Question 1: Compare the following pairs of rational numbers:

(i) and (ii) and (iii) and (iv) and

Answer:

(i) and

First take the LCM of and . LCM

Therefore: and

Hence, we see that or we can say that

(ii) and

First take the LCM of and . LCM

Therefore: and

Hence, we see that or we can say that

(iii) and

First take the LCM of and . LCM

Therefore: and

Hence we see that or we can say that

(iv) and

First take the LCM of and . LCM

Therefore: and

Hence we see that or we can say that

Question 2: Arrange in ascending order:

(i) (ii)

(iii) (iv)

Answer:

(i)

LCM of

The fractions can be written as

Therefore}, the order would be

or

(ii)

Note:

LCM of

The fractions can be written as

Therefore}, the order would be

or

(iii)

Note:

LCM of

The fractions can be written as

Therefore}, the order would be

or

(iv)

Note:

LCM of

The fractions can be written as

Therefore}, the order would be

or

Question 3: Represent each of these numbers on a Number Line:

(i) (ii) (iii) (iv) (v)

Answer:

(i) : Divide the unit length between and in equal parts and then mark

(ii) : Divide the unit length between and in equal parts and then mark

(iii) : Divide the unit length between and in equal parts and then mark

(iv) : Divide the unit length between and in equal parts and then mark

(v) : Divide the unit length between and in equal parts and then mark

Question 4: Find the additive inverse of:

(i) (ii) (iii) (iv) (v) (vi)

Answer:

(i)

. Therefore the additive inverse is

(ii)

. Therefore the additive inverse is

(iii)

. Therefore the additive inverse is

(iv)

. Therefore the additive inverse is

(v)

. Therefore the additive inverse is

(vi)

. Therefore the additive inverse is

Question 5: Find the sum:

(i) + (ii) + (iii) + (iv) +

(v) + + + (vi) + + +

Answer:

(i) +

(ii) +

(iii) + (Note: LCM of and is )

(iv) + = = (Note: LCM of and is )

(v) + + + = (Note: LCM of , and is )

(vi) + + + = (Note: LCM of , and is )

Question 6: Subtract:

(i) from (ii) from (iii) from (iv) from

Answer:

(i) from

(ii) from

(iii) from

(iv) from

Question 7: The sum of two rational numbers is . If one of them is then find the other.

Answer:

Question 8: What number should be added to to get

Answer:

Question 9: What number should be subtracted from to get

Answer:

Question 10: Find the products:

(i) (ii) (iii) (iv)

Answer:

(i) (ii) (iii) (iv)

Question 11: Find the quotient:

(i) (ii) (iii) (iv)

Answer:

(i) (ii)

(iii) (iv)

Question 12: The product of two rational numbers is . If one of the number is , then find the other.

Answer:

Question 13: By what number must be divided to get ?

Answer:

Question 14: Find a rational number between each of the following pairs of rational numbers.

(i) and (ii) and (iii) and (iv) 2 and

Answer:

(i) and

First take the LCM of and which is .

Convert the numbers with as the denominator.

Hence we get and

Therefore the rational numbers between and are or

(ii) and 2 Or and

Therefore the rational numbers between and 2 are or

(iii) and

First take the LCM of and which is .

Convert the numbers with as the denominator.

Hence we get and

Therefore the rational numbers between and are or

(iv) 2 and Or and

Therefore the rational number between and are or

Question 15: Find three rational numbers between:

(i) and (ii) and

Answer:

(i) and Or and

Therefore the rational numbers between and are and and Or and and

(ii) and Or and

Therefore the rational numbers between and are

and and or and and

Question 16: Find 5 rational numbers between:

(i) and (ii) and

Answer:

(i) and Or and

The rational numbers are

, , , , Or , , , ,

(ii) and Or and

The rational numbers are , , , , Or , , , ,

Question 17: Determine whether the numbers are rational or irrational:

(i) (ii) (iii) (iv) (v)

(vi) (vii) (viii) (ix) (x)

(xi) (xii) (xiii) (xiv) (xv)

Answer:

(i) : Rational

*Rational numbers* are those *numbers* which can be expressed in the form of where and are integers. So, 1 and 17 are and respectively and they are integers too.! … *Rational numbers* are in the form of and is not equal to zero.

(ii) : Rational

The decimal is a rational number. It is the decimal form of the fraction . *Rational numbers* are in the form of and is not equal to zero

(iii) : Rational

*Rational numbers* are those *numbers* which can be expressed in the form of where and are integers. So, 11 and 13 are and respectively and they are integers too.! … *Rational numbers* are in the form of and is not equal to zero.

(iv) : Rational

can be written as, , where and is any non-zero integer. Hence, 0 is a rational number.

(v) : Irrational

is non terminating recurring, so it is a rational number.

(vi) $latex \frac{4 \times 3 \sqrt{2}}{3

\sqrt{2}} &s=2$ : Rational

Every whole number is a rational number, because any whole number can be written as a fraction. For example, can be written as .

(vii) : Irrational

is an irrational number. Hence is irrational.

*Proof: Lets first assume to be rational.** , where is not equal to .**Here, and are co-primes whose HCF is .** ( squaring both sides )…** ** … … … … … (i) **Here, divides also a ( because, if a prime number divides the square of a positive integer, then it divides the integer itself )**Now, let ( squaring both sides )…** … … … … … (ii) **Substituting Equation (i) in Equation (ii),** ** divides as well as .**Conclusion:**Here, and are both divisible by . But our assumption that their HCF is is being contradicted.**Therefore, our assumption that is rational is wrong. Thus, it is irrational.*

(viii) : Irrational

*Proof: Let is a rational number.**Squaring both sides, ** Irrational number.**Now if is rational, then is also rational and hence which contradicts our initial assumption. Hence $latex *

(ix) : Rational

*Rational numbers* are those *numbers* which can be expressed in the form of where and are integers. So, 2 and 5 are and respectively and they are integers too.! … *Rational numbers* are in the form of and is not equal to zero.

(x) : Irrational

The basic answer to this question is that is irrational because it represents the ratio of the circumference of a circle to its diameter and that ratio is irrational.

is irrational. It cannot be represented as a ratio of integers (with non zero denominator). is just an approximation of . It is used to simplify the problems and achieve a result as less deviated as possible.

(xi) Rational

can be written as, , where and are non-zero integer. For example Hence, -12 is a rational number.

(xii) : Rational

It is in a repeating decimal and hence can be represented in the form of and is not equal to zero.

(xiii) : Irrational

is irrational,so it’s inverse (which is the same as ) is irrational. All unresolved roots are known as an irrational number since is an unresolved root which is also known as surds, so it is an irrational number.

(xiv) : Rational

can be represented in the form of and is not equal to zero.

(xv) : Irrational

is an irrational number as proved above. Hence it’s inverse is irrational number.

Question / Answer 19: State whether True or False:

(i) Every real number is either rational or irrational: True

(ii) Every real number can be represented on a number line: True

(iii) There exists and integer which is not a rational number: False

(iv) There exist a point on a number line which do not represent any real number: False

(v) An infinite number of rational numbers can be inserted between any two rational numbers: True

(vi) The multiplicative inverse of any rational number a is 1/a : False

Question / Answer 20: Fill in the blanks

(i) 0 is a rational number that is its own additive inverse.

(ii) 0 is a rational number that does not have a multiplicative inverse.

(iii) 1 and -1 are two rational numbers which are equal it their own reciprocal.

(iv) The product of a rational number with its reciprocal is 1 .

(v) The reciprocal of a negative number is negative.

(vi) The multiplicative inverse of a rational number is is a .

(vii) Number of irrational number between any two rational number is infinite.

Question 21: Arrange in ascending order

(i) (ii)

Answer:

(i)

First take everything within under root sign. That way we can compare the numbers easily.

The numbers would then be

Now arrange in ascending order

Or

(ii)

First take everything within under root sign. That way we can compare the numbers easily.

The numbers would then be

Now arrange in ascending order

Or

Question 22: Write the rationalizing factors of the following:

(i) (ii) (iii)

(iv) (v) (vi)

Answer:

(i)

Therefore rationalizing factor is

(ii)

Therefore rationalizing factor is

(iii)

Therefore rationalizing factor is

(iv)

Therefore rationalizing factor is

(v)

Therefore rationalizing factor is

(vi)

Therefore rationalizing factor is

Question 23: Rationalize the denominator of each of the following:

(i) (ii) (iii) (iv)

(v) (vi) (vii) (viii)

Answer:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

Question 24: Insert 5 rational numbers between:

(i) and (ii) and (iii) and

Answer:

(i) and

(ii) and

First take everything within the root sign. So we need to find rational numbers between and

Hence the numbers are

(iii) and

We can make the numbers as square root. So we need to find rational numbers between and

Hence the numbers are

Question & Answer 25: State True or False:

(i) : False

(ii) : True

(iii) : True

(iv) is a rational number: True (the value is which is a rational number)

(v) is irrational number: False (the value is which is a rational number)

(vi) is a rational number: True (the value is 6 which is a rational number)

(vii) Is a rational number. True (the value is which is a rational number)