Samacheer Kalvi Class 6 Maths Solutions Term 1 Chapter 1 Numbers Exercise 1.5

Question 1. Fill in the blanks.
(i) The difference between the smallest natural number and the smallest whole number is .........
(ii) 17 \( \times \) ......... = 34 \( \times \) 17
(iii) When ......... is added to a number, it remains the same.
(iv) Division by ......... is not defined.
(v) Multiplication by ......... leaves a number unchanged.
Answer:
(i) The difference between the smallest natural number and the smallest whole number is 1.
(ii) 17 \( \times \) 34 = 34 \( \times \) 17
(iii) When Zero is added to a number, it remains the same.
(iv) Division by Zero is not defined.
(v) Multiplication by One leaves a number unchanged.
In simple words: This question tests your basic understanding of whole numbers and their properties. Natural numbers start from 1, while whole numbers include 0. Zero is special because it doesn't change a number when added, but makes division undefined. One is special because it doesn't change a number when multiplied.

🎯 Exam Tip: Remember the definitions of natural numbers (1, 2, 3...) and whole numbers (0, 1, 2, 3...) as this is a common trick. Also, clearly state "zero" or "one" for the identities and division rule.

Question 2. Say True or False.
1. 0 is the identity for multiplication of whole numbers.
2. The Sum of two whole numbers is always less than their product.
3. Both addition and multiplication are associative for whole numbers.
4. Both addition and multiplication are commutative for whole numbers.
5. Multiplication is distributive over addition for whole numbers.
Answer:
1. False (Zero is the additive identity, not multiplicative. One is the multiplicative identity.)
2. False (For example, \( 1+1=2 \) and \( 1 \times 1=1 \). Also, \( 0+5=5 \) and \( 0 \times 5=0 \). The sum is not always less than the product.)
3. True (This means how you group numbers in addition or multiplication does not change the result.)
4. True (This means the order of numbers in addition or multiplication does not change the result.)
5. True (You can multiply a number by a sum, or multiply it by each part of the sum separately and then add, getting the same answer.)
In simple words: This question checks your knowledge of number properties. Zero is for adding without changing, one is for multiplying without changing. Associative means grouping doesn't matter, and commutative means order doesn't matter. Distributive means you can spread out multiplication over addition.

🎯 Exam Tip: To check True/False for number properties, always try with small, simple whole numbers like 0, 1, 2, and 3. Sometimes, special cases like 0 and 1 can prove a statement false.

Question 3. Name the property being illustrated in each of the cases given below.
1. \( 75 + 34 = 34 + 75 \)
2. \( (12 \times 4) \times 8 = 12 \times (4 \times 8) \)
3. \( 50 + 0 = 50 \)
4. \( 50 \times 1 = 50 \)
5. \( 50 \times 42 = 50 \times 40 + 50 \times 2 \)
Answer:
1. Commutativity for addition (The order of numbers in addition is changed, but the sum remains the same.)
2. Associativity for multiplication (The way numbers are grouped in multiplication is changed, but the product remains the same.)
3. Zero is the additive identity (Adding zero to any number keeps the number unchanged.)
4. One is the multiplicative identity (Multiplying any number by one keeps the number unchanged.)
5. Distributivity of multiplication over addition (Multiplication is distributed over the sum, meaning \( a \times (b+c) = a \times b + a \times c \).)
In simple words: Each example shows a basic rule of how numbers work. Commutative means you can swap numbers. Associative means you can group them differently. Additive identity is zero, and multiplicative identity is one. Distributive property lets you multiply a number by parts of a sum.

🎯 Exam Tip: Pay close attention to the operation (addition, multiplication) and how the numbers are rearranged or grouped. This helps you identify the correct property. The use of parentheses is a strong clue for associativity, while changing order directly indicates commutativity.

Question 4. Use the properties of whole numbers and simplify.
(i) \( 50 \times 102 \)
(ii) \( 500 \times 689 - 500 \times 89 \)
(iii) \( 4 \times 132 \times 25 \)
(iv) \( 196 + 34 + 104 \)
Answer:
(i) \( 50 \times 102 \)
\( = 50 \times (100 + 2) \) (Using distributive property, we break 102 into an easier sum.)
\( = (50 \times 100) + (50 \times 2) \)
\( = 5000 + 100 \)
\( = 5100 \)

(ii) \( 500 \times 689 - 500 \times 89 \)
\( = 500 \times (689 - 89) \) (Using distributive property in reverse, we factor out the common number 500.)
\( = 500 \times 600 \)
\( = 300000 \)

(iii) \( 4 \times 132 \times 25 \)
\( = 4 \times (132 \times 25) \) (Using associative property to group numbers that multiply easily.)
\( = 4 \times (25 \times 132) \) (Using commutative property to reorder for easier calculation.)
\( = (4 \times 25) \times 132 \)
\( = 100 \times 132 \)
\( = 13200 \)

(iv) \( 196 + 34 + 104 \)
\( = (196 + 34) + 104 \) (Grouping the first two numbers first.)
\( = 230 + 104 \)
\( = 334 \)
Also,
\( = 196 + (34 + 104) \) (Using associative property, grouping the last two numbers first for easier mental math.)
\( = 196 + 138 \)
\( = 334 \)
In simple words: We use number rules to make calculations easier. For multiplication, we can split numbers or group them in different ways. For addition, we can also group numbers differently to find sums more quickly. The distributive property is very useful for both multiplication and subtraction.

🎯 Exam Tip: Always look for pairs of numbers that multiply or add to simple values like 10, 100, or 1000. This often involves using the associative or commutative properties. The distributive property is key when you see a common factor.

Objective Type Questions

Question 5. \( (53 + 49) \times 0 \) is
(a) 102
(b) 0
(c) 1
(d) \( 53 + 49 \times 0 \)
Answer: (b) 0
In simple words: When you multiply any number by zero, the answer is always zero. It does not matter how big or small the number is.

🎯 Exam Tip: Remember the "zero property of multiplication": any number multiplied by zero equals zero. This is a fundamental rule in mathematics.

Question 6. \( \frac{59}{1} \) is
(a) 1
(b) 0
(c) \( \frac{1}{59} \)
(d) 59
Answer: (d) 59
In simple words: When you divide any number by 1, the number stays the same. The number 1 acts as the identity for division.

🎯 Exam Tip: Any number divided by 1 is the number itself. This is similar to the multiplicative identity property where multiplying by 1 leaves the number unchanged.

Question 7. The product of a non-zero whole number and its successor is always
(a) an even number
(b) an odd number
(c) zero
(d) none of the options
Answer: (a) an even number
In simple words: A successor is the number right after another number. If you take any number that is not zero and multiply it by the next number in line, the answer will always be an even number. This is because one of the two numbers will always be even.

🎯 Exam Tip: Consecutive integers always include one even number. Since any number multiplied by an even number results in an even number, the product of a number and its successor will always be even.

Question 8. The whole number that does not have a predecessor is
(a) 10
(b) 0
(c) 1
(d) none of the options
Answer: (b) 0
In simple words: A predecessor is the number that comes just before another number. Among whole numbers, zero is the first number, so no whole number comes before it. Natural numbers start from 1, so 1 has no predecessor in natural numbers, but 0 has no predecessor in whole numbers.

🎯 Exam Tip: Understand the difference between natural numbers (1, 2, 3...) and whole numbers (0, 1, 2, 3...). For whole numbers, 0 is the smallest, hence it has no predecessor within that set.

Question 9. Which of the following expressions is not zero?
(a) \( 0 \times 0 \)
(b) \( 0 + 0 \)
(c) \( \frac{2}{0} \)
(d) \( \frac{0}{2} \)
Answer: (c) \( \frac{2}{0} \)
Dividing by 0 is not defined.
In simple words: When you multiply zero by zero, or add zero to zero, the answer is zero. If you divide zero by any other number, the answer is still zero. But, if you try to divide any number by zero, it's not possible to find an answer; it's called "undefined".

🎯 Exam Tip: Remember the rule: division by zero is undefined. This is a critical concept to grasp, as it's different from dividing zero by another number (which results in zero).

Question 10. Which of the following is not true?
(a) \( (4237 + 5498) + 3439 = 4237 + (5498 + 3439) \)
(b) \( (4237 \times 5498) \times 3439 = 4237 \times (5498 \times 3439) \)
(c) \( 4237 + 5498 \times 3439 = (4237 + 5498) \times 3439 \)
(d) \( 4237 \times (5498 + 3439) = (4237 \times 5498) + (4237 \times 3439) \)
Answer: (c) \( 4237 + 5498 \times 3439 = (4237 + 5498) \times 3439 \)
In simple words: Option (a) shows the associative property for addition, which is true. Option (b) shows the associative property for multiplication, which is also true. Option (d) shows the distributive property, which is true. Option (c) incorrectly applies the distributive property to addition with multiplication, making it false.

🎯 Exam Tip: Carefully review the properties of addition and multiplication. The associative property allows you to change the grouping of numbers, while the distributive property links multiplication with addition/subtraction. Option (c) mixes operations in a way that is not a valid property.