
  
  Each cake gets divided into 5 parts and Anil gets one part from each cake i.e. $\frac{1}{5}+\frac{1}{5}=\frac{2}{5}$.
• Find the missing numbers: (a) 5 glasses of juice shared equally among 4 friends is the same as glasses of juice shared equally among 8 friends. So, $\frac{5}{4}=\frac{\square }{8}$ $\square$ (b) 4 kg of potatoes divided equally in 3 bags is the same as 12 kgs of potatoes divided equally in __ bags. So, $\frac{4}{3}=\frac{12}{\square }$ $\square$ (c) 7 rotis divided among 5 children is the same as rotis divided among children. So, $\frac{7}{5}=\frac{\square }{\square }$. $\square$ Sol. (a) $\frac{5}{4}=\frac{\square }{8}$ $\square$ Since $4\times 2=8$ So, $5\times 2=10$ $\Rightarrow \frac{5}{4}=\frac{10}{8}$ (b) $\frac{4}{3}=\frac{12}{\square }$ Since $4\times 3=12$ So, $3\times 3=9$ $\frac{4}{3}=\frac{12}{9}$ (c) $\frac{7}{5}=\frac{\square }{\square }$ $\square$ $7\times 2=14;7\times 3=21;7\times 4=28$ $5\times 2=10;5\times 3=15;5\times 4=20$ $\frac{7}{5}=\frac{14}{10};\frac{7}{5}=\frac{21}{15};\frac{7}{5}=\frac{28}{10}$
• Find equivalent fractions for the given pairs of fractions such that the fractional units are the same. (a) $\frac{7}{2}$ and $\frac{3}{5}$ (b) $\frac{8}{3}$ and $\frac{5}{6}$ (c) $\frac{3}{4}$ and $\frac{3}{5}$ (d) $\frac{6}{7}$ and $\frac{8}{5}$ (e) $\frac{9}{4}$ and $\frac{5}{2}$ (f) $\frac{1}{10}$ and $\frac{2}{9}$ (g) $\frac{8}{3}$ and $\frac{11}{4}$ (h) $\frac{13}{6}$ and $\frac{1}{9}$ Sol. (a) The smallest common multiple of 2 and 5 is 10. $\frac{7\times 5}{2\times 5}=\frac{35}{10};\frac{3\times 2}{5\times 2}=\frac{6}{10}$ (b) The smallest common multiple of 3 and 6 is 6 . $\frac{8\times 2}{3\times 2}=\frac{16}{6}=\frac{5\times 1}{6\times 1}=\frac{5}{6}$ (c) The smallest common multiple of 4 and 5 is 20 . $\frac{3\times 5}{4\times 5}=\frac{15}{20};\frac{3\times 4}{5\times 4}=\frac{12}{20}$ (d) The smallest common multiple of 7 and 5 is 35 . $\frac{6\times 5}{7\times 5}=\frac{30}{35};\frac{8\times 7}{5\times 7}=\frac{42}{35}$ (e) The smallest common multiple of 4 and 2 is 4 . $\frac{9\times 1}{4\times 1}=\frac{9}{4}=\frac{5\times 2}{2\times 2}=\frac{10}{4}$ (f) The smallest common multiple of 10 and 9 is 90 . $\frac{1\times 9}{10\times 9}=\frac{9}{90};\frac{2\times 10}{9\times 10}=\frac{20}{90}$ (i) The smallest common multiple of 3 and 4 is 12 . $\frac{8\times 4}{3\times 4}=\frac{32}{12}=\frac{11\times 3}{4\times 3}=\frac{33}{12}$ (h) The smallest common multiple of 6 and 9 is 18. $\frac{13\times 3}{6\times 3}=\frac{39}{18};\frac{1\times 2}{9\times 2}=\frac{2}{18}$
• Express the following fractions in lowest terms: (a) $\frac{17}{51}$ (b) $\frac{64}{144}$ (c) $\frac{126}{147}$ (d) $\frac{525}{112}$ Sol. (a) Since, 17 is common factor of 17 and 51, So, $\frac{17÷17}{51÷17}=\frac{1}{3}$ (b) $\frac{64}{144}\Rightarrow \frac{64÷2}{144÷2}=\frac{32}{72}\Rightarrow \frac{32÷2}{72÷2}=\frac{16}{36}$ $\Rightarrow \frac{16÷2}{36÷2}=\frac{8}{18}\Rightarrow \frac{8÷2}{18÷2}=\frac{4}{9}$ (c) Since 3 is common factor of 126 and 147, $\frac{126÷3}{147÷3}=\frac{42}{49}\Rightarrow \frac{42÷7}{49÷7}=\frac{6}{7}$ (d) Since 7 as common multiple of 525 and 112, $\frac{525÷7}{112÷7}=\frac{75}{16}$

7.7 Comparing Fractions

• Compare the following fractions and justify your answers: (a) $\frac{8}{3},\frac{5}{2}$ (b) $\frac{4}{9},\frac{3}{7}$ (c) $\frac{7}{10},\frac{9}{14}$ (d) $\frac{12}{5},\frac{8}{5}$ (e) $\frac{9}{4},\frac{5}{2}$ Sol. (a) The smallest common multiple of 3 and $2,4,6$. $\frac{8\times 2}{3\times 2}=\frac{16}{6};\frac{5\times 3}{2\times 3}=\frac{15}{6}$ Since $\frac{16}{6}>\frac{15}{6}$ $\frac{8}{3}>\frac{5}{2}$ (b) The smallest common multiple of 9, 7 is 63. $\frac{4}{9}\times \frac{7}{7}=\frac{28}{63};\frac{3}{7}\times \frac{9}{9}=\frac{27}{63}$ $\because \frac{28}{63}>\frac{27}{63}$ Now $\frac{4}{9}>\frac{3}{7}$ (c) The smallest common multiple of 10,14 , is 70 . $\frac{7\times 7}{10\times 7}=\frac{49}{70};\frac{9}{14}\times \frac{5}{5}=\frac{45}{70}$ $\because \frac{49}{70}>\frac{45}{70}$ $\Rightarrow \frac{7}{10}>\frac{9}{14}$ (d) $\because$ Both fraction have same denominators, $\frac{12}{5}>\frac{8}{5}$ (e) The smallest common multiple of 4, 2 is 4 . $\frac{9}{4}\times \frac{1}{2}=\frac{9}{4};\frac{5\times 2}{2\times 2}=\frac{10}{4}$ $\because \frac{9}{4}<\frac{10}{4}$ $\Rightarrow \frac{9}{4}<\frac{5}{2}$.
• Write the following fractions in ascending order. (a) $\frac{7}{10},\frac{11}{15},\frac{2}{5}$ (b) $\frac{19}{24},\frac{5}{6},\frac{7}{12}$ Sol. (a) The smallest common multiple of $10,15,5$, is 30 . $\begin{array}{ll}& \frac{7\times 3}{10\times 3}=\frac{21}{30};\frac{11\times 2}{15\times 2}=\frac{22}{30};\frac{2\times 4}{5\times 6}=\frac{12}{30}\\ \because & \frac{12}{30}<\frac{21}{30}<\frac{22}{30}\\ \Rightarrow & \frac{2}{5}<\frac{7}{10}<\frac{11}{15}\text{ (ascending order) }\end{array}$ (b) The smallest common multiple of $24,6,12$, is 24 . $\begin{array}{rl}& \frac{19\times 1}{24\times 1}=\frac{19}{24};\frac{5\times 4}{6\times 4}=\frac{20}{24};\frac{7\times 2}{12\times 2}=\frac{14}{24}\\ \Rightarrow & \frac{7}{12}<\frac{19}{24}<\frac{5}{6}\text{ (ascending order) }\end{array}$
• Write the following fractions in descending order. (a) $\frac{25}{16},\frac{7}{8},\frac{13}{4},\frac{17}{32}$ (b) $\frac{3}{4},\frac{12}{5},\frac{7}{12},\frac{5}{4}$ Sol. (a) The smallest common multiple of $16,8,9,32$ is 32 $\begin{array}{rl}& \frac{25\times 2}{16\times 2}=\frac{50}{32};\frac{7\times 4}{8\times 4}=\frac{28}{32};\frac{13\times 8}{4\times 8}=\frac{104}{32};\frac{17\times 1}{32\times 1}=\frac{17}{32}\\ & \because \frac{104}{32}>\frac{50}{32}>\frac{28}{32}>\frac{17}{32}\\ & \Rightarrow \frac{13}{4}>\frac{25}{16}>\frac{7}{8}>\frac{17}{32}\text{ (descending order) }\end{array}$ (b) The smallest common multiple of $4,5,12$, is 60 . $\frac{3\times 15}{4\times 15}=\frac{45}{60};\frac{12\times 12}{5\times 12}=\frac{144}{60};\frac{7\times 5}{12\times 5}=\frac{35}{60};\frac{5\times 15}{4\times 15}=\frac{75}{60}$ $\because \frac{144}{60}>\frac{75}{60}>\frac{45}{60}>\frac{35}{60}$ $\Rightarrow \frac{12}{5}>\frac{5}{4}>\frac{3}{4}>\frac{7}{12}$ (descending order)

7.8 Addition and Subtraction of Fractions

• Add the following fractions using Brahmagupta's method: (a) $\frac{2}{7}+\frac{5}{7}+\frac{6}{7}$ (b) $\frac{3}{4}+\frac{1}{3}$ (c) $\frac{2}{3}+\frac{5}{6}$ (d) $\frac{2}{3}+\frac{2}{7}$ (e) $\frac{2}{3}+\frac{4}{5}$ (g) $\frac{4}{5}+\frac{2}{3}$ (h) $\frac{3}{5}+\frac{5}{8}$ (i) $\frac{9}{2}+\frac{5}{4}$ (j) $\frac{8}{3}+\frac{2}{7}$ (k) $\frac{3}{4}+\frac{1}{3}+\frac{1}{5}$ (i) $\frac{2}{5}+\frac{4}{5}+\frac{3}{7}$ (m) $\frac{9}{2}+\frac{5}{4}+\frac{7}{6}$ Sol. (a) $\frac{2}{7}+\frac{5}{7}+\frac{6}{7}=\frac{2+5+6}{7}=\frac{13}{7}$ (b) $\frac{3}{4}\times \frac{3}{3}=\frac{9}{12};\frac{1}{3}\times \frac{4}{4}=\frac{4}{12}$ $\frac{3}{4}+\frac{1}{3}=\frac{9}{12}+\frac{4}{12}=\frac{9+4}{12}=\frac{13}{12}$ (c) $\frac{2}{3}\times \frac{2}{2}=\frac{4}{6};\frac{5\times 1}{6\times 1}=\frac{5}{6}$ $\frac{2}{3}+\frac{5}{6}=\frac{4}{6}+\frac{5}{6}=\frac{4+5}{6}=\frac{10}{6}=\frac{10÷2}{6÷2}=\frac{5}{3}$ (d) $\frac{2\times 7}{3\times 7}=\frac{14}{21};\frac{2\times 3}{7\times 3}=\frac{6}{21}$ $\frac{2}{3}+\frac{2}{7}=\frac{14}{21}+\frac{6}{21}=\frac{14+6}{21}=\frac{20}{21}$ (e) $\frac{3}{4}+\frac{1}{3}+\frac{1}{5}$ The smallest common multiple of $3,4,5$ is 60 . $\frac{3\times 15}{4\times 15}=\frac{45}{60};\frac{1\times 20}{3\times 20}=\frac{20}{60};\frac{1\times 12}{5\times 12}=\frac{12}{60}$ $\frac{3}{4}+\frac{1}{3}+\frac{1}{5}=\frac{45}{60}+\frac{20}{60}+\frac{12}{60}=\frac{45+20+12}{60}=\frac{77}{60}$ (f) $\frac{2\times 5}{3\times 5}=\frac{10}{15};\frac{4\times 3}{5\times 3}=\frac{12}{15}$ $\frac{2}{3}+\frac{4}{5}=\frac{10}{15}+\frac{12}{15}=\frac{10+12}{15}=\frac{22}{15}$ (g) $\frac{4\times 3}{5\times 3}=\frac{12}{15};\frac{2\times 5}{3\times 5}=\frac{10}{15}$ $\frac{4}{5}+\frac{2}{3}=\frac{12}{15}+\frac{10}{15}=\frac{12+10}{15}=\frac{22}{15}$ (h) The smallest common multiple of 5 and 8 is 40 $\frac{3\times 8}{5\times 8}=\frac{24}{40};\frac{5\times 5}{8\times 5}=\frac{25}{40}$ $\frac{24}{40}+\frac{25}{40}=\frac{24+25}{40}=\frac{49}{40}$ (i) $\frac{9\times 2}{2\times 2}=\frac{18}{4};\frac{5\times 1}{4\times 1}=\frac{5}{4}$ $\frac{18}{4}+\frac{5}{4}=\frac{18+5}{4}=\frac{23}{4}$ (j) $\frac{8\times 7}{3\times 7}=\frac{56}{21};\frac{2\times 3}{7\times 3}=\frac{6}{21}$ $\frac{56}{21}+\frac{6}{21}=\frac{56+6}{21}=\frac{62}{21}$ (k) The smallest common multiple of $3,4,5$ is 60 . $\frac{3\times 15}{4\times 15}=\frac{45}{60};\frac{1\times 20}{3\times 20}=\frac{20}{60};\frac{1\times 12}{8\times 12}=\frac{12}{60}$ $\frac{45}{60}+\frac{20}{60}+\frac{12}{60}=\frac{45+20+12}{60}=\frac{77}{60}$ (1) The smallest common multiple of $3,5,7$ is 105 . $\frac{2\times 35}{3\times 35}=\frac{70}{105};\frac{4\times 21}{5\times 21}=\frac{84}{105};\frac{3\times 15}{7\times 15}=\frac{45}{105}$ $\frac{70}{105}+\frac{84}{105}+\frac{45}{105}=\frac{70+84+45}{105}=\frac{199}{105}$ (m) The smallest common multiple of $2,4,6$, is 12 . $\begin{array}{rl}& \frac{9\times 6}{2\times 6}=\frac{54}{12};\frac{5\times 3}{4\times 3}=\frac{15}{12};\frac{7\times 2}{6\times 2}=\frac{14}{12}\\ & \frac{54}{12}+\frac{15}{12}+\frac{14}{12}=\frac{54+15+14}{12}=\frac{83}{12}\end{array}$
• Rahim mixes $\frac{2}{3}$ litres of yellow paint with $\frac{3}{4}$ litres of blue paint to make green paint. What is the volume of green paint he has made? Sol. Volume of green paint $=\frac{2}{3}+\frac{3}{4}$ $\frac{2\times 4}{3\times 4}=\frac{8}{12};\frac{3\times 3}{4\times 3}=\frac{9}{12}$ $\frac{2}{3}+\frac{3}{4}=\frac{8}{12}+\frac{9}{12}=\frac{8+9}{12}=\frac{17}{12}$.
• Geeta bought $\frac{2}{5}$ meter of lace and Shamim bought $\frac{3}{4}$ meter of the same lace to put a complete border on a table cloth whose perimeter is 1 meter long. Find the total length of the lace they both have bought. Will the lace be sufficient to cover the whole border? Sol. Total length of the lace Geeta and Shamim have bought $=\frac{2}{5}+\frac{3}{4}$ The smallest common multiple of 5 and 4 is 20 . $\frac{2\times 4}{5\times 4}=\frac{8}{20};\frac{3\times 5}{4\times 5}=\frac{15}{20}$ $\frac{8}{20}+\frac{15}{20}=\frac{23}{20}\mathrm{m}=1\frac{3}{20}\mathrm{m}$ $=1\mathrm{m}+\frac{3}{20}\times 100\mathrm{cm}$ $=1\mathrm{m}15\mathrm{cm}$. Since perimeter of border on table cloth as 1 m . Here, length of lack is greater than perimeter, So, the required length of lace will be sufficient to cover the whole border.
• $\frac{5}{8}-\frac{3}{8}$ Sol. $\frac{5}{8}-\frac{3}{8}=\frac{5-3}{8}=\frac{2}{8}=\frac{2÷2}{8÷2}=\frac{1}{4}$
• $\frac{7}{9}-\frac{5}{9}$ Sol. $\frac{7}{9}-\frac{5}{9}=\frac{7-5}{9}=\frac{2}{9}$
• $\frac{10}{27}-\frac{1}{27}$ Sol. $\frac{10}{27}-\frac{1}{27}=\frac{10-1}{27}=\frac{9}{27}=\frac{9÷9}{27÷9}=\frac{1}{3}$
• Carry out the following subtractions using Brahmagupta's method: (a) $\frac{8}{15}-\frac{3}{15}$ (b) $\frac{2}{5}-\frac{4}{15}$ (c) $\frac{5}{6}-\frac{4}{9}$ (d) $\frac{2}{3}-\frac{1}{2}$ Sol. (a) $\frac{8}{15}-\frac{3}{15}$ $\Rightarrow \frac{8-3}{15}\{\because$ same denominators $\}$ $\Rightarrow \frac{5}{15}$ (b) $\frac{2}{5}-\frac{4}{15}$ Converting given fraction having same denominator into equivalent fraction, $\frac{2\times 3}{5\times 3}=\frac{6}{15};\frac{4\times 1}{15\times 1}=\frac{4}{15}$ $\Rightarrow \frac{6}{15}-\frac{4}{15}=\frac{6-4}{15}=\frac{2}{15}$. (c) $\frac{5}{6}-\frac{4}{9}$ Converting given fraction into equivalent fraction having same denominator, $\frac{5\times 3}{6\times 3}=\frac{15}{18};\frac{4\times 2}{9\times 2}=\frac{8}{18}$ $\Rightarrow \frac{15}{18}-\frac{8}{18}=\frac{15-8}{18}=\frac{7}{18}$. (d) $\frac{2}{3}-\frac{1}{2}$ Converting given fraction into equivalent fraction having same denominator, $\frac{2\times 2}{3\times 2}=\frac{4}{6};\frac{1}{2}\times \frac{3}{3}=\frac{3}{6}$ $\Rightarrow \frac{4}{6}-\frac{3}{6}=\frac{4-3}{6}=\frac{1}{6}$
• Subtract as indicated: (a) $\frac{13}{4}$ from $\frac{10}{3}$ (b) $\frac{18}{5}$ from $\frac{23}{3}$ (c) $\frac{29}{7}$ from $\frac{45}{7}$ Sol. (a) $\frac{13}{4}$ from $\frac{10}{3}\Rightarrow \frac{10}{3}-\frac{13}{4}$ $\begin{array}{rl}& \frac{10\times 4}{3\times 4}=\frac{40}{12};\frac{13\times 3}{4\times 3}=\frac{39}{12}\\ & \Rightarrow \frac{40}{12}-\frac{39}{2}=\frac{1}{12}\end{array}$ (b) $\frac{18}{5}$ from $\frac{23}{3}\Rightarrow \frac{23}{3}-\frac{18}{5}$ $\begin{array}{rl}& \frac{23\times 5}{3\times 5}=\frac{115}{15};\frac{18\times 3}{5\times 3}=\frac{54}{15}\\ & \Rightarrow \frac{115}{15}-\frac{54}{15}=\frac{115-54}{15}=\frac{61}{15}\end{array}$ (c) $\frac{29}{7}$ from $\frac{45}{7}\Rightarrow \frac{45}{7}-\frac{29}{7}$ $\frac{45-29}{7}=\frac{16}{7}$
• Solve the following problems: (a) Jaya's school is $\frac{7}{10}\mathrm{km}$ from her home. She takes an auto for $\frac{1}{2}\mathrm{km}$ from her home daily, and then walks the remaining distance to reach her school. How much does she walk daily to reach the school? (b) Jeevika takes $\frac{10}{3}$ minutes to take a complete round of the park and her friend Namit takes $\frac{13}{4}$ minutes to do the same. Who takes less time and by how much? Sol. (a)
  
  Distance between home and school $=\frac{7}{10}\mathrm{km}$. Distance travelled by auto $=\frac{1}{2}\mathrm{km}$. So, remaining distance travelled by walk. $\Rightarrow \frac{7}{10}-\frac{1}{2}$ $\frac{7\times 1}{10\times 1}=\frac{7}{10};\frac{1}{2}\times \frac{5}{5}=\frac{5}{10}$ $\Rightarrow \frac{7}{10}-\frac{1}{2}=\frac{7}{10}-\frac{5}{10}=\frac{7-5}{10}=\frac{2}{10}$ $=\frac{2÷2}{10÷2}=\frac{1}{5}\mathrm{km}$. (b) Time taken by Jeevika $=\frac{10}{3}$ minutes. Time taken by Namit $=\frac{13}{4}$ minutes. Now, $\frac{10\times 4}{3\times 4}=\frac{40}{12};\frac{13}{4}\times \frac{3}{3}=\frac{39}{12}$ Comparing $\frac{40}{12}$ and $\frac{39}{12}$, $\frac{40}{12}>\frac{39}{12}$ {For same denominator, fraction having greater numerator is greater} So, $\frac{10}{3}>\frac{13}{4}$ Or we can say Namit takes less time. Also, $\frac{10}{3}-\frac{13}{4}=\frac{40}{12}-\frac{39}{12}$ $=\frac{40-39}{12}=\frac{1}{12}$ minutes Namit takes less time by $\frac{1}{12}$ minutes.

5.0Key Features and Benefits of Chapter 7: Fractions

• Introduces fractions as parts of a whole and equal shares, helping students understand how quantities can be divided and represented in everyday situations.
• Uses visual models and diagrams to explain fractions, making it easier for students to grasp concepts through pictures and examples.
• Helps students learn how to identify, compare, and arrange fractions, improving their understanding of numerical relationships.
• Explains mixed fractions and equivalent fractions, showing how different fraction forms can represent the same value.
• Develops skills to perform basic operations like addition and subtraction of fractions in a clear and systematic way.
• Builds a strong foundation for future topics in mathematics, including ratios, decimals, and algebra.