NCERT Class 6 Mathematics for Blind and Visually Impaired Students.

When we look into various operations on numbers closely, we notice several properties of whole numbers. These properties help us to understand the numbers better. Moreover, they make calculations under certain operations very simple.

Do This

Let each one of you in the class take any two whole numbers and add them.
Is the result always a whole number?

Your additions may be like this:

7	+	8	=	15, a whole number
5	+	15	=	10, a whole number
0	+	15	=	15, a whole number
---	+	---	=	------
---	+	---	=	------

Try with five other pairs of numbers. Is the sum always a whole number?

Did you find a pair of whole numbers whose sum is not a whole number?

Hence, we say that sum of any two whole numbers is a whole number i.e. the collection of whole numbers is closed under addition. This property is known as the closure property for addition of whole numbers.

Are the whole numbers closed under multiplication too? How will you check it?

Your multiplications may be like this :

7	×	8	=	56, a whole number
5	×	15	=	25, a whole number
0	×	15	=	0, a whole number
---	×	---	=	------
---	×	---	=	------

The multiplication of two whole numbers is also found to be a whole number again. We say that the system of whole numbers is closed under multiplication.

Closure property : Whole numbers are closed under addition and also under multiplication.

Think, discuss and write

1. The whole numbers are not closed under subtraction. Why?
Your subtractions may be like this :

6	−	2	=	4, a whole number
7	−	8	=	?, not a whole number
5	−	4	=	1, a whole number
3	−	9	=	Not a whole number

Take some more examples of your own and confirm.

2. Are the whole numbers closed under division? No. Observe this table :

8	÷	4	=	2, a whole number
5	÷	7	=	5/7 , not a whole number
12	÷	3	=	4, a whole number
6	÷	5	=	6/5, not a whole number

Justify it by taking a few more examples of your own.

Division by zero

Division by a number means subtracting that number repeatedly.
Let us find 8 ÷ 2.

8 – 2 = 6
6 – 2 = 4
4 – 2 = 2
2 – 2 = 0

After how many moves did we reach 0?
In four moves

So, we write 8 ÷ 2 = 4.

Using this, find
24 ÷ 8;
16 ÷ 4.

Let us now try 2 ÷ 0.
2 − 0 = 2
2 − 0 = 2
2 − 0 = 2
2 − 0 = 2
.. ..
.. ..

In every move we get 2 again !
Will this ever stop?
No. We say 2 ÷ 0 is not defined.

Let us try 7 ÷ 0
7 − 0 = 7
7 − 0 = 7
7 − 0 = 7
.. ..
.. ..

Again, we never get 0 at any stage of subtraction.
We say 7 ÷ 0 is not defined.

Check it for 5 ÷ 0, 16 ÷ 0.

Division of a whole number by 0 is not defined.

Commutativity of addition and multiplication

What do the following number line diagrams say?

In both the cases we reach 5. So, 3 + 2 is same as 2 + 3.

Similarly, 5 + 3 is same as 3 + 5.

Try it for 4 + 6 and 6 + 4.

Is this true when any two whole numbers are added?

Check it. You will not get any pair of whole numbers for which the sum is different when the order of addition is changed.

You can add two whole numbers in any order

We say that addition is commutative for whole numbers. This property is known as commutativity for addition.

Discuss with your friends

You have a small party at home. You want to arrange 6 rows of chairs with 8 chairs in each row
for the visitors. The number of chairs you will need is 6 × 8. You find that the room is not wide
enough to accommodate rows of 8 chairs. You decide to have 8 rows of chairs with 6 chairs in each row. How many chairs do you require now? Will you require more number of chairs?

Is there a commutative property of multiplication?
Multiply numbers 4 and 5 in different orders.
You will observe that 4 × 5 = 5 × 4.
Is it true for the numbers 3 and 6; 5 and 7 also?

You can multiply two whole numbers in any order.

We say multiplication is commutative for whole numbers.
Thus, addition and multiplication are commutative for whole numbers.

Verify:

(i) Subtraction is not commutative for whole numbers. Use at least three different pairs of numbers to verify it.

(ii) Is (6 ÷ 3) same as (3 ÷ 6)?
Justify it by taking few more combinations of whole numbers.

Associativity of addition and multiplication

Observe the following diagrams :
(a) (2 + 3) + 4 = 5 + 4 = 9

(b) 2 + (3 + 4) = 2 + 7 = 9

In (a) above, you have added 2 and 3 first and then added 4 to the sum and in (b) you have added 3 and 4 first and then added 2 to the sum.

Are not the results same?

We also have,
(5 + 7) + 3 = 12 + 3 = 15
and 5 + (7 + 3) = 5 + 10 = 15.

So, (5 + 7) + 3 = 5 + (7 + 3)

This is associativity of addition for whole numbers.
Check it for the numbers 2, 8 and 6.

 

Example 1:

Add the numbers 234, 197 and 103.

 

Solution:
234 + 197 + 103
= 234 + (197 + 103)
= 234 + 300 = 534

Play this game

You and your friend can play this.

You call a number from 1 to 10. Your friend now adds to this number any number from 1 to 10. Then it is your turn. You both play alternately. The winner is the one who reaches 100 first. If you always want to win the game, what will be your strategy or plan?

Observe the multiplication fact illustrated by the following diagrams (Fig 2.1).

Fig 2.1 a

Fig 2.1 b

Count the number of dots in Fig 2.1 (a) and Fig 2.1 (b). What do you get? The number of dots is the same. In Fig 2.1 (a), we have 2 × 3 dots in each box. So, the total number of dots is
(2 × 3) × 4 = 24.

In Fig 2.1 (b), each box has 3 × 4 dots, so in all there are 2 × (3 × 4) = 24 dots.
Thus, (2 × 3) × 4 = 2 × (3 × 4). Similarly, you can see that (3 × 5) × 4 = 3 × (5 × 4)

Try this for
(5 × 6) × 2 and 5 × (6 × 2);
(3 × 6) × 4 and 3 × (6 × 4).

This is associative property for multiplication of whole numbers.

Think on and find :
Which is easier and why?
(a) (6 × 5) × 3 or 6 × (5 × 3)
(b) (9 × 4) × 25 or 9 × (4 × 25)

 

Example 2

Find 14 + 17 + 6 in two ways.

 

Solution:
(14 + 17) + 6 = 31 + 6 = 37,

14 + 17 + 6
= 14 + 6 + 17
= (14 + 6) + 17
= 20 + 17 = 37

Here, you have used a combination of associative and commutative properties for addition.

Do you think using the commutative and the associative property has made the calculation easier?

The associative property of multiplication is very useful in the following types of sums.

Help yourself

Find :
7 + 18 + 13;
16 + 12 + 4

 

Example 3

Find 12 × 35.

 

Solution:
12 × 35
= (6 × 2) × 35
= 6 × (2 × 35)
= 6 × 70 = 420.

In the above example, we have used associativity to get the advantage of multiplying the smallest even number by a multiple of 5.

 

Example 4

Find 8 × 1,769 × 125

 

Solution:
8 × 1,769 × 125
= 8 × 125 × 1,769
(What property do you use here?)

= (8 × 125) × 1769
= 1000 × 1769
= 1,769,000.

Help yourself

Find :
25 × 8358 × 4 ;
625 × 3759 × 8

Think, discuss and write

Is (16 ÷ 4) ÷ 2 = 16 ÷ (4 ÷ 2)?
Is there an associative property for division? No.

Discuss with your friends.
Think of (28 ÷ 14) ÷ 2 and 28 ÷ (14 ÷ 2).

Distributivity of multiplication over addition

NCERT Class 6 Mathematics text book for blind and low vision students.

Do This

Take a graph paper of size 6 cm by 8 cm having squares of size 1 cm × 1 cm.

How many squares do you have in all?

Is the number 6 × 8?

Now cut the sheet into two pieces of sizes 6 cm by 5 cm and 6 cm by 3 cm, as shown in the figure.

Number of squares : Is it 6 × 5?

Number of squares : Is it 6 × 3?

In all, how many squares are there in both the pieces?
Is it (6 × 5) + (6 × 3)?
Does it mean that 6 × 8 = (6 × 5) + (6 × 3)?
But, 6 × 8 = 6 × (5 + 3)
Does this show that 6 × (5 + 3) = (6 × 5) + (6 × 3)?

Similarly, you will find that 2 × (3 + 5) = (2 × 3) + (2 × 5)

This is known as distributivity of multiplication over addition.

find using distributivity :
4 × (5 + 8) ;
6 × (7 + 9);
7 × (11 + 9).

Think, discuss and write

Observe the following multiplication and discuss whether we use here the idea of distributivity of multiplication over addition.

425×136
_____
2550 ← 425 × 6 (multiplication by 6 ones)
12750 ← 425 × 30 (multiplication by 3 tens)
42500 ← 425 × 100 (multiplication by 1 hundred)
______
57800 ← 425 × (6 + 30 + 100)

 

Example 5

The school canteen charges Rs. 20 for lunch and Rs. 4 for milk for each day. How much money do you spend in 5 days on these things?

 

Solution:
This can be found by two methods.

Method 1 :

Find the amount for lunch for 5 days.
Find the amount for milk for 5 days.
Then add i.e.
Cost of lunch = 5 × 20 = Rs. 100
Cost of milk = 5 × 4 = Rs. 20
Total cost = Rs. (100 + 20) = Rs. 120

Method 2 :

Find the total amount for one day.
Then multiply it by 5 i.e.
Cost of (lunch + milk) for one day = Rs. (20 + 4) = 24
Cost for 5 days = Rs. 5 × (20 + 4) = Rs. (5 × 24)
= Rs. 120.

The example shows that
5 × (20 + 4) = (5 × 20) + (5 × 4)

This is the principle of distributivity of multiplication over addition.

 

Example 6

Find 12 × 35 using distributivity.

 

Solution:
12 × 35 = 12 × (30 + 5)
= 12 × 30 +12 × 5
= 360 + 60 = 420

Help yourself

Find
15 × 68;
17 × 23;
69 × 78 + 22 × 69
using distributive property.

Identity (for addition and multiplication)

How is the collection of whole numbers different from the collection of natural numbers? It is just the presence of 'zero' in the collection of whole numbers. This number 'zero' has a special role in addition. The following table will help you guess the role.

7	+	0	=	7
5	+	0	=	5
0	+	15	=	15
0	+	26	=	26
0	+	.. ..	=	.. ..

When you add zero to any whole number what is the result?

It is the same whole number again! Zero is called an identity for addition of whole numbers or additive identity for whole numbers.

Zero has a special role in multiplication too. Any number when multiplied by zero becomes zero!

For example, observe the pattern :

{5 × 6 = 30
5 × 5 = 25
5 × 4 = 20
5 × 3 = 15
5 × 2 = ...
5 × 1 = ...
5 × 0 = ?}

Observe how the products decrease. Do you see a pattern? Can you guess the last step? Is this pattern true for other whole numbers also? Try doing this with two different whole numbers.

You came across an additive identity for whole numbers. A number remains unchanged when added to zero. Similar is the case for a multiplicative identity for whole numbers. Observe this table.

7	×	1	=	7
5	×	1	=	5
1	×	12	=	12
1	×	100	=	100
1	×	.. ..	=	.. ..

We are right. 1 is the identity for multiplication of whole numbers or multiplicative identity for whole numbers.

EXERCISE 2.2

Q1. Find the sum by suitable rearrangement:
(a) 837 + 208 + 363
(b) 1962 + 453 + 1538 + 647

A1.
(a) 1,408
(b) 4,600

Q2. Find the product by suitable rearrangement:
(a) 2 × 1768 × 50
(b) 4 × 166 × 25
(c) 8 × 291 × 125
(d) 625 × 279 × 16
(e) 285 × 5 × 60
(f) 125 × 40 × 8 × 25

A2.
(a) 176,800
(b) 16,600
(c) 291,000
(d) 2,790,000
(e) 85,500
(f) 1,000,000

Q3. Find the value of the following:
(a) 297 × 17 + 297 × 3
(b) 54,279 × 92 + 8 × 54,279
(c) 81,265 × 169 – 81,265 × 69
(d) 3,845 × 5 × 782 + 769 × 25 × 218

A3.
(a) 5,940
(b) 5,427,900
(c) 8,126,500
(d) 19,225,000

Q4. Find the product using suitable properties.
(a) 738 × 103
(b) 854 × 102
(c) 258 × 1008
(d) 1005 × 168

A4.
(a) 76,014
(b) 87,108
(c) 260,064
(d) 168,840

Q5. A taxidriver filled his car petrol tank with 40 litres of petrol on Monday. The next day, he filled the tank with 50 litres of petrol. If the petrol costs Rs. 44 per litre, how much did he spend in all on petrol?

A5. Rs.3,960

Q6. A vendor supplies 32 litres of milk to a hotel in the morning and 68 litres of milk in the evening. If the milk costs Rs. 45 per litre, how much money is due to the vendor per day?

A6. Rs. 4,500

Q7. Match the following:

(i)	425 × 136 = 425 × (6 + 30 +100)	(a) Commutativity under multiplication.
(ii)	2 × 49 × 50 = 2 × 50 × 49	(b) Commutativity under addition.
(iii)	80 + 2005 + 20 = 80 + 20 + 2005	(c) Distributivity of multiplication over addition.

A7.
(i) →(c)
(ii) → (a)
(iii) → (b)