Rational Numbers

Rational Numbers – A number which can be written in form of

where p and q are integers and q≠0.

is a rational number

where p and q are 5 and 8 respectively and q≠0.

A no is a rational number if and only if p and q are integers and q≠0.

(a) Data insufficient

(c) Data insufficient

(d)It is mentioned in the option that p≠0 which violates the

definition of ration of rational number because p can be 0.

+ () =

In the given expression the addition of two rational numbers is given and the result obtained is also a rational number.

(b) Incorrect as (a) is true and proved.

(c) Incorrect the operation performed is addition not Multiplication (Irrelevant option).

(d) Incorrect no commutative operation is performed here (Irrelevant option).

Any rational number “a” is not defined in the condition when ,thus rational numbers are not closed under division.

(a) True because the addition of two rational number is a rational number.

(b) True because subtraction of two rational number is a rational number.

(c) True because multiplication of two rational number is a rational number.

The question is incorrect.

In place of it shoulb be .

⇒Now the question becomes ) {Here a = ,b = }

⇒ Now the operation performed is a + b = b + a

⇒This is commutative property of rational numbers

⇒ Hence option a is answer.

(b)Incorrect the number obtained on the right side is not the added value, It has been changed from a + b to b + a.

(c)Incorrect the property is an example of commutativity not associativity.

{Associative property for addition - a + (b + c) = (a + b) + c}.

(d)Incorrect it’s not an example of distribution or distributive Property.

{Distributive property - a×(b + c) = a×b + a×c

The question is incorrect.

The correction required is in option1 ,infront of it should be either (-) sign or ( + ) sign.

Let only take one case of ( + ) sign infront of .

⇒The equation now becomes

⇒This equation follow associative property of rational numbers i.e.

a + (b + c) = (a + b) + c.

⇒Hence,(a) is answer.

Other options are irrelevant they do not follow the associative property.

Whenever a number is added to Zero (0) the number obtained is a the same rational number to which it is added.

(b) Incorrect because if we subtract a rational number from 0 the number obtained will not be the same number.

(c) Incorrect because if we multiply 0 with a rational number the number obtained will be 0 and not the number to which 0 was multiplied.

(d) Incorrect because when a number is divided by 0 the result will be undefined.

When a rational number is multiplied by 1 the number obtained will be the same number,hence one (1) is the identity for addition of rational numbers.

(a)Incorrect because when 1 is added to a number the result obtained will not be the same number to which 1 was added.

(b) Incorrect because when 1 is subtracted from a number the result obtained will not be the same number.

(d)When a number is divided by 1 the number will be the reciprocal of the number and will not have the same value.

(a)Incorrect as,

⇒

(c) Incorrect as,

(d)Incorrect as,

Multiplicative inverse of a negative rational number is a negative rational number.

Example:

is a rational number so its multiplicative inverse is so that their multiplication will be 1.

⇒ Thus the multiplicative inverse is negative rational number.

(a) Incorrect because if multiplicative inverse will be positive the number obtained will have a negative sign and the result obtained will not be 1.

(c ) Incorrect because if 0 is multiplied to something the result will also be zero.

(d)Incorrect because if 1 is multiplied to a number the result obtained will not be 1.

Here,

x + 0 = 0 + x = x

i.e. when 0 is added to x or x is added to 0 the result obtained will be the same number that is x

⇒0 is the identity for addition of rational numbers.

B. Incorrect, additive inverse of x will be –x not 0.

(c) Incorrect, multiplicative inverse of x will be and not 0.

(d) Incorrect, reciprocal of x will be and not 0.

Multiplicative inverse of is i.e. the reciprocal of and hence their multiplication will be 1.

(a)Incorrect because their multiplication will be .

(b)Incorrect because their multiplication will be .

(d)Incorrect because their multiplication will be -1.

-(-x) will be x,as when negative sign is put outside of any expression the sign of expression gets changed.

(a)Incorrect as per the explanation.

(c)Incorrect as it’s the rexiprocal of x(Irrelevant option).

(d)Incorrect as it’s the reciprocal of x with negative sign(Irrelevant option).

⇒ (Reciprocal of )

⇒

⇒The multiplicative inverse is .Answer

x + 0 = x,(as Zero is identity function for addition of rational functions).

The reciprocal of 1 is = 1

⇒ The reciprocal of 1 is 1.

Other options are irrelevant.

The reciprocal of –1 is = -1.

Other options are irrelevant.

The reciprocal of 0 is

⇒ AS, is undefined

⇒Not defined.Answer.

The reciprocal of

⇒

Reciprocal of

⇒

(Equation 1)

⇒ Reciprocal of (Equation 2)

⇒from equation 1 and equation 2 the reciprocal of y is x.

=

⇒Reciprocal of

⇒Reciprocal of

Distributive property of multiplication over addition for rational numbers is

⇒ a×(b + c) = a×b + a×c (condition 1)

Only option (a) satisfies the given condition.

let the two rational numbers are and .

The rational number between them are , etc.

Similarly, we can write for , etc.

Thus there could be infinite no of rational numbers between any two rational numbers.

Let the value of x and y rational numbers be and

⇒x and y are 2 and 4 respectively.

Since ,number 3 lies between 2 and 4.

⇒Thus,answer is option(a).

Let the value of x and y rational numbers be and

⇒x and y are 2 and 4 respectively.

Since , number 3 lies between 4 and 2.

(a)Incorrect because 4-2 = 2

⇒and it do not lie between 4 and 2.

(c)Incorrect because

⇒ and it do not lie between 4 and 2.

(d)Incorrect because

⇒ and it do not lie between 4 and 2.

(Multiply both num and den. By 9)

(Multiply bith num. and den. By 5)

⇒

⇒ - ( 1)

⇒ - (2)

Comparing (1) and (2)

(2) > (1)

⇒>

Positive rational number

let the positive rational no be a

⇒reciprocal of a = which is also a positive no. as both of 1 and a are positive.

Negative rational number.Answer

Let a is a positive rational no.

⇒ -a is a negative rational number

⇒ Reciprocal of rational number –a is () = ()

⇒ Reciprocal of rational number is

⇒ Reciprocal of a rational number is negative as a is positive but

-1 is a negative value.

No

Reciprocal of zero = ()

⇒Reciprocal of zero has undefined value.

1 and -1

(i) For value 1

⇒Reciprocal of 1 is ()

⇒Reciprocal of 1 is 1.

(ii) For value -1

⇒Reciprocal of -1 is ()

⇒Reciprocal of -1 is -1.

()

Y =

⇒Reciprocal of y² = ().

reciprocal of ×() =

⇒Reciprocal of

⇒Reciprocal = .Answer

.

LHS = (213×657)^-1 =

⇒(213×657)^-1 = ×

RHS = 213^-1×657^-1 = ×

-1

1 + (-1) = 0

⇒Negative of 1 is -1.

××

×() = ×× (Distribution of multiplication over addition)

Greater -3

⇒ =

⇒The other number is

⇒>

⇒> -3.

Infinite

let the two rational numbers are and .

The rational number between them are , etc.

Similarly, we can write for , etc.

Thus there could be infinite no of rational numbers between any two rational numbers.

The rational numbers and are on the opposite sides of zero on the number line.

The negative of a negative rational number is always a positive rational number.

-(-x) = + x

Rational numbers can be added or multiplied in any order.

Reciprocal of a rational number is a number which when multiplied by that number gives 1.Hence,the reciprocal of is .

Multiplicative inverse is same as Reciprocal of a number, which is a number which when multiplied by that number gives 1.

The multiplicative inverse of is .

The rational number 10.11 in the fromis .

The two rational numbers lying between –2 and –5 with denominator as 1 are and .

False

The given statement false because 0 is a whole number and y (denominator) cannot be equal to 0.

False

P can be equal to 0 as 0 can be written as .

True

Rational numbers are those which are written in ,fraction form. So, s cannot be equal to 0 as it would cease to be fraction.

False

Clearly, we have to find whether lies between and 1

Writing

And

Clearly,

Hence, 5/6 lies between 2/3 and 1

False, .

Thus it does not lie between and 1.

True

is equal to which lies between –3 and –4.

True,

True

Multiplicative inverse is same as Reciprocal of a number, which is a number which when multiplied by that number gives 1.

False

Multiplicative inverse of a number is reciprocal of that number which does not affect the sign of the number

multiplicative inverse of is -.

False

Additive inverse of a number is the number that, when added to that number yields zero.

Additive inverse of is .

True

Additive inverse of a number is the number that, when added to that number yields zero.

False

Clearly, for any number say x, it cannot be written as x + 1 = x

Example, say x = 2,

Then 2 + 1 ≠ 2

i.e. 3≠ 2

Hence, False.

False

Additive inverse of a number is the number that, when added to that number yields zero.

False

Reciprocal of is .

True

Assume x = 1 and y = -1,

then x + y = 1 + (-1)

= 1-1

= 0

True

True

False

The negative of 1 is -1.

False

True

Associative Property of Multiplication.

False

Anything multiplied by 0 is equal to 0.

True

Associative property of multiplication and addition.

True

Distributive property of rational numbers.

True

Reciprocal of 1 is which is equal to 1.

True

False

x + (–1) = x-1

False

Assuming values of x and y as 2 and 3, and -3 and -2 respectively, we see that x-y is a negative number.

True

Assume x = -2 and y = -3,

Then x + y = -2 + (-3)

= -2-3

= -5

False

We can insert infinitely many rational numbers between any two rational numbers.

False

To add or subtract two rational numbers, you first need to change your rational numbers so that they share a common denominator.

Therefore, they are closed under subtraction.

False

Only Addition and multiplication are commutative.

False

is equal to -0.75 which is greater than -2.

True

A rational number is number which can be expressed in or fraction form. We can write 0 as or .

False

There are infinitely many positive rational numbers from till infinity.

False

A rational number is number which can be expressed in or fraction form.

True

There are infinitely many positive rational numbers between and .

False

The reciprocal of x^–1, which is -, is x.

Reciprocal of a rational number is a number which when multiplied by that number gives 1.

False

Rational number is positive which lies on the right side on the number line.

False

Rational number is negative which lies on the left side on the number line.

False

Rational number is positive so it lies on the right side of the number line.

True

True

A rational number can always be expressed in or fraction form.

True

Every integer can be written as (fraction form). Hence, it is rational number.

True

True

True

Assume values of x and y as 3,2 and -2,-3, we see that x-y is a positive rational number.

False

There are infinitely many negative rational numbers below 0. Hence, 0 is not the smallest rational number.

True

All whole numbers are integers, but not all integers are whole numbers. -2 is an integer but not a whole number.

True

Every whole number can be expressed in fraction form. Hence, there are rational numbers.

False

0 is both a whole number and a rational number.

True,

True, Associative property of Multiplication.

Interger is a number that can be written without a fractional component. For example, 21, 4, 0, and −20 are integers

Any number that can be written as p/q where p and q are integers provided (q≠ 0)

The rational numbers are:

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

(i) ⇒

⇒

⇒

(ii) ⇒

⇒

⇒

(iii) ⇒

⇒

⇒

When two rational numbers are added, their sum is always a rational number. For example,

, which is also a rational number.

Therefore, rational numbers are closed under addition.

When two rational numbers are subtracted, the result is always a rational number. For example, , which is also a rational number.

Therefore, rational numbers are closed under subtraction.

When two rational numbers are multiplied, their product is always a rational number. For example, , which is also a rational number.

Therefore, rational number are closed under multiplication.

As for any rational number a, a ÷ 0 is not defined, therefore not all rational numbers are closed under division. We can say that except zero, all rational numbers are closed under division. For example,

(i) , which is a rational number.

(ii) , which is a rational number.

To show: x + y = y + x

⇒

⇒

⇒

x + y = y + x

To show: x + y = y + x

⇒

⇒

x + y = y + x

To show: x + y = y + x

⇒

⇒

⇒

x + y = y + x

To show: x + y = y + x

⇒

⇒

⇒

x + y = y + x

Distributive Property

⇒

⇒

⇒

Distributive Property

⇒

⇒

⇒

Distributive Property:
a(b+c) = ab + ac

⇒

⇒

⇒

⇒

Associative Property of Multiplication: According to this property you can multiply regardless of how the numbers are grouped (by using brackets)

can be written as:

To verify Commutative property of Multiplication = x × y = y × z

⇒

⇒

To verify Commutative property of Multiplication = x × y = y × z

⇒

⇒

To verify Commutative property of Multiplication = x × y = y × z

⇒

⇒

To verify Commutative property of Multiplication = x × y = y × z

⇒

⇒

The property used is associative Property of multiplication.

(a) x = 1, and

⇒ x × (y × z) = (x × y) × z

⇒

⇒

(b)

⇒ x × (y × z) = (x × y) × z

⇒

⇒

(c)

⇒ x × (y × z) = (x × y) × z

⇒

⇒

Using the property,

x × (y + z) = x × y + x × z

⇒

Now, R.H.S

Hence, L.H.S = R.H.S

Using the property,

x × (y + z) = x × y + x × z

⇒

Now, R.H.S

Hence, L.H.S = R.H.S

Using the property,

x × (y + z) = x × y + x × z

⇒

Now, R.H.S

Hence, L.H.S = R.H.S

x × (y + z) = x × y + x × z

⇒

Now, R.H.S

Hence, L.H.S = R.H.S

(a) Using distributive property over addition

(b) Using distributive property over addition

(c) Using distributive property over addition

(d) Using distributive property over addition

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

Rational number which does not belong in the group is = which is non-recurring.

Whereas , are recurring.

Cost of metres of wire is = Rs.

cost of 1 metre of wire is = Rs.

cost of 1 metre of wire is = Rs. 38

Given, Distance = km

Time = hours

Speed =

⇒ Speed =

⇒ Speed =

⇒ Speed = 85 kmph

16 shirts of equal size can be made out of = 24m of cloth

⇒ 1 shirt of equal size can be made out of = m of cloth

1 shirt of equal size can be made out of = m of cloth

Let total money be x.

⇒

⇒

⇒

Size of rope = m =

Size of small pieces = m =

⇒ Number of small pieces =

⇒ Number of small pieces =

Number of small pieces = 24

Number of students above average =

Number of students average =

Number of students below average =

Number of students below average =

= 28

Let total number of students be x.

Given,

of total number of students of a school come by car

of students come by bus to school

Number of students who walk to school =

=

=

Number of students walking on their own = of

=

=

Given, Number of students walking on their own = 224

⇒

⇒

Let Huma’s share be x.

Seema’s share =

Hubna’s share = times Seema’s share = 5/3 times Seema's share

=

Given, Total amount = Rs.2016

⇒

⇒

⇒

⇒

Huma’s share = Rs.864

Seema’s share = = Rs.432

Hubna’s share =

= Rs.720

Let mother’s contribution be x.

Elder daughter’s share =

Younger daughter’s share =

Given Total amount = Rs.62000

⇒

⇒

⇒

⇒

Associative Property

(i) Commutative Property of Multiplication

(ii) Distributive Property

(iii) Associative Property of Addition

(iv) Commutative Property of Addition

(v) Commutative Property of Multiplication

(i) ⇒ =

Multiplicative Inverse of is .

(ii) ⇒ =

Multiplicative Inverse of is .

Given:

Therefore, in arranging in the descending order i.e. bigger number to smaller number, we get,

Or

Let the other number be x.

Given, product of two rational numbers is , and one of the numbers is .

⇒

⇒

Let the other number be x.

⇒

⇒

Let the other number be x.

⇒

⇒

Let the other number be x.

Given, product of two rational numbers is ,and one of the numbers is .

⇒

⇒

No as for a rational number to have multiplicative inverse as -1 is which is basically -1.

There can be infinitely many rational numbers between 0 and 1.

Given numbers 1 and 0.

Rational number between 2 numbers a and b is .

Rational number between 2 numbers 1 and 0 is

=

rational number between 2 numbers 1 and 1/ 2 is

= = 3/4

rational number between 2 numbers 1/ 2 and 0 is

=

Rational number between = 5 / 8

Absolute value of two rational numbers is the distance between those numbers when plotted on the number line.

Two rational number whose absolute value is = are -0.1 and 0.1 as when plotted on the number the distance between these two numbers is equal to .

Length of rope = 40 metres

Length of one piece is metre

Number of pieces =

⇒ Number of pieces =

⇒ Number of pieces =

Number of pieces = 12

Let length of each piece be x.

Length of rope = metres = metres

Number of pieces = 12

Number of pieces =

⇒

⇒

⇒

Length of one piece is metre.

(i) ⇒

⇒

⇒

(ii) ⇒

⇒

Temperature in Himachal Pradesh =

To convert it to degree Fahrenheit ():

⇒

⇒

⇒

⇒

⇒

⇒

⇒

Additive inverse of 7 is =

Multiplicative inverse of 7 is =

Additive inverse of a number is the number that, when added to that number yields zero.

Additive inverse of is =

Multiplicative inverse is same as Reciprocal of a number,which is a number which when multiplied by that number gives 1.

Multiplicative inverse of is =

Product of additive inverse and multiplicative inverse of is =

= -1

Wingspan of Albatross = m

Wingspan of Sea Gull = m

Wingspan of Albatross is longer than wingspan of Sea Gull by =

=

=

Wingspan of Golden eagle = m

Wingspan of Blue Jay = m

Wingspan of Golden eagle is longer than wingspan of Blue Jay by =

=

=

Diameter of circle = cm

Length of aluminium strip = cm

Width of aluminium strip = cm

Number of circles that can be cut = Length of aluminium strip/Diameter of circle =

= 7

Area of aluminium strip = Length Width

cm²

Area of circle =

Wastage of aluminum strip =

Given, 1 tablespoon is equivalent to = cup.

One fruit salad recipe requires = cup of sugar.

Another recipe for the same fruit salad requires = 2 tablespoons of sugar = cups of sugar

= cups of sugar

First recipe requires = cups of sugar

= cups of sugar

= cups of sugar

(a) Distance covered by Soni = km

Distance covered by Nancy = km

Soni covered =

=

= km more than Nancy

(b) Distance covered by Seema = km

Distance covered by Megha = km

Total Distance =

=

= km

(c) Distance covered by Nancy = km

Distance covered by Megha = km

Nancy walked farther.

Distance between Himgaon and Rawalpur = km = km

Distance between Sonapur and Ramgarh = km = km

On subtracting distance between Sonapur and Ramgarh from distance between Himgaon and Rawalpur = km

= km

= km

= km

We observe that distance between Himgaon and Rawalpur is more than distance between Himgaon and Rawalpur.

Distance between Himgaon to Sonapur = km

Distance between Sonapur to Rawalpur = km

Total distance covered =

=

=

= km

(a) Amount of paper recycled = , which is less than

(b) Paper and Glass

(c) The quantity of aluminium cans recycled is more than half of the quantity of aluminium cans.

(d)

Given widths (in cm) = 97.28,, ,97.94

=

Now arranging in ascending order:

Given, Roller Coaster at an amusement park is m high.

We know, the height of new roller coaster is times the height of the existing roller coaster.

Height of new roller coaster = metre

= metre

May =

June =

July =

August =

Length of skirt = cm = cm

Length of hem = cm = cm

Total length of skirt =

=

=

= 39 cm

Allowance of each = Rs.1260

Left over with Manavi = Allowance - Saving - Spend at mall = 1260 - 1/2. 1260 - 1/4. 1260 =

=

=

= Rs. 315

Left over with Kuber = Allowance - Saving - Spend at mall = 1260 - 1/4. 1260 - 3/5. 1260 =

=

=

= Rs. 84