RBSE Solutions Class 7 Maths Chapter 1 Integers Exercise 1.3

Question 1. Following are the properties of multiplication of integers and opposite to them are the examples. Match the correct pair.
(i) \( (-4) \times (5) = 5 \times (-4) \)
(ii) \( (-4) \times [(-3) + (-2)] = (-4) \times (-3) + (-4) \times (-2) \)
(iii) -4 (an integer) +7 (another integer), product \( (-4) \times (+ 7) = (-28) \) an integer property
(iv) \( (4) \times [(-7) \times (5)] = [(-4) \times (-7)] \times (5) \)

(a) Associative Property
(b) Commutative property
(c) Distributive property
(d) Closure property
Answer:
(i) \( (-4) \times (5) = 5 \times (-4) \) is the **Commutative Property**.
(ii) \( (-4) \times [(-3) + (-2)] = (-4) \times (-3) + (-4) \times (-2) \) is the **Distributive Property**.
(iii) -4 (an integer) +7 (another integer), product \( (-4) \times (+ 7) = (-28) \) an integer property is the **Closure Property**.
(iv) \( (4) \times [(-7) \times (5)] = [(-4) \times (-7)] \times (5) \) is the **Associative Property**.
In simple words: This question asks us to match mathematical examples with the correct property of multiplication for integers. We look at how the numbers are arranged or what happens when they are multiplied to find the right property. Each example shows a specific rule in action.

🎯 Exam Tip: Remember the definitions of each property: Commutative (order doesn't matter), Associative (grouping doesn't matter), Distributive (multiplication spreads over addition/subtraction), and Closure (result stays in the same set). Comparing the given equation to these definitions helps in matching them correctly.

Question 2. Fill in the blanks keeping in view the properties of multiplication of integers :
(i) \( 26 \times (- 48) = (- 48) \times \text{____} \) Commutative
(iii) \( 100 \times [(-4) \times (-52)] = [100 \times (- 4)] \times \text{____} \) Associative
Answer:
(i) \( 26 \times (- 48) = (- 48) \times \mathbf{26} \)
(iii) \( 100 \times [(-4) \times (-52)] = [100 \times (- 4)] \times \mathbf{(-52)} \)
In simple words: We used the given property names to figure out the missing numbers in the equations. The commutative property lets us swap numbers, and the associative property lets us group numbers differently without changing the result.

🎯 Exam Tip: Always identify the property (commutative, associative, etc.) being used first, as this will guide you to find the correct number to fill in the blank. The property name is a big hint!

Question 3. Find product by using appropriate property:
(i) \( 26 \times (- 48) + (-48) \times (-56) \)
(ii) \( 8 \times (78) \times (-125) \)
(iii) \( 9 \times (50 - 2) \)
(iv) \( 999 \times 45 \)
Answer:
(i) We use the distributive property to simplify the expression.
\( 26 \times (- 48) + (-48) \times (-56) \)
\( = (-48) \times 26 + (-48) \times (-56) \) (Rearrange terms using commutative property)
\( = (-48) \times [26 + (-56)] \) (Factor out \( -48 \) using distributive property)
\( = (-48) \times [26 - 56] \)
\( = (-48) \times (-30) \)
\( = 1440 \)

(ii) We can rearrange the terms to make multiplication easier using the associative property.
\( 8 \times 78 \times (-125) \)
\( = [8 \times (-125)] \times 78 \) (Group 8 and -125 together)
\( = -1000 \times 78 \)
\( = -78000 \)

(iii) We use the distributive property of multiplication over subtraction.
\( 9 \times (50 - 2) \)
\( = 9 \times 50 - 9 \times 2 \)
\( = 450 - 18 \)
\( = 432 \)

(iv) We can rewrite 999 as \( (1000 - 1) \) to use the distributive property.
\( 999 \times 45 \)
\( = (1000 - 1) \times 45 \)
\( = 1000 \times 45 - 1 \times 45 \)
\( = 45000 - 45 \)
\( = 44955 \)
In simple words: We used different math properties like distributive and associative to make the multiplication problems easier to solve. By changing how we group or split the numbers, we can simplify the calculations. This makes finding the final product much faster.

🎯 Exam Tip: Always look for opportunities to apply properties like the distributive property (e.g., \( a \times (b+c) = a \times b + a \times c \)) or associative property (e.g., \( (a \times b) \times c = a \times (b \times c) \)) to simplify complex calculations and avoid large multiplications directly. For example, multiplying by 100 or 1000 is much simpler.

Question 4. Identify True/False. Correct the false statements and write :
(i) Multiplication of integers is closed.
(ii) Division of integers is closed.
(iii) Division of integers is not commutative but multiplication is commutative.
(iv) Multiplication of integers, is distributive over addition.
(v) Division of integers is distributive on subtraction.
Answer:
(i) **True**. When you multiply any two integers, the result is always another integer. For example, \( 3 \times (-5) = -15 \), which is an integer.

(ii) **False**. Division of integers is not closed. For example, \( 5 \div 2 = 2.5 \), which is not an integer. The closure property means the result of an operation on numbers within a set must also be in that set.

(iii) **True**. Division of integers is not commutative because changing the order changes the result (e.g., \( 6 \div 3 \neq 3 \div 6 \)). However, multiplication of integers is commutative (e.g., \( 3 \times 5 = 5 \times 3 \)).

(iv) **True**. Multiplication of integers is distributive over addition. This means for any integers a, b, and c, \( a \times (b + c) = (a \times b) + (a \times c) \).

(v) **False**. Division of integers is not distributive on subtraction. This means \( a \div (b - c) \neq (a \div b) - (a \div c) \). For example, \( 12 \div (6-2) = 12 \div 4 = 3 \), but \( (12 \div 6) - (12 \div 2) = 2 - 6 = -4 \).
In simple words: We checked each statement about integer properties to see if it was true or false. We learned that multiplication always gives an integer (closed) and the order doesn't matter (commutative), but division often does not. It is important to know which properties apply to which operations.

🎯 Exam Tip: To check if a property holds (especially for closure, commutativity, or distributivity), always try with a few simple positive and negative integer examples. If you find even one counter-example, the property does not hold. For example, for division not being closed, \( 1 \div 2 \) is a clear example.