NCERT Class 7 Science Textbooks for blind and visually impaired students made Screen Readable by Professor T K bansal.

1.5.1 Closure under Multiplication

1. Observe the following table and complete it:

Statement	Inference
(−20) × (−5) = 100	Product is an integer
(−15) × 17 = −255
(−30) × 12 = _____
(−15) × (−23) = _____
(−14) × (−13) = _____
12 × (−30) = _____

What do you observe? Can you find a pair of integers whose product is not an integer?

No. This gives us an idea that the product of two integers is always an integer.

So we can say that integers are closed under multiplication.

In general,
a × b is an integer, for all integers a and b.

Find the product of five more pairs of integers and verify the above statement.

1.5.2 Commutativity of Multiplication

We know that multiplication is commutative for whole numbers.
Can we say, multiplication is also commutative for integers?

Observe the following table and complete it:

Statement 1	Statement 2	Inference
3 × (−4) = −12	(−4) × 3 = −12	3 × (−4) = (− 4) × 3
(−30) × 12 = _____	12 × (−30) = _____
(−15) × (−10) = 150	(−10) × (−15) = 150
(−35) × (−12) = _____	(−12) × (−35) = _____
(−17) × 0 = _____
__________ = _____	(−1) × (−15) = _____

What are your observations?
The above examples suggest multiplication is commutative for integers.
Write five more such examples and verify.

In general,
for any two integers a and b, a × b = b × a

1.5.3 Multiplication by Zero

We know that any whole number when multiplied by zero gives zero.

Observe the following products of negative integers and zero.
These are obtained from the patterns done earlier.

(−3) × 0 = 0
0 × (− 4) = 0
− 5 × 0 = _____
0 × (− 6) = _____

This shows that the product of a negative integer and zero is zero.
In general,
for any integer a, a × 0 = 0 × a = 0

1.5.4 Multiplicative Identity

We know that 1 is the multiplicative identity for whole numbers.

Check that 1 is the multiplicative identity for integers as well.

Observe the following products of integers with 1.
( −3) × 1 = −3
1 × 5 = 5
( −4) × 1 = _____
1 × 8 = _____
1 × ( −5) = _____
3 × 1 = _____
1 × ( −6) = _____
7 × 1 = _____

This shows that 1 is the multiplicative identity for integers also.

In general,
for any integer a we have, a × 1 = 1 × a = a

What happens when we multiply any integer with −1?

Complete the following:
(−3) × (−1) = 3
3 × (−1) = −3
(− 6) × (−1) = _____
(−1) × 13 = _____
(−1) × (−25) = _____
18 × (−1) = _____
What do you observe?

0 is the additive identity whereas 1 is the multiplicative identity for integers.

We get additive inverse of an integer a when we multiply ( −1) to a, i.e. a × ( −1) = ( −1) × a = − a

Can we say −1 is a multiplicative identity of integers? No.

1.5.5 Associativity for Multiplication

Consider −3, −2 and 5.
Look at [(−3) × (−2)] × 5 and (−3) × [(−2) × 5].

In the first case (−3) and (−2) are grouped together and in the second (−2) and 5 are grouped together.

We see that [(−3) × (−2)] × 5 = 6 × 5 = 30
and (−3) × [(−2) × 5] = (−3) × (−10) = 30

So, we get the same answer in both the cases.
Thus, [(−3) × (−2)] × 5 = (−3) × [(−2) × 5]

Look at this and complete the products:
[(7) × (− 6)] × 4 = __________ × 4 = __________
7 × [(− 6) × 4] = 7 × __________ = __________

Is [7 × (− 6)] × 4 = 7 × [(− 6) × 4]?

Does the grouping of integers affect the product of integers? No.

In general, for any three integers a, b and c
(a × b) × c = a × (b × c)

Take any five values for a, b and c each and verify this property.

Thus, like whole numbers, the product of three integers does not depend upon the grouping of integers and this is called the associative property for multiplication of integers.

1.5.6 Distributive Property

We know
16 × (10 + 2) = (16 × 10) + (16 × 2) [Distributivity of multiplication over addition]

Let us check if this is true for integers also.

Observe the following:
(a) (−2) × (3 + 5) = −2 × 8 = −16
and [(−2) × 3] + [(−2) × 5] = (− 6) + (−10) = −16
So, (−2) × (3 + 5) = [(−2) × 3] + [(−2) × 5]

(b) (− 4) × [(−2) + 7] = (− 4) × 5 = −20
and [(− 4) × (−2)] + [(− 4) × 7] = 8 + (−28) = −20
So, (− 4) × [(−2) + 7] = [(− 4) × (−2)] + [(− 4) × 7]

(c) (− 8) × [(−2) + (−1)] = (− 8) × (−3) = 24
and [(− 8) × (−2)] + [(− 8) × (−1)] = 16 + 8 = 24
So, (− 8) × [(−2) + (−1)] = [(− 8) × (−2)] + [(− 8) × (−1)]

Can we say that the distributivity of multiplication over addition is true for integers also?
Yes.

In general, for any integers a, b and c,
a × (b + c) = a × b + a × c

Take atleast five different values for each of a, b and c and verify the above Distributive property.

TRY THESE 1.9

(i) Is 10 × [(6 + (−2)] = 10 × 6 + 10 × (−2)?

(ii) Is (−15) × [(−7) + (−1)] = (−15) × (−7) + (−15) × (−1)?

Now consider the following:

Can we say 4 × (3 − 8) = 4 × 3 − 4 × 8?

Let us check:
4 × (3 − 8) = 4 × (−5) = −20
4 × 3 − 4 × 8 = 12 − 32 = −20

So, 4 × (3 − 8) = 4 × 3 − 4 × 8.

Look at the following:
( −5) × [( − 4) − ( − 6)] = ( −5) × 2 = −10
[( −5) × ( − 4)] − [ ( −5) × ( − 6)] = 20 − 30 = −10
So, ( −5) × [( − 4) − ( − 6)] = [( −5) × ( − 4)] − [ ( −5) × ( − 6)]
Check this for ( −9) × [ 10 − ( −3)] and [( −9) × 10 ] − [ ( −9) × ( −3)]
You will find that these are also equal.

In general, for any three integers a, b and c,
a × (b − c) = a × b − a × c

Take at least five different values for each of a, b and c and verify this property.

TRY THESE 1.10

(i) Is 10 × (6 − (−2)] = 10 × 6 − 10 × (−2)?
(ii) Is (−15) × [(−7) − (−1)] = (−15) × (−7) − (−15) × (−1)?

1.5.7 Making Multiplication Easier

Consider the following:
(i) We can find (−25) × 37 × 4 as
[(−25) × 37] × 4 = (− 925)× 4 = −3700

Or, we can do it this way,
(−25) × 37 × 4 = (−25) × 4 × 37 = [(−25) × 4] × 37 = (−100) × 37 = −3700

Which is the easier way?

Obviously the second way is easier because multiplication of (−25) and 4 gives −100 which is easier to multiply with 37. Note that the second way involves commutativity and associativity of integers.

So, we find that the commutativity, associativity and distributivity of integers help to make our calculations simpler. Let us further see how calculations can be made easier using these properties.

(ii) Find 16 × 12
16 × 12 can be written as 16 × (10 + 2).
16 × 12 = 16 × (10 + 2) = 16 × 10 + 16 × 2 = 160 + 32 = 192

(iii) (−23) × 48 = (−23) × [50 − 2] = (−23) × 50 − (−23) × 2 = (−1150) − (− 46)
= −1104

(iv) (−35) × (−98) = (−35) × [(−100) + 2] = (−35) × (−100) + (−35) × 2
= 3500 + (−70) = 3430

(v) 52 × (− 8) + (−52) × 2
(−52) × 2 can also be written as 52 × (−2).
Therefore, 52 × (− 8) + (−52) × 2 = 52 × (− 8) + 52 × (−2)
= 52 × [(− 8) + (−2)] = 52 × [(−10)] = −520

TRY THESE 1.11

 

Find (− 49) × 18; (−25) × (−31); 70 × (−19) + (−1) × 70 using distributive property.

 

EXAMPLE 2

Find each of the following products:
(i) (−18) × (−10) × 9
(ii) (−20) × (−2) × (−5) × 7
(iii) (−1) × (−5) × (− 4) × (− 6)

SOLUTION:

(i) (−18) × (−10) × 9 = [(−18) × (−10)] × 9 = 180 × 9 = 1620

(ii) (−20) × (−2) × (−5) × 7 = − 20 × (−2 × −5) × 7 = [−20 × 10] × 7 = − 1400

(iii) (−1) × (−5) × (− 4) × (− 6) = [(−1) × (−5)] × [(− 4) × (− 6)] = 5 × 24 = 120

EXAMPLE 3

Verify (−30) × [13 + (−3)] = [(−30) × 13] + [(−30) × (−3)]

SOLUTION:

(−30) × [13 + (−3)] = (−30) × 10 = −300
[(−30) × 13] + [(−30) × (−3)] = −390 + 90 = −300
So, (−30) × [13 + (−3)] = [(−30) × 13] + [(−30) × (−3)]

EXAMPLE 4

In a class test containing 15 questions, 4 marks are given for every correct answer and (−2) marks are given for every incorrect answer.
(i) Gurpreet attempts all questions but only 9 of her answers are correct. What is her total score?
(ii) One of her friends gets only 5 answers correct. What will be her score?

SOLUTION:

(i) Marks given for one correct answer = 4
So, marks given for 9 correct answers = 4 × 9 = 36
Marks given for one incorrect answer = − 2
So, marks given for 6 = (15 − 9) incorrect answers = (−2) × 6 = −12
Therefore, Gurpreet’s total score = 36 + ( −12) = 24

(ii) Marks given for one correct answer = 4
So, marks given for 5 correct answers = 4 × 5 = 20
Marks given for one incorrect answer = (−2)
So, marks given for 10 (=15 − 5) incorrect answers = (−2) × 10 = −20
Therefore, her friend’s total score = 20 + ( −20) = 0

EXAMPLE 5

Suppose we represent the distance above the ground by a positive integer and that below the ground by a negative integer, then answer the following:
(i) An elevator descends into a mine shaft at the rate of 5 metre per minute. What will be its position after one hour?
(ii) If it begins to descend from 15 m above the ground, what will be its position after 45 minutes?

SOLUTION:

(i) Since the elevator is going down, so the distance covered by it will be represented by a negative integer.

Change in position of the elevator in one minute = − 5 m

Position of the elevator after 60 minutes = (−5) × 60 = − 300 m, i.e., 300 m below down from the starting position of elevator.

(ii) Change in position of the elevator in 45 minutes = (−5) × 45 = −225 m, i.e., 225 m below ground level.

So, the final position of the elevator = −225 + 15 = −210 m, i.e., 210 m below ground level.

EXERCISE 1.3

Q1. Find each of the following products:
(a) 3 × (−1)
(b) (−1) × 225
(c) (−21) × (−30)
(d) (−316) × (−1)
(e) (−15) × 0 × (−18)
(f) (−12) × (−11) × (10)
(g) 9 × (−3) × (− 6)
(h) (−18) × (−5) × (− 4)
(i) (−1) × (−2) × (−3) × 4
(j) (−3) × (−6) × (−2) × (−1)

A1. (a) −3
(b) −225
(c) 630
(d) 316
(e) 0
(f) 1320
(g) 162
(h) −360
(i) −24
(j) 36

Q2. Verify the following:
(a) 18 × [7 + (−3)] = [18 × 7] + [18 × (−3)]
(b) (−21) × [(− 4) + (− 6)] = [(−21) × (− 4)] + [(−21) × (− 6)]

Q3. (i) For any integer a, what is (−1) × a equal to?
(ii) Determine the integer whose product with (−1) is
(a) −22
(b) 37
(c) 0

A3. (i) − a
(ii) (a) 22
(b) −37 (c) 0

Q4. Starting from (−1) × 5, write various products showing some pattern to show (−1) × (−1) = 1.

A4. −1 × 5 = −5,
−1 × 4 = − 4 = − 5 + 1,
− 1 × 3 = − 3 = − 4 + 1,
−1 × 2 = − 2 = − 3 + 1,
− 1 × 1 = − 1 = − 2 + 1,
− 1 × 0 = 0 = − 1 + 1
so, − 1 × (−1) = 0 + 1 = 1.

Q5. Find the product using suitable properties:
(a) 26 × (− 48) + (− 48) × (−36)
(b) 8 × 53 × (−125)
(c) 15 × (−25) × (− 4) × (−10)
(d) (− 41) × 102
(e) 625 × (−35) + (− 625) × 65
(f) 7 × (50 − 2)
(g) (−17) × (−29)
(h) (−57) × (−19) + 57

A5. (a) 480
(b) − 53000
(c) − 15000
(d) − 4182
(e) − 62500
(f) 336
(g) 493
(h) 1140

Q6. A certain freezing process requires that room temperature be lowered from 40°C at the rate of 5°C every hour. What will be the room temperature 10 hours after the process begins?

A6. − 10°C

Q7. In a class test containing 10 questions, 5 marks are awarded for every correct answer and (−2) marks are awarded for every incorrect answer and 0 for questions not attempted.
(i) Mohan gets four correct and six incorrect answers. What is his score?
(ii) Reshma gets five correct answers and five incorrect answers, what is her score?
(iii) Heena gets two correct and five incorrect answers out of seven questions she attempts. What is her score?

A7. (i) 8
(ii) 15
(iii) 0

Q8. A cement company earns a profit of Rs. 8 per bag of white cement sold and a loss of Rs. 5 per bag of grey cement sold.

(a) The company sells 3,000 bags of white cement and 5,000 bags of grey cement in a month. What is its profit or loss?
(b) What is the number of white cement bags it must sell to have neither profit nor loss, if the number of grey bags sold is 6,400 bags.

A8. (a) Loss of Rs 1000
(b) 4000 bags

Q9. Replace the blank with an integer to make it a true statement.
(a) (−3) × _____ = 27
(b) 5 × _____ = −35
(c) _____ × (− 8) = −56
(d) _____ × (−12) = 132

A9. (a) − 9
(b) − 7
(c) 7
(d) − 11