NCERT Class 7 Solutions: Integers (Chapter 1) Exercise 1.3–Part 2

Q-3

  1. For any integer a, what is (1)×a equal to?

  2. Determine the integer whose product with (1) is:

  1. 22

  2. 37

  3. 0

Solution:

Properties of Multiplication

Commutative Property: When two numbers are multiplied together, the product is the same regardless of the order of the multiplicands. For example 4×2=2×4

Associative Property: When three or more numbers are multiplied, the product is the same regardless of the grouping of the factors. For example (2×3)×4=2×(3×4)

Multiplicative Identity Property: The product of any number and one gives us that number. For example 5×1=5 .

The Multiplication Property of Zero: Zero has a unique rule called the multiplication property. The multiplication property states that the product of any number and zero is zero. Any number, integer, real, rational, irrational, when multiplied by zero, gives us zero as the answer.

Distributive Property: The sum of two numbers times a third number is equal to the sum of each addend times the third number. Multiplication is said to be distributive over addition. For example 4×(6+3)=4×6+4×3 .

Multiplicative Inverse and Multiplicative Identity: A multiplicative inverse or reciprocal for a number x , denoted by 1x or x1 , is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction ab is ba . For the multiplicative inverse of a real number, divide 1 by the number.

Signs in Multiplication

Multiplication of positive and negative numbers

Signs in Multiplication

Multiplication of positive and negative numbers

  1. (1)×a=a (-ve × +ve = -ve)

  2. Solutions:

  1. 22_×(1)=22 (+ve × -ve = -ve)

  2. (37)_×(1)=37 (-ve × -ve = +ve)

  3. 0_×(1)=0 (Multiplication property of zero)

Q-4 Starting from (1)×5 , write various products showing some pattern and demonstrate that (1)×(1)=1

Solution:

  • 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

Note that this proves the consistency of the pattern and how rules of multiplication and addition allow us to reach this consistency. However this should not be taken as a proof of such properties.

Q-5 Find the product, using suitable properties:

  1. 26×(48)+(48)×(36)

  2. 8×53×(125)

  3. 15×(25)×(4)×(10)

  4. (41)×102

  5. 625×(35)+(625)×65

  6. 7×(502)

  7. (17)×(29)

  8. (57)×(19)+57

Solution:

Addition and Multiplication properties in brief.

Summary of Properties of Addition and Multiplication

Addition and Multiplication properties in brief.

  1. 26×(48)+(48)×(36)

    =(48)×26+(48)×(36) (Commutative property of addition, a+b=b+a )

    =(48)×[4636] (Distributive property of multiplication over addition a×(b+c)=a×b+a×c)

    =(48)×(10)

    =480

  2. 8×53×(125)

    We know that numbers with 0s are easier to multiply so, first take the product of 8 (an even number) and 125.

    =8×[53×(125)]

    =8×[(125)×53] (Commutative property of multiplication, a×b=b×a )

    =[8×(125)]×53 (Associative property (a×b)×c=a×(b×c) )

    =[1000]×53

    =53000

  3. [15×(25)]×(4)×(10) , again we know that product of 25 and 4 is 100.

    =[15×[(25)×(4)]]×(10) (Associative property (a×b)×c=a×(b×c) )

    =[15×[100]]×(10)

    = 15×[[100]×(10)] (Associative property (a×b)×c=a×(b×c) )

    =15×(1000)

    =15000

  4. (41)×102

    =(41)×(100+2)

    =(41)×100+(41)×2 (Distributive property, a×(b+c)=a×b+a×c)

    =410082

    =4182

  5. 625×(35)+(625)×65

    =625×[(35)+65] (Distributive property, a×(b+c)=a×b+a×c)

    =625×[100]

    =62500

  6. 7×(502)

    =(7×50)(7×2) (Distributive property, a×(b+c)=a×b+a×c)

    =35014

    =336

  7. (17)×(29)

    =(17)×(30+1)

    =(17)×(30)+(17)×1 (Distributive property, a×(b+c)=a×b+a×c)

    =510+(17)

    =493

  8. (57)×(19)+57

    =(57)×(19)+57×1 (1 is the multiplicative identity)

    =57[19+1] (Distributive property, a×(b+c)=a×b+a×c)

    =57×20

    =1140

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