ML Aggarwal Rational and Irrational Number Exe-1.3 Class 9 ICSE Maths Solutions

ML Aggarwal Rational and Irrational Number Exe-1.3 Class 9 ICSE Maths Solutions. We Provide Step by Step Answer of Exe-1.3 Questions for Rational and Irrational Number as council prescribe guideline for upcoming board exam. Visit official Website CISCE for detail information about ICSE Board Class-9.

ML Aggarwal Rational and Irrational Number Exe-1.3 Class 9 ICSE Maths Solutions

Board ICSE
Publications Avichal Publishig Company (APC)
Subject Maths
Class 9th
Chapter-1 Rational and Irrational
Writer ML Aggarwal
Book Name Understanding
Topics Solution of Exe-1.3 Questions
Edition 2022-2023

Rational and Irrational Number Exe-1.3

ML Aggarwal Class 9 ICSE Maths Solutions

(Page 23)

 Question 1. Locate √10 and √17 on the amber line.

Ans:√10

√10 = √(9 + 1) = √((3)2 + 12)

Now let us construct:

  • Draw a line segment AB = 3cm.
  • At point A, draw a perpendicular AX and cut off AC = 1cm.
  • Join BC.

BC = √10cm

1. Locate √10 and √17 on the amber line

√17

√17 = √(16 + 1) = √((4)2 + 12)

Now let us construct:

  • Draw a line segment AB = 4cm.
  • At point A, draw a perpendicular AX and cut off AC = 1cm.
  • Join BC.

BC = √17cm

1. Locate √10 and √17 on the amber lin

Question 2. Write the decimal expansion of each of the following numbers and say what kind of decimal expansion each has:

(i) 36/100

(ii) 4 1/8

(iii) 2/9

(iv) 2/11

(v) 3/13

(vi) 329/400

Ans:

(i) 36/100

2. Write the decimal expansion of each of the following numbers and say what kind of decimal expansion each has

36/100 = 0.36

It is a terminating decimal.

(ii) 4 1/8

4 1/8 = (4×8 + 1)/8 = 33/8

2. a Write the decimal expansion of each of the following numbers and say what kind of decimal expansion eaach has

33/8 = 4.125

It is a terminating decimal.

(iii) 2/9

2. c Write the decimal expansion of each of the following numbers and say what kind of decimal expansion eaach has

2/9 = 0.222

It is a non-terminating recurring decimal.

(iv) 2/11

2. d Write the decimal expansion of each of the following numbers and say what kind of decimal expansion eaach has

2/11 = 0.181

It is a non-terminating recurring decimal.

(v) 3/13

2. e Write the decimal expansion of each of the following numbers and say what kind of decimal expansion eaach has

 

3/13 = 0.2317692307

It is a non-terminating recurring decimal.

(vi) 329/400

2. f Write the decimal expansion of each of the following numbers and say what kind of decimal expansion eaach has

329/400 = 0.8225

It is a terminating decimal.

Question 3. Without actually performing the king division, State whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

(i) 13/3125

(ii) 17/8

(iii) 23/75

(iv) 6/15

(v) 1258/625

(vi) 77/210

Ans: We know that, if the denominator of a fraction has only 2 or 5 or both factors, it is a terminating decimal otherwise it is non-terminating repeating decimals.

(i) 13/3125

3 a . Without actually performing the king division, State whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion a

3125 = 5 × 5 × 5 × 5 × 5

Prime factor of 3125 = 5, 5, 5, 5, 5 [i.e., in the form of 2n, 5n]

It is a terminating decimal.

(ii) 17/8

3 b . Without actually performing the king division, State whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion a

8 = 2 × 2 × 2

Prime factor of 8 = 2, 2, 2 [i.e., in the form of 2n, 5n]

It is a terminating decimal.

(iii) 23/75

3 c . Without actually performing the king division, State whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion a

75 = 3 × 5 × 5

Prime factor of 75 = 3, 5, 5

It is a non-terminating repeating decimal.

(iv) 6/15

Let us divide both numerator and denominator by 3

6/15 = (6 ÷ 3) / (15 ÷ 3)

= 2/5

Since the denominator is 5.

It is a terminating decimal.

(v) 1258/625

3 e . Without actually performing the king division, State whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion a

625 = 5 × 5 × 5 × 5

Prime factor of 625 = 5, 5, 5, 5 [i.e., in the form of 2n, 5n]

It is a terminating decimal.

(vi) 77/210

Let us divide both numerator and denominator by 7

77/210 = (77 ÷ 7) / (210 ÷ 7)

= 11/30

3 f . Without actually performing the king division, State whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion a

30 = 2 × 3 × 5

Prime factor of 30 = 2, 3, 5

It is a non-terminating repeating decimal.


ML Aggarwal Rational and Irrational Number Exe-1.3 Class 9 ICSE Maths Solutions

page 24

Question 4. Without actually performing the long division, find if 987/10500 will have terminating or non-terminating repeating decimal expansion. Give reasons for your answer.

Ans: Given:

The fraction 987/10500

Let us divide numerator and denominator by 21, we get

987/10500 = (987 ÷ 21) / (10500 ÷ 21)

= 47/500

So,

The prime factors for denominator 500 = 2 × 2 × 5 × 5 × 5

Since it is of the form: 2n, 5n

Hence it is a terminating decimal.

Question 5. Write the decimal expansions of the following numbers which have terminating decimal expansions:

(i) 17/8

(ii) 13/3125

(iii) 7/80

(iv) 6/15

(v) 2²×7/54

(vi) 237/1500

Ans: 

(i) 17/8

5.a Write the decimal expansions of the following numbers which have terminating decimal expansions

17/8 = 2.125

(ii) 13/3125

5.b Write the decimal expansions of the following numbers which have terminating decimal expansions

 

13/3125 = 0.00416

(iii) 7/80

5.c Write the decimal expansions of the following numbers which have terminating decimal expansions

7/80 = 0.0875

(iv) 6/15

Let us divide both numerator and denominator by 3, we get

6/15 = (6 ÷ 3) / (15 ÷ 3)

= 2/5

Since the denominator is 5,

It is terminating decimal.

5.d Write the decimal expansions of the following numbers which have terminating decimal expansions

6/15 = 0.4

(v) (2²×7)/54

We know that the denominator is 54

It is a terminating decimal.

(2²×7)/54 = (2 × 2 × 7) / (5 × 5 × 5 × 5)

= 28/625

5.e Write the decimal expansions of the following numbers which have terminating decimal expansions

28/625 = 0.0448

It is a terminating decimal.

(vi) 237/1500

Let us divide both numerator and denominator by 3, we get

237/1500 = (237 ÷ 3) / (1500 ÷ 3)

= 79/500

Since the denominator is 500,

Its factors are, 500 = 2 × 2 × 5 × 5 × 5

= 22 × 53

It is terminating decimal.

5.f Write the decimal expansions of the following numbers which have terminating decimal expansions

237/1500= 79/500 = 0.1518

Question 6. Write the denominator of the rational number 257/5000 in the form 2× 5where m, n is non-negative integers. Hence, write its decimal expansion on without actual division.

Ans: Given:

The fraction 257/5000

Since the denominator is 5000,

The factors for 5000 are:

6. Write the denominator of the rational number  in the form  where m, n is non-negative integers. Hence, write its decimal expansion on without actual division

5000 = 2 × 2 × 2 × 5 × 5 × 5 × 5

= 23 × 54

257/5000 = 257/(23 × 54)

It is a terminating decimal.

So,

Let us multiply both numerator and denominator by 2, we get

257/5000 = (257 × 2) / (5000 × 2)

= 514/10000

= 0.0514

Question 7. Write the decimal expansion of 1/7. Hence, write the decimal expression of? 2/7, 3/7, 4/7, 5/7 and 6/7.

Ans: 

Given:

The fraction: 1/7

7. Write the decimal expansion of . Hence, write the decimal expression of and

 

1/7 = 0.142857142857

Since it is recurring,

7 b. Write the decimal expansion of . Hence, write the decimal expression of and

Question 8. Express the following numbers in the form p/q’. Where p and q are both integers and q≠0;

8. Express the following numbers in the form p Where p and q are both integers and 

Since there is one repeating digit after the decimal point,

Multiplying by 10 on both sides, we get

10x = 3.3333…

Now, subtract both the values,

9x = 3

x = 3/9

= 1/3

8. bExpress the following numbers in the form p Where p and q are both integers and 

Since there is one repeating digit after the decimal point,

Multiplying by 10 on both sides, we get

10x = 52.2222…

Now, subtract both the values,

9x = 52 – 5

9x = 47

x = 47/9

8. c Express the following numbers in the form p Where p and q are both integers and 

Let x = 0.404040

Since there is two repeating digit after the decimal point,

Multiplying by 100 on both sides, we get

100x = 40.404040…

Now, subtract both the values,

99x = 40

x = 40/99

0.404040… = 40/99

8. d Express the following numbers in the form p Where p and q are both integers and 

Since there is one non-repeating digit after the decimal point,

Multiplying by 10 on both sides, we get

10x = 4.7777

Since there is one repeating digit after the decimal point,

Multiplying by 10 on both sides, we get

100x = 47.7777

Now, subtract both the values,

90x = 47 – 4

90x = 43

x = 43/90

8. e Express the following numbers in the form p Where p and q are both integers and 

Since there is one non-repeating digit after the decimal point,

Multiplying by 10 on both sides, we get

10x = 1.343434

Since there is two repeating digit after the decimal point,

Multiplying by 100 on both sides, we get

1000x = 134.343434

Now, subtract both the values,

990x = 133

x = 133/990

8. f Express the following numbers in the form p Where p and q are both integers and 

Since there is three repeating digit after the decimal point,

Multiplying by 1000 on both sides, we get

1000x = 1.001001

Now, subtract both the values,

999x = 1

x = 1/999

Question 9. Classify the following numbers as rational or irrational:

(i) √23

(ii) √225

(iii) 0.3796

(iv) 7.478478

(v) 1.101001000100001…

9. Classify the following numbers as rational or irrational

Ans: 

(i) √23

Since, 23 is not a perfect square,

√23 is an irrational number.

(ii) √225

√225 = √(15)2 = 15

Since, 225 is a perfect square,

√225 is a rational number.

(iii) 0.3796

0.3796 = 3796/1000

Since, the decimal expansion is terminating decimal.

0.3796 is a rational number.

(iv) 7.478478

Let x = 7.478478

Since there is three repeating digit after the decimal point,

Multiplying by 1000 on both sides, we get

1000x = 7478.478478…

Now, subtract both the values,

999x = 7478 – 7

999x = 7471

x = 7471/999

7.478478 = 7471/999

Hence, it is neither terminating nor non-terminating or non-repeating decimal.

7.478478 is an irrational number.

(v) 1.101001000100001…

Since number of zero’s between two consecutive ones are increasing. So it is non-terminating or non-repeating decimal.

1.101001000100001… is an irrational number.

9. Classify the following numbers as rational or irrational

Let x = 345.0456456

Multiplying by 10 on both sides, we get

10x = 3450.456456

Since there is three repeating digit after the decimal point,

Multiplying by 1000 on both sides, we get

1000x = 3450456.456456…

Now, subtract both the values,

10000x – 10x = 3450456 – 345

9990x = 3450111

x = 3450111/9990

Since, it is non-terminating repeating decimal.

Question 10. The following real numbers have decimal expansions as given below. In each case, state whether they are rational or not. If they are rational and expressed in the form p/q, where p, q are integers, q≠ 0 and p, q are co-prime, then what can you say about the prime factors of q?

(i) 37.09158

10. The following real numbers have decimal expansions as given below. In each case, state whether they are rational or not. If they are rational and expressed in the form p/q, where p, q are integers, q≠ 0 and p, q are co-prime, then what can you say about the prime factors of

(iii) 8.9010010001…

(iv) 2.3476817681…

Ans: 

(i) 37.09158

We know that

It has terminating decimal

Here

It is a rational number and factors of q will be 2 or 5 or both.

10. The following real numbers have decimal expansions as given below. In each case, state whether they are rational or not. If they are rational and expressed in the form p/q, where p, q are integers, q≠ 0 and p, q are co-prime, then what can you say about the prime factors of

We know that

It has non-terminating recurring decimals

Here

It is a rational number.

(iii) 8.9010010001…

We know that

It has non-terminating, non-recurring decimal.

Here

It is not a rational number.

(iv) 2.3476817681…

We know that

It has non-terminating, recurring decimal.

Here

It is a rational number and the factors of q are prime factors other than 2 and 5.

Question 11. Insert an irrational number between the following.

(i) 1/3 and ½

(ii) -2/5 and ½

(iii) 0 and 0.1

Ans: 

(i) One irrational number between 1/3 and ½

11. Insert an irrational number between the following

½ = 0.5

So there are infinite irrational numbers between 1/3 and ½.

One irrational number among them can be 0.4040040004…

(ii) One irrational number between -2/5 and ½

11. b Insert an irrational number between the following

½ = 0.5

So there are infinite irrational numbers between -2/5 and ½.

One irrational number among them can be 0.1010010001…

(iii) One irrational number between 0 and 0.1

There are infinite irrational numbers between 0 and 1.

One irrational number among them can be 0.06006000600006…

Question 12. Insert two irrational numbers between 2 and 3.

Ans: 2 is expressed as √4

And 3 is expressed as √9

So, two irrational numbers between 2 and 3 or √4 and √9 are √5, √6

Question 13. Write two irrational numbers between 4/9 and 7/11.

Ans: 4/9 is expressed as 0.4444…

7/11 is expressed as 0.636363…

So, two irrational numbers between 4/9 and 7/11 are 0.4040040004… and 0.6060060006…

Question 14. Find one rational number between √2 and √3.

Ans: √2 is expressed as 1.4142…

√3 is expressed as 1.7320…

So, one rational number between √2 and √3 is 1.5.

Question 15. Find two rational numbers between 2√3 and √15.

Ans: √12 = √(4×3) = 2√3

Since, 12 < 12.25 < 12.96 < 15

So, √12 < √12.25 < √12.96 < √15

Hence, two rational numbers between √12 and √15are [√12.25, √12.96] or [√3.5, √3.6].

Question 16. Insert irrational numbers between √5 and √7.

Ans: Since, 5 < 6 < 7

So, irrational number between √5 and √7 is √6.

Question 17. Insert two irrational numbers between √3 and √7.

Ans: Since, 3 < 4 < 5 < 6 < 7

So,

√3 < √4 < √5 < √6 < √7

But √4 = 2, which is a rational number.

So,

Two irrational numbers between √3 and √7 are √5 and √6.

—  : End of ML Aggarwal Rational and Irrational Number Exe-1.3 Class 9 ICSE Maths Solutions :–

Return to :- ML Aggarawal Maths Solutions for ICSE  Class-9

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