Figure it Out (Page 152):
Fill in the blanks with fractions.
1. Three guavas together weigh 1 kg. If they are roughly of the same size, each guava will roughly weigh ____kg.
Answer:
The weight of 3 guavas of the same size = 1 kg.
The weight of 1 guava of the same size = 1/3 kg.
Three guavas together weigh 1 kg. If they are roughly of the same size, each guava will roughly weigh 1/3 kg.
2. A wholesale merchant packed 1 kg of rice in four packets of equal weight. The weight of each packet is ___ kg.
Answer:
The weight of 4 packets of rice of equal weight = 1 kg.
The weight of 1 packet of rice of equal weight = ¼ kg.
A wholesale merchant packed 1 kg of rice in four packets of equal weight. The weight of each packet is 1/4 kg.
3. Four friends ordered 3 glasses of sugarcane juice and shared it equally among themselves. Each one drank ____ glass of sugarcane juice.
Answer:
Number of glasses that 4 friends drank = 3.
Number of glasses that 1 friend drank = ¾.
Four friends ordered 3 glasses of sugarcane juice and shared it equally among themselves. Each one drank ¾ glass of sugarcane juice.
4. The big fish weighs 1/2 kg. The small one weighs 1/4 kg. Together they weigh ____ kg.
Answer:
Weight of the big fish = ½ kg.
Weight of the small fish = ¼ kg.
Together they weight = ½ + ¼ = 2/4 + ¼ = (2 + 1)/4 = ¾ kg.
The big fish weighs 1/2 kg. The small one weighs 1/4 kg. Together they weigh 3/4 kg.
In-text Questions (Page 153):
5. 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.
Answer:
| Fractions in Words | Fraction | Fraction with same Denominator |
| One and a half | 1 ½ = 3/2 | 6/4 |
| three quarters | ¾ | 3/4 |
| one and a quarter | 1 ¼ = 5/4 | 5/4 |
| half | ½ | 2/4 |
| quarter | ¼ | 1/4 |
| two and a half | 2 ½ = 5/2 | 10/4 |
The fraction words from the smallest to the biggest are as follows:
Quarter < half < three quarters < one and a quarter < One and a half < two and a half
In-text Questions (Page 154):
By dividing the whole chikki into 6 equal parts in different ways, we get 1/6 chikki pieces of different shapes. Are they of the same size?
Answer:
Yes, 1/6 chikki pieces of different shapes are of the same size because each piece = 1/6 of the whole piece.
Figure it Out (Page 155):
The figures below show different fractional units of a whole chikki. How much of a whole chikki is each piece?
Answer:
In-text Questions (Page 157):
Answer:
Figure it Out (Page 158):
1. Continue this table of ½ for 2 more steps.
Answer:
2. Can you create a similar table for ¼?
Answer:
3. Make 1/3 using a paper strip. Can you use this to also make 1/6?
Answer:
- Gently fold one end towards the center without creasing fully.
- Bring the other end overlapping the first fold so that all three sections look equal.
- Adjust as needed to ensure equal parts.
- Once satisfied, press the folds to make sharp creases.
We use 1/3 using a paper strip as shown below:
1. Take one of the 1/3 sections from the previous step.
2. Fold it into 2 equal parts.
3. Unfold it, and each smaller section now represents 1/6 of the whole strip.
4. Draw a picture and write and addition statement as above to show:
a. 5 times ¼ of a roti b. 9 times ¼ of a roti
Answer:
a. 5 times ¼ of a roti
The picture is shown below:
The addition statement is: (¼ + ¼ + ¼ + ¼ + ¼) = 5/4.
b. 9 times ¼ of a roti
The addition statement is: (¼ + ¼ + ¼ + ¼ + ¼ + ¼ + ¼ + ¼ + ¼) = 9/4.
5. Match each fractional unit with the correct picture:
Answer:
Figure 159:
Now, can you find the lengths of the various blue lines shown below? Fill in the boxes as well.
1. 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.
Answer:
2. 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.
Answer:
3. Now, a unit is divided into 8 equal parts. Write the appropriate fractions in your notebook.
Answer:
A unit divided into 8 equal parts. The appropriate fractions are: 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8 and 8/8.
Figure it Out (Page 160):
1. On a number line, draw lines of lengths 1/10, 3/10, and 4/5.
Answer:
2. Write five more fractions of your choice and mark them on the number line.
Answer:
3. How many fractions lie between 0 and 1? Think, discuss with your classmates, and write your answer.
Answer:
Countless or infinite number of fractions lie between 0 and 1.
Explanation: You can always find an infinite number of fractions between any two fractions.
4. 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.
Answer:
5. Write the fraction that gives the lengths of the black lines in the respective boxes.
Answer:
Figure it Out (Page 161):
Did you notice something common between the fractions that are greater than 1?
Answer:
In all the fractions that are less than 1 unit, the numerator is smaller than the denominator. For example, in the fraction 2/3, the numerator 2 is smaller than the denominator 3.
In the fractions that are more than 1 unit, the numerator is larger than the denominator. For example, in the fraction 5/2, the numerator 5 is larger than the denominator 2.
Figure it Out (Page 162):
1. How many whole units are there in 7/2 ?
Answer:
7/2
= ½ + ½ + ½ + ½ + ½ + ½ + ½
= 1 + 1 + 1 + ½
= 3 + ½
Therefore, there are 3 whole units in 7/2.
2. How many whole units are there in 4/3 and in 7/3 ?
Answer:
4/3
= 1/3 + 1/3 + 1/3 + 1/3
= 1 + 1/3
Therefore, there is 1 whole unit in 4/3.
7/3
= 1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3
= 1 + 1 + 1/3
= 2 + 1/3
Therefore, there are 2 whole units in 7/3.
Figure it Out (Page 162):
1. Figure out the number of whole units in each of the following fractions:
a. 8/3 b. 11/5 c. 9/4
Answer:
a. 8/3
8/3
= (1/3 + 1/3 + 1/3) + (1/3 + 1/3 + 1/3) + 1/3 + 1/3
= 3 × 1/3 + 3 × 1/3 + 2/3
= 1 + 1 + 2/3
= 2 + 2/3
= 2 2/3
b. 11/5
11/5
= (1/5 + 1/5 + 1/5 + 1/5 + 1/5) + (1/5 + 1/5 + 1/5 + 1/5 + 1/5) + 1/5
= 5 × 1/5 + 5 × 1/5 + 1/5
= 1 + 1 + 1/5
= 2 + 1/5
= 2 1/5
c. 9/4
9/4
= (¼ + ¼ + ¼ + ¼) + (¼ + ¼ + ¼ + ¼) + ¼
= 4 × ¼ + 4 × ¼ + ¼
= 1 + 1 + ¼
= 2 + ¼
= 2 ¼
2. Can all fractions greater than 1 be written as such mixed numbers?
Answer:
Yes, all fractions greater than 1 can be written as mixed numbers.
Reason: A mixed number / mixed fraction contains a whole number (called the whole part) and a fraction that is less than 1 (called the fractional part).
3. Write the following fractions as mixed fractions (e.g., 9/2 = 4 1/2):
a. 9/2 b. 9/5 c. 21/19 d. 47/9 e. 12/11 f. 19/6
Answer:
a. 9/2
9/2
= ½ + ½ + ½ + ½ + ½ + ½ + ½ + ½ + ½
= (½ + ½) + (½ + ½) + (½ + ½) + (½ + ½) + ½
= 1 + 1 + 1 + 1 + ½
= 4 + ½
= 4 ½
b. 9/5
9/5
= 1/5 + 1/5 + 1/5 + 1/5 + 1/5 + 1/5 + 1/5 + 1/5 + 1/5
= (1/5 + 1/5 + 1/5 + 1/5 + 1/5) + 4 × (1/5)
= 5 × (1/5) + 4 × (1/5)
= 1 + 4/5
= 1 4/5
c. 21/19
21/19
= (19 + 2)/19
= 19/19 + 2/19
= 1 + 2/19
= 1 2/19
d. 47/9
47/9
= (9 + 9 + 9 + 9 + 9 + 2)/9
= 9/9 + 9/9 + 9/9 + 9/9 + 9/9 + 2/9
= 1 + 1 + 1 + 1 + 1 + 2/9
= 5 + 2/9
= 5 2/9
e. 12/11
12/11
= (11 + 1)/11
= 11/11 + 1/11
= 1 + 1/11
= 1 1/11
f. 19/6
19/6
= (6 + 6 + 6 + 1)/6
= 6/6 + 6/6 + 6/6 + 1/6
= 1 + 1 + 1 + 1/6
= 3 + 1/6
= 3 1/6
Figure it Out (Page 163):
Write the following mixed numbers as fractions:
a. 3 1/4 b. 7 2/3 c. 9 4/9
d. 3 1/6 e. 2 3/11 f. 3 9/10
Answer:
a. 3 ¼
3 ¼
= (¼ + ¼ + ¼ + ¼) + (¼ + ¼ + ¼ + ¼) + (¼ + ¼ + ¼ + ¼) + ¼
= (4 × ¼) + (4 × ¼) + (4 × ¼) + ¼
= 13/4
b. 7 2/3
7 2/3
= (1/3 + 1/3 + 1/3) + (1/3 + 1/3 + 1/3) + (1/3 + 1/3 + 1/3) + (1/3 + 1/3 + 1/3) + (1/3 + 1/3 + 1/3) + (1/3 + 1/3 + 1/3) + (1/3 + 1/3 + 1/3) + 2/3
= (3 × 1/3) + (3 × 1/3) + (3 × 1/3) + (3 × 1/3) + (3 × 1/3) + (3 × 1/3) + (3 × 1/3) + 2/3
= 23/3
c. 9 4/9
9 4/9
= 9 + 4/9
= 9 × 9/9 + 4/9
= 81/9 + 4/9
= 85/9
d. 3 1/6
3 1/6
= 3 + 1/6
= 3 × 6/6 + 1/6
= 18/6 + 1/6
= 19/6
e. 2 3/11
2 3/11
= 2 + 3/11
= 2 × 11/11 + 3/11
= 22/11 + 3/11
= 25/11
f. 3 9/10
3 9/10
= 3 + 9/10
= 3 × 10/10 + 9/10
= 30/10 + 9/10
= 39/10
Figure it Out (Page 164):
Answer the following questions after looking at the fraction :
1. Are the lengths 1/2 and 3/6 equal?
Answer:
Looking at the fraction wall, we see that the lengths ½ and 3/6 are equal.
2. Are 2/3 and 4/6 equivalent fractions? Why?
Answer:
Yes, 2/3 and 4/6 are equivalent fractions.
Reason: Looking at the fraction wall, we can see that 2/3 and 4/6 denote the same length.
3. How many pieces of length 1/6 will make a length of 1/2 ?
Answer:
Looking at the fraction wall, 3 lengths of 1/6 will make a length of ½.
4. How many pieces of length 1/6 will make a length of 1/3 ?
Answer:
Looking at the fraction wall, we can see that 2 pieces of length 1/6 will make a length of 1/3.
Figure it Out (Page 165):
1. Are 3/6, 4/8, 5/10 equivalent fractions? Why?
Answer:
Yes, 3/6, 4/8, 5/10 are equivalent fractions because they are all reduce to ½.
2. Write two equivalent fractions for 2/6.
Answer:
Two equivalent fractions for 2/6 are 1/3 and 3/9.
3. 4/6 = ____ = ____ = ____ = ………… (Write as many as you can)
Answer:
4/6 = 8/12 = 16/24 = 20/30
Figure it Out (Page 166):
1. 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!
Answer:
Fraction of roti each child gets is ¼ + ¼ + ¼ = 3/4.
Division fact: 3 ÷ 4 = ¾
Addition fact: ¾ + ¾ + ¾ + ¾ = 12/4 = 3
Multiplication fact: 4 × ¾ = 3
2. 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.
Answer:
When 2 rotis are equally shared between 4 children, each child gets ½ a roti. The required picture is shown below:
Division fact: 2 ÷ 4 = 2/4 = ½
Addition fact: 2/4 + 2/4 + 2/4 + 2/4 = 8/4 = 2
Multiplication fact: 4 × 2/4 = 2
3. Anil was in a group where 2 cakes were divided equally among 5 children. How much cake would Anil get?
Answer:
Each cake was divided equally among 5 children.
Therefore,
Part of first cake that Anil got = 1/5.
Part of second cake that Anil got = 1/5.
Therefore, part of cake that Anil got = 1/5 + 1/5 = 2/5.
In-text Questions (Page 168):
Equally divide the rotis in the situations shown below and write down the share of each child. Are the shares in each of these cases the same? Why?
Answer:
The shares of each child in each of these cases are shown below:
Figure it Out (Page 168):
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, 5/4 = ?/8.
b. 4 kg of potatoes divided equally in 3 bags is the same as 12 kgs of potatoes divided equally in ___ bags.
So, 4/3 = 12/?
c. 7 rotis divided among 5 children is the same as ____ rotis divided among _____ children.
So, 7/5 = ?/?.
Answer:
a. 5 glasses of juice shared equally among 4 friends is the same as 10 glasses of juice shared equally among 8 friends.
So, 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 9 bags.
So, 4/3 = 12/9
c. 7 rotis divided among 5 children is the same as 14 rotis divided among 10 children.
So, 7/5 = 14/10.
In-text Questions (Page 172):
Find equivalent fractions for the given pairs of fractions such that the fractional units are the same.
a. 7/2 and 3/5 b. 8/3 and 5/6 c. 3/4 and 3/5 d. 6/7 and 8/5
e. 9/4 and 5/2 f. 1/10 and 2/9 g. 8/3 and 11/4 h. 13/6 and 1/9
Answer:
a. 7/2 and 3/5
7/2 = (7 × 5)/(2 × 5) = 35/10
3/5 = (3 × 2)/(5 × 2) = 6/10
Equivalent fractions with the same fractional units are as follows: 35/10 and 6/10.
b. 8/3 and 5/6
8/3 = (8 × 2)/(3 × 2) = 16/6
5/6 = 5/6
Equivalent fractions with the same fractional units are as follows: 16/6 and 5/6.
c. 3/4 and 3/5
3/4 = (3 × 5)/(4 × 5) = 15/20
3/5 = (3 × 4)/(5 × 4) = 12/20
Equivalent fractions with the same fractional units are as follows: 15/20 and 12/20.
d. 6/7 and 8/5
6/7 = (6 × 5)/(7 × 5) = 30/35
8/5 = (8 × 7)/(5 × 7) = 56/35
Equivalent fractions with the same fractional units are as follows: 30/35 and 56/35.
e. 9/4 and 5/2
9/4 = 9/4
5/2 = (5 × 2)/(2 × 2) = 10/4
Equivalent fractions with the same fractional units are as follows: 9/4 and 10/4.
f. 1/10 and 2/9
1/10 = (1 × 9)/(10 × 9) = 9/90
2/9 = (2 × 10)/(9 × 10) = 20/90
Equivalent fractions with the same fractional units are as follows: 9/90 and 20/90.
g. 8/3 and 11/4
8/3 = (8 × 4)/(3 × 4) = 32/12
11/4 = (11 × 3)/(4 × 3) = 33/12
Equivalent fractions with the same fractional units are as follows: 32/12 and 33/12.
h. 13/6 and 1/9
13/6 = (13 × 3)/(6 × 3) = 39/18
1/9 = (1 × 2)(9 × 2) = 2/18
Equivalent fractions with the same fractional units are as follows: 39/18 and 2/18.
Figure it Out (Page 173):
Express the following fractions in lowest terms:
a. 17/51 b. 64/144 c. 126/147 d. 525/112
Answer:
a. 17/51
17/51
= (17 ÷ 17)/(51 ÷ 17)
= 1/3
b. 64/144
64/144
= (64 ÷ 2)/(144 ÷ 2)
= 32/72
= (32 ÷ 4)/(72 ÷ 4)
= 8/18
= (8 ÷ 2)/(18 ÷ 2)
= 4/9
c. 126/147
126/147
= (126 ÷ 3)/(147 ÷ 3)
= 42/49
= (42 ÷ 7)(49 ÷ 7)
= 6/7
d. 525/112
525/112
= (525 ÷ 7)/(112 ÷ 7)
= 75/16
Figure it Out (Page 174):
1. Compare the following fractions and justify your answers:
a. 8/3 , 5/2 b. 4/9 , 3/7 c. 7/10 , 9/14 d. 12/5 , 8/5 e. 9/4 , 5/2
Answer:
a. 8/3 , 5/2
We find equivalent fractions with the same denominator.
Common denominator of equivalent fractions = LCM of 3, 2 = 6.
Therefore,
8/3
= (8 × 2)/(3 × 2)
= 16/6
5/2
= (5 × 3)/(2 × 3)
= 15/6
Since clearly 16/6 > 15/6, we can say that 8/3 > 5/2.
b. 4/9 , 3/7
We find equivalent fractions with the same denominator.
Common denominator of equivalent fractions = LCM of 9, 7 = 63.
Therefore,
4/9 = (4 × 7)/(9 × 7) = 28/63
3/7 = (3 × 9)/(7 × 9) = 27/63
Since clearly 28/63 > 27/63, we can say that 4/9 > 3/7.
c. 7/10 , 9/14
We find equivalent fractions with the same denominator.
Common denominator of equivalent fractions = LCM of 10, 14 = 70.
Therefore,
7/10 = (7 × 7)/(10 × 7) = 49/70
9/14 = (9 × 5)/(14 × 5) = 45/70
Since clearly 49/70 > 45/70, we can say that 7/10 > 9/14.
d. 12/5 , 8/5
The above fractions have common denominator = 5.
Clearly, 12/5 > 8/5.
e. 9/4 , 5/2
We find equivalent fractions with the same denominator.
Common denominator of equivalent fractions = LCM of 4, 2 = 4.
Therefore,
9/4 = 9/4
5/2 = (5 × 2)/(2 × 2) = 10/4
Since clearly 10/4 > 9/4, we can say that 5/2 > 9/4.
2. Write the following fractions in ascending order.
a. 7/10 , 11/15 , 2/5 b. 19/24 , 5/6 , 7/12
Answer:
a. 7/10 , 11/15 , 2/5
We find equivalent fractions with the same denominator.
Common denominator of equivalent fractions = LCM of 10, 15 and 5 = 30.
Therefore,
7/10 = (7 × 3)/(10 × 3) = 21/30
11/15 = (11 × 2)/(15 × 2) = 22/30
2/5 = (2 × 6)/(5 × 6) = 12/30
Since, 12/30 < 21/30 < 22/30, the fractions arranged in ascending order are 2/5 < 7/10 < 11/15.
b. 19/24 , 5/6 , 7/12
We find equivalent fractions with the same denominator.
Common denominator of equivalent fractions = LCM of 24, 6 and 12 = 24.
Therefore,
19/24 = 19/24
5/6 = (5 × 4)/(6 × 4) = 20/24
7/12 = (7 × 2)/(12 × 2) = 14/24
Since, 14/24 < 19/24 < 20/24, the fractions arranged in ascending order are 7/12 < 19/24 < 5/6.
3. Write the following fractions in descending order.
a. 25/16 , 7/8 , 13/4 , 17/32 b. 3/4 , 12/5 , 7/12 , 5/4
Answer:
a. 25/16 , 7/8 , 13/4 , 17/32
We find equivalent fractions with the same denominator.
Common denominator of equivalent fractions = LCM of 16, 8, 4, 32 = 32.
Therefore,
25/16 = (25 × 2)/(16 × 2) = 50/32
7/8 = (7 × 4)/(8 × 4) = 28/32
13/4 = (13 × 8)/(4 × 8) = 104/32
17/32 = 17/32
Since clearly, 104/32 > 50/32 > 28/32 > 17/32, the fractions written in descending order are:
13/4 > 25/16 > 7/8 > 17/32.
b. 3/4 , 12/5 , 7/12 , 5/4
We find equivalent fractions with the same denominator.
Common denominator of equivalent fractions = LCM of 4, 5, 12, 4 = 60.
Therefore,
¾ = (3 × 15)/(4 × 15) = 45/60
12/5 = (12 × 12)/(5 × 12) = 144/60
7/12 = (7 × 5)/(12 × 5) = 35/60
5/4 = (5 × 15)/(4 × 15) = 75/60
Since clearly, 144/60 > 75/60 > 45/60 > 35/60, the fractions written in descending order are:
12/5 > 5/4 > 3/4 > 7/12.
In-text Questions (Page 177):
Try adding 4/7 + 6/7 using a number line. Do you get the same answer?
Answer:
We divide the part of the number line between 0 and 1 into 7 units.
We can see that successively adding 4/7 and 6/7 gives us 10/7 = 13/7.
Hence, we get the same answer.
Figure it Out (Page 179):
1. Add the following fractions using Brahmagupta’s method:
a. 2/7 + 5/7 + 6/7 b. 3/4 + 1/3 c. 2/3 + 5/6 d. 2/3 + 2/7 e. 3/4 + 1/3 + 1/5
f. 2/3 + 4/5 g. 4/5 + 2/3 h. 3/5 + 5/8 i. 9/2 + 5/4 j. 8/3 + 2/7
k. 3/4 + 1/3 + 1/5 l. 2/3 + 4/5 + 3/7 m. 9/2 + 5/4 + 7/6
Answer:
a. 2/7 + 5/7 + 6/7
2/7 + 5/7 + 6/7
= (2 + 5 + 6)/7
= 13/7
= (7 + 6)/7
= 7/7 + 6/7
= 1 + 6/7
= 1 6/7
b. 3/4 + 1/3
The lowest common multiple of 4 and 3 is 12.
Then we see that:
3/4 = (3 × 3)/(4 × 3) = 9/12 and
1/3 = (1 × 4)/(4 × 3) = 4/12
Therefore,
3/4 + 1/3
= 9/12 + 4/12
= (9 + 4)/12
= 13/12
= (12 + 1)/12
= 12/12 + 1/12
= 1 + 1/12
= 1 1/12
c. 2/3 + 5/6
The lowest common multiple of 3, 6 = 6.
Then we see that:
2/3 = (2 × 2)/(3 × 2) = 4/6 and
5/6 = 5/6
Therefore,
2/3 + 5/6
= 4/6 + 5/6
= (4 + 5)/6
= 9/6
= (9 ÷ 3)/(6 ÷ 3)
= 3/2
(2 + 1)/2
= 2/2 + ½
= 1 + ½
= 1 ½
d. 2/3 + 2/7
The lowest common multiple of 3, 7 = 21.
Then we see that:
2/3 = (2 × 7)/(3 × 7) = 14/21 and
2/7 = (2 × 3)/(7 × 3) = 6/21
Therefore,
2/3 + 2/7
= 14/21 + 6/21
= (14 + 6)/21
= 20/21
e. 3/4 + 1/3 + 1/5
The lowest common multiple of 4, 3, 5 = 60.
Then we see that:
3/4 = (3 × 15)/(4 × 15) = 45/60
1/3 = (1 × 20)/(3 × 20) = 20/60
1/5 = (1 × 12)/(1 × 12) = 12/60
Therefore,
3/4 + 1/3 + 1/5
= 45/60 + 20/60 + 12/60
= (45 + 20 + 12)/60
= 77/60
= (60 + 17)/60
= 60/60 + 17/60
= 1 + 17/60
= 1 17/60
f. 2/3 + 4/5
The lowest common multiple of 3, 5 = 15.
Then we see that:
2/3 = (2 × 5)/(3 × 5) = 10/15
4/5 = (4 × 3)/(5 × 3) = 12/15
Therefore,
2/3 + 4/5
= 10/15 + 12/15
= (10 + 12)/15
= 22/15
= (15 + 7)/15
= 15/15 + 7/15
= 1 + 7/15
= 1 7/15
g. 4/5 + 2/3
The lowest common multiple of 5, 3 = 15.
Then we see that:
4/5 = (4 × 3)/(5 × 3) = 12/15
2/3 = (2 × 5)/(3 × 5) = 10/15
Therefore,
4/5 + 2/3
= 12/15 + 10/15
= (12 + 10)/15
= 22/15
= (15 + 7)/15
= 15/15 + 7/15
= 1 + 7/15
= 1 7/15
h. 3/5 + 5/8
The lowest common multiple of 5, 8 = 40.
Then we see that:
3/5 = (3 × 8)/(5 × 8) = 24/40
5/8 = (5 × 5)/(8 × 5) = 25/40
Therefore,
3/5 + 5/8
= 24/40 + 25/40
= (24 + 25)/40
= 49/40
= (40 + 9)/40
= 40/40 + 9/40
= 1 + 9/40
= 1 9/40
i. 9/2 + 5/4
The lowest common multiple of 2, 4 = 4.
Then we see that:
9/2 = (9 × 2)/(2 × 2) = 18/4
5/4 = 5/4
Therefore,
9/2 + 5/4
= 18/4 + 5/4
= 23/4
= (5 × 4 + 3)/4
= (5 × 4)/4 + 3/4
= 5 + ¾
= 5 ¾
j. 8/3 + 2/7
The lowest common multiple of 3, 7 = 21.
Then we see that:
8/3 = (8 × 7)/(3 × 7) = 56/21
2/7 = (2 × 3)/(7 × 3) = 6/21
Therefore,
8/3 + 2/7
= 56/21 + 6/21
= (56 + 6)/21
= 62/21
= (21 × 2 + 20)/21
= (21 × 2)/21 + 20/21
= 2 + 20/21
= 2 20/21
k. 3/4 + 1/3 + 1/5
The lowest common multiple of 4, 3, 5 = 60.
Then we see that:
3/4 = (3 × 15)/(4 × 15) = 45/60
1/3 = (1 × 20)/(3 × 20) = 20/60
1/5 = (1 × 12)/(5 × 12) = 12/60
Therefore,
3/4 + 1/3 + 1/5
= 45/60 + 20/60 + 12/60
= (45 + 20 + 12)/60
= 77/60
= (60 + 17)/60
= 60/60 + 17/60
= 1 + 17/60
= 1 17/60
l. 2/3 + 4/5 + 3/7
The lowest common multiple of 3, 5, 7 = 105.
Then we see that:
2/3 = (2 × 35)/(3 × 35) = 70/105
4/5 = (4 × 21)/(5 × 21) = 84/105
3/7 = (3 × 15)/(7 × 15) = 45/105
Therefore,
2/3 + 4/5 + 3/7
= 70/105 + 84/105 + 45/105
= (70 + 84 + 45)/105
= 199/105
= (105 + 94)/105
= 105/105 + 94/105
= 1 + 94/105
= 1 94/105
m. 9/2 + 5/4 + 7/6
The lowest common multiple of 2, 4, 6 = 12.
Then we see that:
9/2 = (9 × 6)/(2 × 6) = 54/12
5/4 = (5 × 3)/(4 × 3) = 15/12
7/6 = (7 × 2)/(6 × 2) = 14/12
Therefore,
9/2 + 5/4 + 7/6
= 54/12 + 15/12 + 14/12
= (54 + 15 + 14)/12
= 83/12
= (12 × 6 + 11)/12
= (12 × 6)/12 + 11/12
= 6 + 11/12
= 6 11/12
2. 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?
Answer:
Volume of yellow paint = 2/3 litres.
Volume of blue paint = 3/4 litres.
Volume of green paint = (2/3 + 3/4) litres.
The lowest common multiple of 3, 4 = 12.
Then we see that:
2/3 = (2 × 4)/(3 × 4) = 8/12
3/4 = (3 × 3)/(4 × 3) = 9/12
Therefore,
2/3 + 3/4
= 8/12 + 9/12
= (8 + 9)/12
= 17/12
= (12 + 5)/12
= 12/12 + 5/12
= 1 + 5/12
= 1 5/12 litres
Volume of green paint he has made = 1 5/12 litres.
3. 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?
Answer:
Length of lace Geeta bought = 2/5 meter.
Length of lace Shamim bought = 3/4 meter.
Total length of the lace they both have bought = (2/5 + 3/4) meters.
2/5 + 3/4
= (2 × 4)/(5 × 4) + (3 × 5)/(4 × 5)
= 8/20 + 15/20
= (8 + 15)/20
= 23/20
= (20 + 3)/20
= 20/20 + 3/20
= 1 + 3/20
= 1 3/20
Perimeter of the table cloth = 1 meter.
Therefore, the total length of the lace they both have bought is more than sufficient to cover the whole border of the table cloth.
Figure it Out (Page 181):
1. 5/8 – 3/8
Answer:
5/8 – 3/8
= (5 – 3)/8
= 2/8
= (2 ÷ 2)/(8 ÷ 2)
= 1/4
2. 7/9 – 5/9
Answer:
7/9 – 5/9
= (7 – 5)/9
= 2/9
3. 10/27 – 1/27
Answer:
10/27 – 1/27
= (10 – 1)/27
= 9/27
= (9 ÷ 9)/(27 ÷ 9)
= 1/3
Figure it Out (Page 182):
1. Carry out the following subtractions using Brahmagupta’s method:
a. 8/15 – 3/15 b. 2/5 – 4/15 c. 5/6 – 4/9 d. 2/3 – ½
Answer:
a. 8/15 – 3/15
8/15 – 3/15
= (8 – 3)/15
= 5/15
= (5 ÷ 5)/(15 ÷ 5)
= 1/3
b. 2/5 – 4/15
2/5 – 4/15
= (2 × 3)/(5 × 3) – 4/15
= 6/15 – 4/15
= (6 – 4)/15
= 2/15
c. 5/6 – 4/9
5/6 – 4/9
= (5 × 3)/(6 × 3) – (4 × 2)/(9 × 2)
= 15/18 – 8/18
= (15 – 8)/18
= 7/18
d. 2/3 – 1/2
2/3 – 1/2
= (2 × 2)/(3 × 2) – (1 × 3)/(2 × 3)
= 4/6 – 3/6
= (4 – 3)/6
= 1/6
2. Subtract as indicated:
a. 13/4 from 10/3 b. 18/5 from 23/3 c. 29/7 from 45/7
Answer:
a. 13/4 from 10/3
10/3 – 13/4
= (10 × 4)/(3 × 4) – (13 × 3)/(4 × 3)
= 40/12 – 39/12
= (40 – 39)/12
= 1/12
b. 18/5 from 23/3
23/3 – 18/5
= (23 × 5)/(3 × 5) – (18 × 3)/(5 × 3)
= 115/15 – 54/15
= (115 – 54)/15
= 61/15
= (4 × 15 + 1)/15
= (4 × 15)/15 + 1/15
= 4 + 1/15
= 4 1/15
c. 29/7 from 45/7
45/7 – 29/7
= (45 – 29)/7
= 16/7
= (2 × 7 + 2)/7
= (2 × 7)/7 + 2/7
= 2 + 2/7
= 2 2/7
3. Solve the following problems:
a. Jaya’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?
b. Jeevika takes 10/3 minutes to take a complete round of the park and her friend Namit takes 13/4 minutes to do the same. Who takes less time and by how much?
Answer:
a. Distance of Jaya’s school from here home = 7/10 km.
Distance covered by Jaya via auto = 1/2 km.
Therefore, the remaining distance she covered by walking = (7/10 – 1/2) km.
7/10 – 1/2
= 7/10 – 1/2
= 7/10 – (1 × 5)/(2 × 5)
= 7/10 – 5/10
= (7 – 5)/10
= 2/10
= (2 ÷ 2)/(10 ÷ 2)
= 1/5
b. Time Jeevika takes to take a complete round of the park = 10/3 minutes.
Time Namit takes to take a complete round of the park = 13/4 minutes.
We compare the fractions 10/3 and 13/4 by taking the LCM of 3, 4 = 12.
10/3 = (10 × 4)/(3 × 4) = 40/12
13/4 = (13 × 3)/(4 × 3) = 39/12
Since clearly, 40/12 > 39/12, we can say that 10/3 > 13/4.
Therefore, Namit takes less time to take a complete round of the park.
Now,
40/12 – 39/12
= (40 – 39)/12
= 1/12
Therefore, Namit takes less time by 1/12 minutes to take a complete round of the park.
Puzzle (Page 185):
2. Can you find four different fractional units that add up to 1?
It turns out that this problem has six solutions! Can you find at least one of them? Can you find them all? You can try using similar reasoning as in the cases of two and three fractional units – or find your own method!
Once you find one solution, try to divide a circle into parts like in the figure above to visualize it!
Answer:
First Solution:
¼ + ¼ + ¼ + ¼ = 1
We increase the first ¼ to 1/2 and the second ¼ to 1/3.
Now,
1/2 + 1/3
= (1 × 3)/(2 × 3) + (1 × 2)/(3 × 2)
= 3/6 + 2/6
= (3 + 2)/6
= 5/6
We see,
1 – 5/6 = 1/6
1/6 = (1 × 5)/(6 × 5) = 5/30 = (2 + 3)/30 = 2/30 + 3/30 = 1/15 + 1/10
Therefore,
¼ + ¼ + ¼ + ¼ = 1/2 + 1/3 + 1/15 + 1/10 = 1 (First solution)
The rest of the solutions are left for you to find as an exercise!
