NCERT Solutions: Fractions

Table of Contents
1. Page 152 & 153
2. Page 155
3. Page 158
4. Page 159
5. Page 160
6. Page 162
7. Page 163
8. Page 164
9. Page 165
10. Page 166
11. Page 168
12. Page 172
13. Page 173
14. Page 174
15. Page 179
16. Page 181
17. Page 182
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Page 152 & 153

Figure it Out

Fill in the blanks with fractions.
Q1: Three guavas together weigh 1 kg. If they are roughly of the same size, each guava will roughly weigh ______kg.
Ans: 1/3

Q2: A wholesale merchant packed 1 kg of rice in four packets of equal weight. The weight of each packet is ____ kg.
Ans: 1/4

Q3: Four friends ordered 3 glasses of sugarcane juice and shared it equally among themselves. Each one drank ____ glass of sugarcane juice.
Ans: 3/4

Q4: The big fish weighs 1/2 kg. The small one weighs 1/4 kg. Together they weigh ____ kg.
Ans: 3/4


Q5: 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.
Ans: One and a half means = 1 + 1/2 = 3/2, Three quarters means = 3/4, One and a quarter means = 1 + 1/4 = 5/4, Half means = 1/2, Quarter means = 1/4, Two and a half means = 2 + 1/2 = 5/2
Arrange these fractions from smallest to largest = 1/4 < 1/2 < 3/4 < 5/4 < 3/2 < 5/2.
So, let's arrange these fraction words in order of size from smallest to larges,
Quarter < Half < Three quarters < One and a quarter < One and a half < Two and a half

Page 155

Q: The figures below show different fractional units of a whole chikki. How much of a whole chikki is each piece?
Ans:

Page 158

Q1: Continue this table of 1/2 for 2 more steps.
Ans:

Q2: Can you create a similar table for 1/4?
Ans: 

Q3: Make 1/3 using a paper strip. Can you use this to also make 1/6?
Ans: 
Q4: Draw a picture and write an addition statement as above to show:

(a) 5 times 1/4 of a roti
Ans: 

(b) 9 times 1/4 of a roti
Ans: 

Q5: Match each fractional unit with the correct picture:
Ans:

Page 159 

Now, can you find the lengths of the various blue lines shown below? Fill in the boxes as well.
Q1: Here, the fractional unit is dividing a length of 1 unit into three equal parts. Write the fraction that gives the length of the blue line in the box or in your notebook.
Ans: Here number line OR is divided into three equal parts OP, PQ and QR.
Hence length of pink line = OP + PQ = 1/3 + 1/3 = 2/3

Q2: Here, a unit is divided into 5 equal parts. Write the fraction that gives the length of the blue lines in the respective boxes or in your notebook.

Ans: Here number line OT = 1 unit is divided into five equal parts OP, PQ, QR, RS and ST.
Hence length of pink line OQ = OP + PQ =  1/5 + 1/5 = 2/5
Now, length of pink line OS = OP + PQ + QR + RS =  1/5 + 1/5 + 1/5 + 1/5 = 4/5
Hence, OQ = 2/5 or OS = 4/5

Q3: Now, a unit is divided into 8 equal parts. Write the appropriate fractions in your notebook.
Ans: Here number line OH is divided into 8 equal parts OA, AB, BC, CD, DE, EF, FG and GH.

Also, OA = 1/8, OB = 2/8, OC = 3/8, OH = 8/8 = 1
​

Page 160

Q1: On a number line, draw lines of lengths 1/10, 3/10, and 4/5.
Ans:

Q2: Write five more fractions of your choice and mark them on the number line.
Ans: Five fractions are = 1/5, 2/5, 5/10, 7/10 and 9/10

Q3: How many fractions lie between 0 and 1? Think, discuss with your classmates, and write your answer.
Ans: There are an infinite number of fractions between 0 and 1.
Example: 3/5, 7/10, 3/4, 1/2etc.

Q4: 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 1/2. So the blue line is 1/2 units long. Write the fraction that gives the length of the black line in the box.
Ans: Length of black line = 1/2 + 1/2 + 1/2 = 3/2

Q5: Write the fraction that gives the lengths of the black lines in the respective boxes.
Ans:

Page 162

Q1: How many whole units are there in 7/2?
Ans: 

7 There are 3 whole units in 7/2.

Q2: How many whole units are there in 4/3 and in 7/3?
Ans: 
(i)
There are 1 whole unit in 4/3.
(ii) 
There are 2 whole units in 7/3.

Figure it Out

Q1: Figure out the number of whole units in each of the following fractions:

(a) 8/3
Ans: 
There are 2 whole units in 8/3.

(b) 11/5
Ans:
There are 2 whole units in 11/5.

(c) 9/4
Ans:
There are 2 whole units in 9/4.

Q2: Can all fractions greater than 1 be written as such mixed numbers? 
Ans: Yes, all fractions greater than 1 can be written as mixed numbers. A mixed number consists of a whole number and a proper fraction (fraction less than 1).

Q3: Write the following fractions as mixed fractions (e.g., )

Page 163

Q: Write the following mixed numbers as fractions:

Ans: 

Page 164

Answer the following questions after looking at the fraction wall:
Q1: Are the lengths 1/2 and 3/6 equal?
Ans: Yes, here lengths 1/2 and 3/6 = 1/2 
Lengths are equal.

Q2: Are 2/3 and 4/6 equivalent fractions? Why?
Ans: Yes, lengths 2/3 and 4/6 = 1/3 are equivalent fractions, as they have the same length.

Q3: How many pieces of length 1/6 will make a length of 1/2?
Ans:
Total no. of pieces =
Hence, three pieces of length 1/6 will make a length of 1/2.

Q4: How many pieces of length 1/6 will make a length of 1/3?
Ans: 

Total no. of pieces =

Hence, two pieces of length 1/6 will make a length of 1/3.

Page 165

Figure it Out

Q1: Are 3/6, 4/8, 5/10 equivalent fractions? Why?
Ans: 

Yes, 3/6, 4/8, and 5/10 are equivalent fractions because they all simplify to 1/2.

Q2: Write two equivalent fractions for 2/6.
Ans: 

Therefore, 4/12, 6/18 are two equivalent fractions for 2/6

Q3: (Write as many as you can)
Ans:

Page 166

Figure it Out

Q1: 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!
Ans: As 3 rotis are divided equally among 4 children:

(i) Each roti is divided into 4 equal parts.
(ii) Each child receives $\frac{1}{4}\backslash frac\{1\}\{4\}$ share from each roti.

Fraction of roti each child gets = $\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{3}{4}\backslash frac\{1\}\{4\}\; +\; \backslash frac\{1\}\{4\}\; +\; \backslash frac\{1\}\{4\}\; =\; \backslash frac\{3\}\{4\}$

 
Division fact:
3 wholes divided into 4 parts
3 ÷ 4 = 3/4
Addition fact:
Four times 3/4 added gives 3 wholes
= 3/4 + 3/4 + 3/4 + 3/4
= 12/4 = 3
Multiplication fact:
4 parts of 3/4 make 3 wholes
4 × 3/4 = 3

Q2: 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.
Ans: 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:
2 wholes divide in 4 parts
2 ÷ 4 or 2/4 = 12
Addition fact:
2/4 + 2/4 + 2/4 + 2/4 = 8/4 = 2
Multiplication fact:
4 × 2/4 = 2

Q3: Anil was in a group where 2 cakes were divided equally among 5 children. How much cake would Anil get?
Ans: Anil is in a group where 2 cakes were divided equally among 5 children.

Each cake gets divided into 5 parts and Anil gets one part from each cake i.e. 1/5 + 1/5 = 2/5

Page 168

Figure it Out

Q: 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. 

Ans: Here, the amount of juice each friend gets when 5 glasses are shared among 4 friends =  number of glasses  number of friends = 5/4
Now to determine how many glasses of juice would be needed to give each of the 8 friends the same amount = 8 × 5/4 = 10 glasses
So, 10 glasses of juice shared equally among 8 friends is the same as 5 glasses of juice shared equally among 4 friends.
∴ 5/4 = 10/8

b. 4 kg of potatoes divided equally in 3 bags is the same as 12 kgs of potatoes divided equally in ____________ bags.

Ans: Let x is the number of bags for 12 kg of potatoes, where each bag has the same amount of potatoes then
12 kg/x bags = 4/3 kg per bag
⇒ 12 × 3 = 4 × x
⇒ 36 = 4x
⇒ x = 36/4
⇒ x = 9
∴ 4/3 = 12/9

c. 7 rods divided among 5 children is the same as rods divided among children. So, 7/5 = ____________

Ans: Dividing 7 rotis among 4 children gives 7 each child = 7/5 of a roti. We can find an equivalent fraction by multiplying both the numerator and the denominator by the same number. For example, multiplying both by 2.

So, 7 rotis divided among 5 children is the same as 14 rotis divided among 10 children

∴ 7/5 = 14/10

Page 172

Q: Find equivalent fractions for the given pairs of fractions such that the fractional units are the same.

a. 7/2 and 3/5
Ans: Given fractions are 7/2 and 3/5
Here, the denominators are 2 and 5.
And least common multiple of 2 and 5 is 10.
Hence for both fractions let's have same denominator of 10.
Now for 7/2 multiply both the numerator and the denominator by 5.

And for 3/5 multiply both the numerator and the denominator by 2, we get,

Hence, the equivalent fractions with the same denominator are:
35/10 and 6/10

b. 8/3 and 5/6
Ans: Given fractions are 8/3 and 5/6
Here, the denominators are 3 and 6.
And least common multiple of 3 and 6 is 6.
Now for 8/3 multiply both the numerator and the denominator by 2.

5/6 already have a denominator 6.
Hence, the equivalent fractions with the same denominator are:
16/6 and 5/6

c. 3/4 and 3/5
Ans: Given fractions are 3/4 and 3/5
Here, the denominators are 4 and 5.
And least common multiple of 4 and 5 is 20.
Now for 3/4 multiply both the numerator and the denominator by 5.

And for 3/5 multiply both the numerator and the denominator by 4, we get

So, the equivalent fractions with the same denominator are:
15/20 and 12/20

d. 6/7 and 8/5
Ans: Given fractions are 6/7 and 8/5
Here, the denominators are 7 and 5.
And least common multiple of 7 and 5 is 35.
Now for 6/7 multiply both the numerator and the denominator by 5.

And for 8/5 multiply both the numerator and the denominator by 7, we get

So, the equivalent fractions with the same denominator are:
30/35 and 56/35

e. 9/4 and 5/2
Ans: Given fractions are 9/4 and 5/2
Here, the denominators are 4 and 2.
And least common multiple of 4 and 2 is 4.
Now for 5/2 multiply both the numerator and the denominator by 2.

and 9/4 already have a denominator 4
So, the equivalent fractions with the same denominator are:
9/4 and 10/4

f. 1/10 and 2/9
Ans: Given fractions are and 1/10 and 2/9
Here, the denominators are 10 and 9.
And least common multiple of 10 and 9 is 90.
Now for 1/10 multiply both the numerator and the denominator by 9.

And for 2 multiply both the numerator and the denominator by 10, we get

So, the equivalent fractions with the same denominator are:'
9/90 and 20/90

g. 8/3 and 11/4
Ans: Given fractions are 8/3 and 11/4
Here, the denominators are 3 and 4.
And least common multiple of 3 and 4 is 12.
Now for 8/3 multiply both the numerator and the denominator by 4.

And for 11/4 multiply both the numerator and the denominator by 3, we get

So, the equivalent fractions with the same denominator are:
32/12 and 33/12

h. 13/6 and 1/9
Ans: Given fractions are 13/6 and 1/9
Here, the denominators are 6 and 9.
And least common multiple of 6 and 9 is 18.
Now for 13/6 multiply both the numerator and the denominator by 3.

And for 1/9 multiply both the numerator and the denominator by 2, we get

So, the equivalent fractions with the same denominator are:
39/18 and 2/18

Page 173

Q: Express the following fractions in lowest terms:

a. 17/51
Ans: Here 51 is divisible by 17 then 51/17 = 3

which is the lowest term.

b. 64/144
Ans: Here 64 and 144 are both multiples of 16. So we divide both by 16.

c. 126/147
Ans: Here, 126/147

[∵ HCF of 126 and 147 is 7 × 3 = 21]

d. 525/112
Ans: Here 525 and 112 are both multiples of 7, we divide both by 7.

Page 174

Figure it Out

Q1: Compare the following fractions and justify your answers:

a. 8/3, 5/2
Ans: L.C.M of 3 and 2 is 6.

b. 4/9, 3/7
Ans: Given fractions are 4/9 and 3/7.
Here LCM of denominators 9 and 7 is 63.
then multiplying and dividing 4/9 by 7 and 3/7 by 9 then

c. 7/10, 9/14 
Ans: Given fractions are 7/10 and 9/14.
Here LCM of denominators 10 and 14 is 70.
then multiplying and dividing 7/10 by 7 and 9/14 by 5 then

d. 12/5, 8/5 
Ans: Given fractions are 12/5, 8/5
Since, the denominators of both the fractions are same, then the fraction with the larger numerator is greater.
Here, 12/5 > 8/5

e. 9/4, 5/2
Ans: Given fractions are 9/4, 5/2.
Here LCM of denominators 4 and 2 is 4.
then multiplying and dividing 5/2 by 2 and
Equivalent fractions with 4 are

9/4 < 10/4 
∴ 9/4 < 5/2

Q2: Write following fractions in ascending order.

a. 7/10, 11/15, 2/5
Ans: The given fractions are 7/10, 11/15, 2/5
Let us find LCM of denominator 10, 15, 5

∴ LCM of 10, 15 and 5 = 2 × 3 × 5 = 30
Now let us make denominator of each fractions as LCM

Hence given fractions in ascending order are: 25,710115

b. 19/24, 5/6, 7/12
Ans: The given fractions are 19/24, 5/6, 7/12
Here LCM of 24, 6, 12 is 24.

On arranging in ascending Order, we get
14/24, 19/24, 20/24
⇒ 7/12, 19/24, 5/6

Q3: Write the following fractions descending order.

a. 25/16, 7/8, 13/4, 17/32
Ans: The given fractions are 25/16, 7/8, 13/4, 17/32
Let us find LCM of denominator 16, 8, 4, 32

LCM of 16, 8,4, 32 = 2 × 2 × 2 × 2 × 2 = 32
Now let us make denominator of each fractions as LCM thus

On arranging in descending order, we get

Hence given fractions in descending order are 13/4, 25/16, 7/8 and 17/32.

b. 3/4, 12/5, 7/12, 5/4
Ans: Given fractions are 3/4, 12/5, 7/12, 5/4
Here LCM of 4, 5, 12, 4 is 60
Now let us make denominator of each fraction as LCM then

Hence fractions in descending order are
12/5 > 5/4 > 3/4 > 7/12

Page 179

Figure it Out

Q1: Add the following fractions using Brahmagupta's method:

a. 
Ans: 

b.
Ans: Here
Here LCM of denominators 4 and 3 is 12
∴ Equivalent fraction of 3/4 with denominators 12 is 9/12 and equivalent fraction of 1/3 with denominators 12 is 4/12

c.
Ans: Given

Now LCM of 3 and 6 is 6.
Expressing as equivalent fractions with denominators 6, we get

d.
Ans: Here
Now LCM of 3 and 7 is 21
Expressing as equivalent fractions with denominators 21, we get

To add $\frac{2}{3}+\frac{2}{7}\backslash frac\{2\}\{3\}\; +\; \backslash frac\{2\}\{7\}$, start by finding the least common denominator (LCD) of the two fractions. The denominators are 3 and 7, and their LCD is 21.

Next, rewrite each fraction with a denominator of 21.

$\frac{2}{3}\backslash frac\{2\}\{3\}$ is equivalent to $\frac{14}{21}\backslash frac\{14\}\{21\}$, because $2\times 7=142\; \backslash times\; 7\; =\; 14$.
$\frac{2}{7}\backslash frac\{2\}\{7\}$ is equivalent to $\frac{6}{21}\backslash frac\{6\}\{21\}$, because $2\times 3=62\; \backslash times\; 3\; =\; 6$.

Now that the fractions have the same denominator, add them together:

$\frac{\mathrm{<img\; src="https://cn.edurev.in/ApplicationImages/Temp/24174782\_a0585bb9-00a7-4012-afea-237cefde9177\_lg.png"\; style="display:block;margin:5px\; auto;text-align:center;width:211px"\; loading="lazy"\; data-gumlet="false"\; alt="Page\; 179">}}{}rac\{14\}\{21\}\; +\; \backslash frac\{6\}\{21\}\; =\; \backslash frac\{14\; +\; 6\}\{21\}\; =\; \backslash frac\{20\}\{21\}.$ 

The final result is $\frac{20}{21}\backslash frac\{20\}\{21\}$

e.
Ans: Here

Now LCM of 4, 3, 5 is 60.
Expressing as equivalent fractions with denominators 60, we get

f.
Ans: Here
Now LCM of 3 and 5 is 15
Expressing as equivalent fractions with denominators 15, we get

g.
Ans: Here

Now LCM of 5 and 3 is 15
Thus expressing as equivalent fractions with denominators 15, we get

h.
Ans: Given
Here LCM of 5 and 8 is 40
Expressing as equivalent fractions with denominators 40, we get

i.
Ans: Here

Now LCM of 2 and 4 is 4.
Expressing as equivalent fractions with denominators 4, we get

j.
Ans: Given
Here LCM of 3 and 7 is 21
Expressing as equivalent fractions with denominators 21, we get

k.
Ans: Here
Now LCM of 4, 3, 5 is 60
Expressing as equivalent fractions with denominators 60, we get

l.
Ans: Here

Now LCM of 3, 5 and 7 is 105.
Expressing as equivalent fractions with denominators 105, we get

m.
Ans: Given
Here LCM of 2, 4, 6 is 12.
Now expressing as equivalent fractions with denominators 12, we get



Q2: Rahim mixes 2/3 litres of yellow paint with 3/4 litres of blue paint to make green paint. What is the volume of green paint he has made?
Ans: Given quantity of yellow paint = 2/3 litres
and quality of blue paint = 3/4 litres
Volume of green paint made
= 2/3 litres + 3/4 litres
Here LCM of 3 and 4 is 12
Now expressing as equivalent fractions with denominator 12, we get




Q3: Geeta bought 2/5 meter of lace and Shamim bought 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?
Ans: Given length of lace bought by Geeta = 2/5 meter
and length of lace bought by Shamim = 3/4 meter
Total length of lace = 2/5 + 3/4
LCM of 5 and 4 is 20
Now expressing as equivalent fractions with denominator 20, we get

Total lace required = Perimeter = 1 m
Hence total lace Geeta and Shamim have together is sufficient to cover the whole border.

Page 181

Figure it Out

Q1:
Ans: 

Q2:
Ans: 

Q3:
Ans: 

Page 182

Figure it Out

Q1: Carry out the following subtractions using Brahmagupta's method:

a.
Ans: Given
Fractional unit for both fractions is 1/15 then

b.
Ans: Given
Here LCM of 5 and 15 is 15. Fractional unit for both fractions should be 1/15

c.
Ans: Given
Hence LCM of 6 and 9 is 18. Fractional unit for both fractions should be 1/18 then

d.
Ans: Given
Here LCM of 3 and 2 is 6. Fractional unit for both fractions should be 1/6

Q2: Subtract as indicated:

a.
Ans: Given y - y
Here, LCM of 3 and 4 is 12.
Fractional unit for both fractions should be 1/12

b.
Ans: Here,

Now, LCM of 3 and 5 is 15.
Fractional unit = 1/15 for both fractions
Hence

c.
Ans: Here fractional = 1/7 for both fractions.




Q3: Solve the following problems:
a. Java's school is 7/10 km from her home. She takes an auto for 1/2 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?
Ans: Given distance between Jaya's school and home is 7/10 km and distance covered by Jaya in auto is 1/2 km.
∴ Distance Jaya covered by walking =

LCM of 10 and 2 is 10.

Hence Jaya walks 1/5 km or 200 meters to reach her school.

b. Jeevika takes 10/3 minutes to take a complete round of the park and her 13 friend Namit takes 13/4 minutes to do the same. Who takes less time and by how much?
Ans: Time taken by Jeevika to cover 1 round of park = 10/3 mm
Time taken by Namit to cover 1 round of 13 park= 13/4 min
To find who takes less time we need to compare 10/3 and 13/4
Here, LCM of 3 and 4 is 12

[Expressing fractions in fractional unit 1/12]

Hence Namit takes less time

Namit takes 1/2 of minutes less than Jeevika.