RBSE Solutions Class 6 Maths Chapter 3 Whole Numbers Exercise 3.2

Question 1. Add the following by arranging in proper order.
(i) 85 + 186 + 15
(ii) 175 + 96 + 25
(iii) 65 + 75 + 35
(iv) 55 + 86 + 45
Answer: We will use the associative property of addition to group numbers that are easier to add first.
(i) \( 85 + 186 + 15 \)
\( (85 + 186) + 15 = 85 + (186 + 15) \)
\( 271 + 15 = 85 + 201 \)
\( 286 = 286 \)
(ii) \( 175 + 96 + 25 \)
\( (175 + 96) + 25 = 175 + (96 + 25) \)
\( 271 + 25 = 175 + 121 \)
\( 296 = 296 \)
(iii) \( 65 + 75 + 35 \)
\( (65 + 75) + 35 = 65 + (75 + 35) \)
\( 140 + 35 = 65 + 110 \)
\( 175 = 175 \)
(iv) \( 55 + 86 + 45 \)
\( (55 + 86) + 45 = 55 + (86 + 45) \)
\( 141 + 45 = 55 + 131 \)
\( 186 = 186 \)
In simple words: We grouped the numbers differently, but the final sum stayed the same. This shows how addition works no matter which numbers you add first.

🎯 Exam Tip: When adding three or more numbers, look for pairs that sum to a round number (like 10, 100) to make calculations easier and faster.

Question 2. Find out the multiplication by proper order.
(i) 4 x 1225 x 25
(ii) 4 × 158 × 125
(iii) 4 x 85 x 25
(iv) 8 x 20 x 125
Answer: We use the associative property of multiplication to group numbers that are easier to multiply first.
(iii) \( 4 \times 85 \times 25 \)
\( (4 \times 85) \times 25 = 85 \times (4 \times 25) \)
\( 340 \times 25 = 85 \times 100 \)
\( 8500 = 8500 \)
(iv) \( 8 \times 20 \times 125 \)
\( (8 \times 20) \times 125 = 20 \times (8 \times 125) \)
\( 160 \times 125 = 20 \times 1000 \)
\( 20000 = 20000 \)
In simple words: When multiplying three or more numbers, we can change the order of multiplication without changing the final product. This makes some calculations simpler, especially when creating easy-to-handle numbers like 100 or 1000.

🎯 Exam Tip: Group numbers strategically, such as \( 4 \times 25 = 100 \) or \( 8 \times 125 = 1000 \), to simplify multiplication problems quickly.

Question 3. Find out the value of each of the following by distributive property.
(i) 185 × 25 + 185 × 75
(ii) 4 x 18 + 4 x 12
(iii) 54279 × 92 + 8 × 54279
(iv) 12 x 8 + 12 × 2
Answer: We apply the distributive law, which says \( a \times b + a \times c = a \times (b + c) \).
(i) \( 185 \times 25 + 185 \times 75 \)
\( = 185 \times (25 + 75) \)
\( = 185 \times 100 \)
\( = 18500 \)
(ii) \( 4 \times 18 + 4 \times 12 \)
\( = 4 \times (18 + 12) \)
\( = 4 \times 30 \)
\( = 120 \)
(iii) \( 54279 \times 92 + 8 \times 54279 \)
\( = 54279 \times (92 + 8) \)
\( = 54279 \times 100 \)
\( = 5427900 \)
(iv) \( 12 \times 8 + 12 \times 2 \)
\( = 12 \times (8 + 2) \)
\( = 12 \times 10 \)
\( = 120 \)
In simple words: The distributive property helps us take a common number out of an addition (or subtraction) problem before multiplying. This often makes the calculation much easier, especially when one part adds up to 10 or 100.

🎯 Exam Tip: Always look for a common factor in expressions like \( a \times b + a \times c \) or \( a \times b - a \times c \) to simplify using the distributive property.

Question 4. Find out the multiplication by using proper property.
(i) 185 x 106
(ii) 208 x 185
(iii) 54 x 102
(iv) 158 × 1008
Answer: We use the distributive method over addition, where a number is broken down into a sum (e.g., \( 106 = 100 + 6 \)) and then multiplied.
(i) \( 185 \times 106 \)
\( = 185 \times (100 + 6) \)
\( = 185 \times 100 + 185 \times 6 \)
\( = 18500 + 1110 \)
\( = 19610 \)
(ii) \( 208 \times 185 \)
\( = 185 \times 208 \)
\( = 185 \times (200 + 8) \)
\( = 185 \times 200 + 185 \times 8 \)
\( = 37000 + 1480 \)
\( = 38480 \)
(iii) \( 54 \times 102 \)
\( = 54 \times (100 + 2) \)
\( = 54 \times 100 + 54 \times 2 \)
\( = 5400 + 108 \)
\( = 5508 \)
(iv) \( 158 \times 1008 \)
\( = 158 \times (1000 + 8) \)
\( = 158 \times 1000 + 158 \times 8 \)
\( = 158000 + 1264 \)
\( = 159264 \)
In simple words: We break one of the numbers into an easy sum, like 100 plus something small, or 1000 plus something small. Then, we multiply the other number by each part of the sum separately and add the results. This makes big multiplications simpler by breaking them into smaller, more manageable steps.

🎯 Exam Tip: Choose to expand the number that is close to a power of 10 (like 100, 1000) to simplify the calculation significantly. This is key for the distributive property.

Question 5. Match the following.

Answer:
(i) (b) - Commutativity of addition (the order of numbers in addition does not change the sum).
(ii) (a) - Commutativity of multiplication (the order of numbers in multiplication does not change the product).
(iii) (d) - Distributive property of multiplication over addition (multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products).
(iv) (c) - Associative property of multiplication (the way numbers are grouped in multiplication does not change the product).
In simple words: Matching means finding the correct rule for each math example. Commutative means you can swap numbers when adding or multiplying. Associative means you can group them differently. Distributive means you can spread out the multiplication over an addition.
🎯 Exam Tip: Remember the keywords: 'Commutative' is about order (swap positions), 'Associative' is about grouping (change parentheses), and 'Distributive' is about spreading multiplication over addition/subtraction.

Question 6. If the multiplication of any two whole numbers is zero, can we say that one or both of the numbers must be zero? Give an example to prove it.
Answer: Yes, if the product of any two whole numbers is zero, then it is always true that one or both of the numbers must be zero. This is called the Zero Product Property.
For example:
\( 0 \times 7 = 0 \)
\( 8 \times 0 = 0 \)
\( 0 \times 0 = 0 \)
In simple words: If two numbers multiply to zero, at least one of them has to be zero. There is no other way to get zero as an answer when you multiply.

🎯 Exam Tip: This is a fundamental property in mathematics; always remember that any number multiplied by zero is zero, and if a product is zero, at least one factor must be zero.

Question 7. If the multiplication of two whole numbers is 1, then can we say that one or both of the numbers are equal to 1? Prove your answer with example.
Answer: If the multiplication of two whole numbers is 1, then yes, we can definitely say that both numbers must be equal to 1. This is because 1 is the multiplicative identity for whole numbers, and it's the only whole number that, when multiplied by itself, results in 1.
For example: \( 1 \times 1 = 1 \).
It is important to note that for numbers that are not whole numbers (like fractions), one or both numbers might not be 1 but could be reciprocals of each other, e.g., \( 5 \times \frac {1}{5} = 1 \). However, since the question specifies "whole numbers," only \( 1 \times 1 \) is possible.
In simple words: If you multiply two whole numbers and get 1, both numbers must be 1. No other whole numbers can do this.

🎯 Exam Tip: Pay close attention to the definition of "whole numbers" (0, 1, 2, 3...) as it restricts the possible answers; fractions or negative numbers would change the answer.

Question 8. Find the product of the following using the distributive method.
(i) 138 × 101
(ii) 125 x 400
(iii) 608 × 35
Answer: We use the distributive property to simplify multiplication by breaking one number into a sum.
(i) \( 138 \times 101 \)
\( = 138 \times (100 + 1) \)
\( = 138 \times 100 + 138 \times 1 \)
\( = 13800 + 138 \)
\( = 13938 \)
(ii) \( 125 \times 400 \)
\( = 125 \times (300 + 100) \)
\( = 125 \times 300 + 125 \times 100 \)
\( = 37500 + 12500 \)
\( = 50000 \)
(iii) \( 608 \times 35 \)
\( = 608 \times (30 + 5) \)
\( = 608 \times 30 + 608 \times 5 \)
\( = 18240 + 3040 \)
\( = 21280 \)
In simple words: To use the distributive method for multiplication, we split one of the numbers into parts that are easier to multiply by. Then, we multiply the other number by each part and add those answers together.

🎯 Exam Tip: Always look for ways to break down numbers into sums involving powers of 10 (like 10, 100, 1000) for the easiest application of the distributive property.

Question 9. Which of the following will not result in zero?
(i) 1 + 0
(ii) 0 × 0
(iii) \( \frac {0}{2} \)
(iv) \( 10 - \frac {10}{2} \)
Answer: (i) 1 + 0
In simple words: When you add 1 and 0, the answer is 1. This is the only option that does not give zero as the result, because adding zero to a number leaves the number unchanged.

🎯 Exam Tip: Remember that adding zero to any number results in that same number, while multiplying by zero always results in zero. Division of zero by a non-zero number is also zero.

Question 10. Choose the correct option and write it in the bracket.
(i) Which of the following has the commutative property of addition?
(a) \( 5 \times 8 = 8 \times 5 \)
(b) \( (2 \times 3) \times 5 = 2 \times (3 \times 5) \)
(c) \( (12 + 8) + 10 = (2 + 8) + 10 \)
(d) \( 15 + 8 = 8 + 15 \)
(ii) Which of the following has commutative property of multiplication.
(a) \( 10 \times 20 = 20 \times 10 \)
(b) \( 10 \times 10 = 20 \times 20 \)
(c) \( (10 \times 20) = 10 \times 1 \)
(d) \( 10 + 20 = 10 \times 20 \)
Answer:
(i) (d) \( 15 + 8 = 8 + 15 \)
(ii) (a) \( 10 \times 20 = 20 \times 10 \)
In simple words: The commutative property means you can change the order of numbers when you add or multiply them, and the answer will stay the same. Option (i)(d) shows this for addition, and option (ii)(a) shows it for multiplication.

🎯 Exam Tip: Distinguish between commutative (order of numbers doesn't matter) and associative (grouping of numbers doesn't matter) properties; both are important for simplifying calculations.