NCERT Solutions Class 9 Mathematics

Exercise 1.3

Q.1:Write the following in decimal form and say what kind of decimal expansion each has:

i36100= 0.36 

ii111= 0.09090909 =0.09

iii338= 4.125 

iv313= 0.230769230769 = 0 .230769

v 211= 0.181818181818 = 0.18 

vi 329400= 0.8225 

Q.2. You know that 17 = 0.142857. Can you predict what the decimal expansion of 27,37,47,57,67 are without actually doing the long division? If so, how?

[Hint: Study the remainders while finding the value of 17 carefully.]

Q.3. Express the following in the form pq where p and q are integers and q ≠ 0.

(i) 0.6 (ii) 0.47 (iii) 0.001

Q.4. Express 0.99999…in the form pq. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Q.5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 117? Perform the division to check your answer.

Q.6. Look at several examples of rational numbers in the form pq (q≠0) where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

Q.7. Write three numbers whose decimal expansions are non-terminating non-recurring.

Q.8. Find three different irrational numbers between the rational numbers 57 and 911.

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

(i) 23 (ii) 225 (iii) 0.3796

(iv) 7.478478… (v) 1.101001000100001…

NCERT Solutions Class 9 Mathematics

Exercise 1.3

Q.1:Write the following in decimal form and say what kind of decimal expansion each has:

i36100= 0.36 

ii111= 0.09090909 =0.09

iii338= 4.125 

iv313= 0.230769230769 = 0 .230769

v 211= 0.181818181818 = 0.18 

vi 329400= 0.8225 

Solution:

Write the following in decimal form and say what kind of decimal expansion each has:

i36100= 0.36 

ii111= 0.09090909 =0.09

iii338= 4.125 

iv313= 0.230769230769 = 0 .230769

v 211= 0.181818181818 = 0.18 

vi 329400= 0.8225 

Q.2. You know that 17 = 0.142857. Can you predict what the decimal expansion of 27,37,47,57,67 are without actually doing the long division? If so, how?

[Hint: Study the remainders while finding the value of 17 carefully.]

Solution:

Yes. We can be done this by simply multiplying the RHS by the numerator, i.e., 2, 3, 4 etc.

27=2×0.142857=0.285714

37=3×0.142857=0.428571

47=4×0.142857=0.571428

57=5×0.142857=0.714285

67=6×0.142857=0.857142

Q.3. Express the following in the form pq where p and q are integers and q ≠ 0.

(i) 0.6 (ii) 0.47 (iii) 0.001

Solution:

(i) 0.6 = 0.666...

Let x = 0.666... (1)

⇒ 10x = 6.666... (2)

Subtracting (1) from (2), we get,

9x = 6

x = 23

(ii) 0.47= 0.4777...

Let x = 0.4777… (1)

⇒ 10x = 4.777… (2)

Subtracting (1) from (2), we get,

9x = 4.3

x = 4.39 = 4390

(iii) 0.001= 0.001001...

Let x = 0.001001... (1)

⇒ 1000x = 1.001001… (2)

Subtracting (1) from (2), we get,

999x = 1

x =1999

Q.4. Express 0.99999…in the form pq. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Solution:

Let x = 0.9999… (1)

10x = 9.9999… (2)

Subtracting (1) from (2), we get,

9x = 9

x = 1

As the number of digits increases in the decimal representation, the difference between 1 and the actual number decreases. For example difference between 1 and 0.999999 is 0.000001 which is negligible. Thus, 0.999… is very near 1.

Q.5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 117? Perform the division to check your answer.

Solution:

The maximum number of digits in the repeating block of digits in the decimal expansion of 1n can be (n-1). Therefore, the maximum number of digits in the repeating block of the decimal expansion of 117 can be 16.

This can be verified by long division method, and we get,

117 = 0.0588235294117647 (16 digits).

Q.6. Look at several examples of rational numbers in the form pq (q≠0) where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

Solution:

If the prime factorisation of the denominator of the given fractions has the power of 2 only or 5 only or both (and no other factors), the decimal expansion will be terminating.

For example,

12 = 0.5, denominator q = 21

78 = 0.875, denominator q = 23

45 = 0.8, denominator q = 51

13 = 0.3333… (denominator contains factor 3)

Q.7. Write three numbers whose decimal expansions are non-terminating non-recurring.

Solution:

Three numbers whose decimal expansions are non-terminating non-recurring are:

0.101001000100001...

0.707007000700007...

0.2102100210002100002100000…

Q.8. Find three different irrational numbers between the rational numbers 57 and 911.

Solution:

We can see that,

57= 0.714285

911= 0.81

Three different irrational numbers can be,

0.72072007200072000072…

0.73073007300073000073…

0.75075007500075000075…

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

(i) 23 (ii) 225 (iii) 0.3796

(iv) 7.478478… (v) 1.101001000100001…

Solution:

(i) 23 = 4.79583152331...

Square roots of all the numbers, except perfect squares are irrational. Hence it is an irrational number.

(ii) 225= 15 

=151

All integers (like 15) can be represented in the pq form, hence it is rational number.

(iii) 0.3796

Since the number is terminating therefore, it is a rational number.

(iv) 7.478478… = 7.478

Since this number is non-terminating recurring, it is a rational number.

(v) 1.101001000100001…

Since the number is non-terminating non-repeating, it is an irrational number.