# 6th Class Mathematics Number System and its Operations Operation on whole Numbers

## Operation on whole Numbers

Category : 6th Class

### Operation on whole Numbers

We are familiar with the basic operations of addition, multiplication, subtraction and division of the whole numbers. We will learn about these operations on whole numbers.

• Closure Property

The sum of two whole numbers is always a whole number. If a and b are two whole numbers, their addition (a + b) is also a whole number.

2 + 10 = 12, 2 and 10 are whole numbers and their sum 12 is also a whole number.

Consider the following two statements:

Statement 1: Subtraction of two whole numbers is always a whole number.

Statement 2: Subtraction of two whole numbers never be a whole number.

Which one of the following options is correct?

(a) Statement 1 is true and B is false

(b) Statement 1 is false and 2 is true

(c) Statement 1 and 2 are false

(d) All of these

(e) None of these

Explanation

Subtraction of two whole numbers is always a whole number.

• Commutative Property

The sum of two & more whole numbers remains same even if the order of the numbers are changed. If a and b are two whole numbers then a + b = b + a. The order of a + b is changed into b + a but the sum remains same. i.e. 4 + 5 = 9, on changing their order 5+4 =9. Therefore, 4+5=5+4.

If the sum of two numbers remains same on changing the order of the numbers then what will be the difference of the numbers if their order is changed?

(a) Difference of the numbers is not same on changing their order

(b) Difference of the numbers is same on changing their order

(c) Cannot be defined

(d) All of these

(e) None of these

Explanation

The result of subtraction of two numbers is changed on changing their order.

• Associative Property

The addition or multiplication of a set of numbers is the same regardless of how the numbers are grouped. The associative property involves 3 or more numbers. If a, b and c are three whole numbers then their sum is same on grouping the numbers in different way, (a + b) + c = a + (b + c). The sum of (3 + 4) + 5 = 12 and the sum of 3 + (4 + 5) =12, therefore, the sum of both groups is equal.

If the sum of $x+(y+z)=m+n$ then the sum of $(y+z)+x$ is?

(a) m - n

(b) n- m

(c) m+n

(d) All of these

(e) None of these

Explanation

According the associative property of addition if the sumof $x+(y+z)=m+n$ then the sum of $~(y+z)+x=m+n.$

Zero (0) is called the additive identity of every whole number. When 0 is added to the whole number its identity does not change or number remains unchanged. If a is a whole number then a + 0 = 0 + a = a/ therefore, the number remains same. The sum of 5 + 0 = 5, 0 +5 =5 and the sum of 6 + 0 = 6, 0 + 6 = 6.

Consider the following two statements:

Statement 1: Zero is the additive identity of every number.

Statement 2: Zero is the additive inverse of every number.

Which one of the following options is correct?

(a) Statement 1 is false and 2 is true

(b) Statement 1 is true and 2 is false

(c) Both the statements are true

(d) All of these

(e) None of these

Explanation

If a number is added with zero the result is the number itself. Therefore zero is called additive identity of every number.

Additive inverse of a is -a and additive inverse of -a is a, therefore, the sum of number with its additive inverse is always zero.

Let us consider a number 5 and its additive inverse -5, 5 + (-5) = 5 - 5 = 0, therefore, the sum of both the number is always zero.

Find the additive inverse of $(3x+2x-x)$?

(a)$2x$

(b)$4x$ (c)$-4x$

(d) All of these

(e) None of these

Explanation

$(3x+2x-x)=5x-x=4x$ and additive inverse o f$~4x=-4x.$

Properties of Subtraction

Subtraction is inverse process of addition, and subtraction of two whole numbers is always a whole numbers even if, a > b or a = b then their subtraction a- b is a whole number, i.e. $5-3\ne 3-5$or $2=-2,$ Therefore, the subtraction of 3 from 5 is 2 but the subtraction of 3  - 5 is  -2 therefore, on reversing the order of the expression, LHS is not equal to RHS. The order of the subtraction of two whole numbers does not change.

If a< b then their subtraction, a - b is not a whole number.

Subtraction is not a commutative or associative, i.e. $8-(6-5)\ne (8-6)-5$ or $8-1\ne 2-5$ or $7\ne -3.$

Which one of the following options is correct?

(a) (0.345 - 0.001) = (0.001 - 0.345)

(b) (0.345-0.001) < (0.001 - 0.345)

(c) (0.345 -  0.001)$\ne$ (0.001 - 0.345)

(d) All of these

(e) None of these

Explanation

According to the properties of subtraction (0.345  - 0.001)$\ne$(0.- 001 - 345)

Properties of Multiplication Closure Property

If a and b are whole numbers, their product $a\times b$ is also a whole number. Let us consider two whole numbers, 3 and 4, their product, 12 is also a whole number.

The sum and difference of two whole numbers is a whole number then the product of two whole numbers is?

(a) Prime number

(b) Whole number

(c) Composite number

(d) All of these

(e) None of these

Explanation

Product of two whole numbers is always a whole number.

• Commutative Property

The product of the whole numbers remains same even if the order of the multiplication is changed. In other words if a and b are whole numbers, their product$a\times b=b\times a$.i.e. $5\times 6=30,$ on changing their order, $6\times 5=30,$Thus $5\times 6=6\times 5.$ $10\times 15=150$and$15\times 10=150,$ thus, $10\times 15=15\times 10.$

Consider the following two statements:

Statement 1: The product of two composite numbers remains same on changing their order.

Statement 2: The product of a pair of co - primes is changed on changing their order.

Which one of the following options is correct?

(a) Statement 1 is true and 2 is false

(b) Statement 1 is false and 2 is true

(c) Both statements are false

(d) All of these

(e) None of these

Explanation

The product of two composite numbers remains same on changing their order.

• Associative Property

The product of more than two numbers remains same by changing the groups of the 8numbers. If a, b and c are three numbers then their product $(a\times b)\times c=a\times (b\times c).$ i.e. $(4\times 5)\times 6=20\times 6=120$ and $4\times (5\times 6)=4\times 30=120.$ Thus $(4\times 5)\times 6=4\times (5\times 6).$

Which one of the following options is correct?

(a) $\left( 3-x \right)\times y=y\times \left( x+3 \right)$

(b )$(3\text{ }-x)\times y=y\times (-x+3)$

(c) $(3-x)\times y\ne y\times (-x+3)$

(d) All of these

(e) None of these

Explanation

According to the associative property, of multiplication

$(3-x)\times y=y\times (-x+3).$

Multiplicative Identity

The product of every whole number with 1 is the number itself.

If a is a whole number then, $a\times 1=1\times a=a$

Multiplication of a number by 1 is the number itself, therefore, the identity of the whole number does not change thus 1 is called multiplicative identity of the whole number, i.e. $5\times 1=1\times 5=5$and $10\times 1=1\times 10=10.$

Which one of the following is called the product of a number and 1?

(a) Multiplicative inverse

(b) Multiplicative identity

(c) Cannot be defined

(d) All of these

(e)  None of these

Explanation

the product of a number with 1 is called multiplicative identity

If a, b, c are three whole numbers then according to the distributive property of addition, $a\times \left( b+c \right)=a\times b+a\times c$or $(b+c)\times a=a\times b+a\times c.$

i.e. $10\times \left( 12+13 \right)=10\times 25=250$ therefore,  $10\times \left( 12+13 \right)\text{ }=10\times 12+10\times 13.$ Group of numbers cannot be changed in the expression,$a\times \left( b+c \right),$ therefore, $a\times (b+c)\ne (a\times b)+c$ or $\text{a }\!\!\times\!\!\text{ (b+c)}\ne \text{(a }\!\!\times\!\!\text{ c)+b}\text{.}$

$10\times (10-5)=10\times 5=50$or$10\times 10-10\times 5=100-50=50,$ Thus $10\times (10-5)~=10\times 10-10\times 5.~~$

For whole numbers, a,b,c,d,e; $a\times (b+c+d+e)=a\times b+a\times c+a\times d+a\times e,$ Thus,   $4\times \left( 1+2+3+4 \right)=4\times 1+4\times 2+4\times 3+4\times 4$$=4+8+12+16=40$

In a class, 20 students are boys and 12 are girls. The number of teachers in the school is the difference of total number of boys and girls students in the school. If the number of chairs in the class is 10 times the difference between the number of boys and girls students in the class, then which one of the following expressions is correct about the number of chairs in the class?

(a)$20\times (12-10)$

(b)$10\times \left( 10-20 \right)$

(c)$10\times \left( 20-12 \right)$

(d) All of these

(e) None of these

Explanation

Number of chairs in the class = 10 times of difference of number of boys and girls students $=10\times \left( 20-12 \right).$

• Multiplication of Whole Numbers by 0

When a whole number is multiplied by 0, it becomes equal to zero. In other words $0\times a=a\times 0=0.$i.e.$0\times 11=0$ and$11\times 0=0$

Simplify:$\left\{ \left( 0.004+1.00 \right)\times \frac{1}{2} \right\}\times 0$

(a) 0

(b) 2

(c) 3

(d) All of these

(e) None of these

Explanation

Product of number with 0 is always 0.

• Multiplicative Inverse

Multiplicative inverse of a number a is$\frac{1}{a}$ and their multiplication is 1.

Let us consider a number 4 and its multiplicative inverse $\frac{1}{4},$ therefore, its multiplication is $4\times \frac{1}{4}=1.$

Simplify the $(34+20)\times 2$ and find the multiplicative inverse of the resulting simplification.

(a) $\frac{2}{108}$

(b) $\frac{1}{108}$

(c) $\frac{3}{108}$

(d) All of these

(e) None of these

Explanation

$(34+20)\times 2=54\times 2=108.$ Multiplicative inverse of 108$=\frac{1}{108}.$

Properties of Division

1.    If a and b are two whole numbers in the form of $a\div b$then it can be expressed by $\frac{a}{b}.$The division of a by b may or may not be a whole number. Where $b\ne 0.$ i.e. $10\div 5=\frac{10}{5}=2$is a whole number. The division of $25\div 15=\frac{25}{15}=\frac{5}{3}=1.666$is not a whole number.

If remainder of a division is greater than 0 then the quotient of complete division is a/an?

(a) Composite number

(b) Decimal number

(c) Whole number

(d) All of these

(e) None of these

Explanation

If remainder of a division is greater than zero then quotient will be a decimal number.

2.  For any non-zero whole number $a\div 1=\frac{a}{1}=a$ and $a\div a=\frac{a}{a}=1$ therefore,the division of every number by 1 is the number itself.

i.e. $6\div 1=\frac{6}{1}=6,6\div 6=\frac{6}{6}=1,10\div 1=\frac{10}{1}=10,\div 10=\frac{10}{10}=1$

Which one of the following options is correct?

(a)  $\left( \frac{4}{3}\div 1 \right)-\frac{4}{3}=1$

(b) $\left( \frac{4}{3}\div 1 \right)-\frac{4}{3}=0$

(c) $\left( \frac{4}{3}\div 1 \right)-\frac{4}{3}=0$

(d) All of these

(e) None of these

Explanation

$\left( \frac{4}{3}\div 1 \right)-\frac{4}{3}=\frac{4}{3}-\frac{4}{3}=0$

3.  If a is a number where, a ^ 0 then 0 a = 0 and a 0 cannot be defined. The division f 0 by 4 or 0 4=0 and $4\div 0$ cannot be defined.

Is this statement true or false $''\left( \frac{x}{2}\div y \right)\div 0=12''$?

(a) True

(b) False

(c) Cannot be defined

(d) All of these

(e) None of these

Explanation

The terms in the bracket is divided by 0 thus, it cannot be defined.

4.  If a, b and care whole numbers then $(a\div b)$ c a (b c) therefore, division is not associative, i.e. $\left( 16\text{ }4 \right)\div 2\text{ }16\text{ }\left( 4\div 2 \right)\text{ }$or$4\div 2\text{ }16\text{ }2\text{ }$or, $2\text{ }8.$

The division of y by $x$ is z then the division of $x$ by y is?

(a)z

(b) Other than z

(c) Less than z

(d) All of these

(e) None of these

Explanation

According to the question

hence,

$\frac{x}{y}\ne z$

5.    If a, b, c, d are whole numbers in the form of a $a\overset{c}{\overline{\left){\frac{b}{d}}\right.}}$then a is called divisor, b isdividend, c is quotient and d is called remainder.

Whereas, Dividend = divisor $x$quotient + remainder. It is also known as division Algorithm. The division of69063 by 35 is: Where, quotient = 1973 and reminder = 8 According to the division algorithm $35\times 1973+8=69055+8=69063$ is the dividend.

If x is dividend and y is remainder then which one of the following options is correct about the quotient?

(a) Quotient is greater than the divisor

(b) Remainder is less than divisor

(c) Dividend is exactly divisible

(d) All of these

(e) None of these